%!PS-Adobe-2.0
%%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software
%%Title: obfuscate.dvi
%%CreationDate: Tue Nov 13 16:26:48 2001
%%Pages: 43
%%PageOrder: Ascend
%%BoundingBox: 0 0 612 792
%%DocumentFonts: CMR17 CMSY10 CMR12 CMSY8 CMBX10 CMR10 CMTI10 CMMI10
%%+ CMBX7 CMSY6 CMR9 CMR6 CMSY9 CMTT9 CMBX12 CMCSC10 CMTT10 CMR8 CMTI9
%%+ CMMI8 CMMI9 CMBX8 MSBM10 CMEX10 MSBM7 CMTI7 CMMI6 LASY10 CMTI8
%%EndComments
%DVIPSWebPage: (www.radicaleye.com)
%DVIPSCommandLine: dvips.exe -P pdf -G0 -t letter obfuscate
%DVIPSParameters: dpi=8000, compressed
%DVIPSSource:  TeX output 2001.11.13:1626
%%BeginProcSet: tex.pro
%!
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S
/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy
setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask
restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
(LaserWriter 16/600)]{A length product length le{A length product exch 0
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end

%%EndProcSet
%%BeginProcSet: alt-rule.pro
%!
% Patch by TVZ
% Makes dvips files draw rules with stroke rather than fill.
% Makes narrow rules more predictable at low resolutions
% after distilling to PDF.
% May have unknown consequences for very thick rules.
% Tested only with dvips 5.85(k).
TeXDict begin
/QV {
  gsave newpath /ruleY X /ruleX X
  Rx Ry gt
  { ruleX ruleY Ry 2 div sub moveto Rx 0 rlineto Ry }
  { ruleX Rx 2 div add ruleY moveto 0 Ry neg rlineto Rx }
  ifelse
  setlinewidth 0 setlinecap stroke grestore
} bind def
end

%%EndProcSet
%%BeginProcSet: texc.pro
%!
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3
1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx
0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp
gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B
/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
(LaserWriter 16/600)]{A length product length le{A length product exch 0
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end

%%EndProcSet
%%BeginProcSet: texps.pro
%!
TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2
index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll
exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics
exch def dict begin Encoding{exch dup type/integertype ne{pop pop 1 sub
dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}
ifelse}forall Metrics/Metrics currentdict end def[2 index currentdict
end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{
dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1
roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def
dup[exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}
if}forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}
def end

%%EndProcSet
%%BeginProcSet: special.pro
%!
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{
Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale
}if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState
save N userdict maxlength dict begin/magscale true def normalscale
currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{
psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath
moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict
begin/SpecialSave save N gsave normalscale currentpoint TR
@SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{
CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx
sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR
}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse
CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury
lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N
/@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end}
repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N
/@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX
currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY
moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X
/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0
1 startangle endangle arc savematrix setmatrix}N end

%%EndProcSet
%%BeginFont: CMTI8
%!PS-AdobeFont-1.1: CMTI8 1.0
%%CreationDate: 1991 Aug 18 21:07:42

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 75 /K put
dup 83 /S put
dup 86 /V put
readonly def
/FontBBox{-35 -250 1190 750}readonly def
/UniqueXX 5000826 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: LASY10
%!PS-AdobeFont-1.1: LASY10 1.001
%%CreationDate: 1992 Oct 23 20:19:17
%%RevisionDate: 2001 Jun 05 20:19:17

% Copyright (C) 1997, 2001 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
 /version (1.001) readonly def
 /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def
 /FullName (LASY10) readonly def
 /FamilyName (LaTeX) readonly def
 /Weight (Medium) readonly def
 /ItalicAngle -14.035 def
 /isFixedPitch false def
end readonly def
/FontName /LASY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 50 /a50 put
readonly def
/FontBBox{-19 -192 944 683}readonly def
/UniqueXX 5011949 def
currentdict end
currentfile eexec
8053514d69233ac976bbf3778e6f5685084745915d7c58013537261b3e454148
ae35a2f8bb486e29ef3d67d963766b945debbf4cb70107996ecb5df5ea03c784
c9efe685090f7f9fdbcd89e5fd86da539bdf5f1a155fcb4b3a7f2509c6e2e3d1
d3a70ec58c115c94c6a0a706f1f48662151281de4bd6683ff4e63d0ecb89496a
9a0e6ac92c1c6287707c27d45a3f34b365e01f92a78fe9baec4e6cdc3a18a2bb
cee2f6a4bfd041f127c00579662975ba6a7e801cfd8e07ce05b929bdcc6b8c17
a22788a76da7a653d175e45dff586f41d5fde040ee32c1c877d71a5edb011412
e52f49ec225236052c68b27b7de41cfedf8bfb7d56c3eed9d3938086cb7b0efd
887e5915763d5f1e13d86819cdb129b84c95c3d2830daacf827e9c799385fdc4
9457303be19de33d6ac97fb33c78abbd6165afeef9861c80d096fed77fb98861
e0de487a399792c9153148f7b659aaf10505d0da9e436dfd611b938d4ffbb1b5
4f896349e3f17d247f404d2894fdb8eed9f1cc6d378d0834d798801e3116100a
789ec2182d7827aac35e52cbe6c7edd833ef62d863476f614aec2dc5e21cf547
c059be793a40a2e8589feb0730125606cbb8a7a5d3ec8a2b157c139588d4ef86
5a33e6dda32ead3ec46d64e5462e06712e0c0c7ddb760543313c85b50b0bdd02
a175c5b94aa9a6ba0dab260883c35341764b9af09caa8c64d3cf14daa5329242
c0208706a8b6a0d59f70e0adcd24edbbb8fc120879b515104eac859b53ee3592
baaece1643f6b12d77c5f0f22b04c55091c7043787daab52a24b801dd703519f
0814bb6fdf4112a02056f04f69ee0f14de15221b9356af87ebf634218828e312
a83599785ffab58131301cffa9a9ad3b72755555d22e616ecbde2fd80c000f12
4381ab90a24107088e4d7329d8e811c7b68d6beb85143085166094db8c335a6e
45efc9b6eecf6596d9094a825f527a0542df8aba05289398ecd4881f92d5b382
cfb0161400ace9a59a416e343d0b76154ee04ed78f60a7d28712e608483fe4e9
e21f28c9ae5937ec262eaa96dc94395af9cbdf9729d8ecb31377de3b1b91161e
43e19c56f51e773d6e873e258757ca5a0446fabdf17c6dee8f8aec7b300c56b9
b5c8002d2dbb772c9499615e3078045efdfe545c48c44bee527a84aa3b971efa
94540f050e47ab56279cc9a041af93efd0123c011d9d9f1dda992cbde23e51b6
82246c7fc9a894dc81d2081c6b505de4663bbee1afc59613a178df71aa366084
0b61930f6c0267e2fa084e6b053753a9d9f77db5b66fb010f80c04e056bd3ead
3cf412d0a6ede124d5e43c6a1615d907d11118cba498ff8338e6ba4e85709f7c
82448f964602f8bad9a9847b21227615e2f0b7ac082197874ec33e81fb6bb4f6
2a0d07448d63d9b6299cd155cbf1fae318a3a99e5dc676322b8143a7cc9b2a74
dda7485a88125d900ab6ecf45ee60e674ee383a406b98abbd1cf035f812feca0
1044c74d45229b4ced25e917631c45cb219cbe3f42354b36f7bcaf7bfe8d3a6a
2c791d9558e616885284
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 12 /beta put
dup 14 /delta put
dup 59 /comma put
dup 71 /G put
dup 75 /K put
dup 98 /b put
dup 107 /k put
dup 110 /n put
dup 114 /r put
dup 116 /t put
dup 120 /x put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueXX 5087381 def
currentdict end
currentfile eexec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%%EndFont 
%%BeginFont: CMTI7
%!PS-AdobeFont-1.1: CMTI7 1.0
%%CreationDate: 1991 Aug 18 21:07:18

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 82 /R put
readonly def
/FontBBox{-27 -250 1268 750}readonly def
/UniqueXX 5000825 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: MSBM7
%!PS-AdobeFont-1.1: MSBM7 2.1
%%CreationDate: 1992 Oct 17 08:30:50

% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 78 /N put
readonly def 
/FontBBox{0 -504 2615 1004}readonly def
/UniqueXX 5032014 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /parenleftbig put
dup 1 /parenrightbig put
dup 2 /bracketleftbig put
dup 3 /bracketrightbig put
dup 12 /vextendsingle put
dup 40 /braceleftBigg put
dup 41 /bracerightBigg put
dup 83 /uniontext put
dup 91 /uniondisplay put
dup 102 /tildewider put
dup 103 /tildewidest put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
dup 110 /braceleftBig put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueXX 5000774 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37

% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 78 /N put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueXX 5031982 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMBX8
%!PS-AdobeFont-1.1: CMBX8 1.0
%%CreationDate: 1991 Aug 20 16:36:07

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 48 /zero put
readonly def
/FontBBox{-59 -250 1235 750}readonly def
/UniqueXX 5000766 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMMI9
%!PS-AdobeFont-1.1: CMMI9 1.100
%%CreationDate: 1996 Jul 23 07:53:55

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 12 /beta put
dup 59 /comma put
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 77 /M put
dup 83 /S put
dup 90 /Z put
dup 107 /k put
readonly def
/FontBBox{-29 -250 1075 750}readonly def
/UniqueXX 5087384 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 12 /beta put
dup 13 /gamma put
dup 14 /delta put
dup 58 /period put
dup 59 /comma put
dup 61 /slash put
dup 67 /C put
dup 68 /D put
dup 70 /F put
dup 71 /G put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 89 /Y put
dup 90 /Z put
dup 96 /lscript put
dup 98 /b put
dup 102 /f put
dup 103 /g put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 116 /t put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueXX 5087383 def
currentdict end
currentfile eexec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%%EndFont 
%%BeginFont: CMTI9
%!PS-AdobeFont-1.1: CMTI9 1.0
%%CreationDate: 1991 Aug 18 21:08:07

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 102 /f put
dup 105 /i put
dup 108 /l put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
readonly def
/FontBBox{-35 -250 1148 750}readonly def
/UniqueXX 5000827 def
currentdict end
currentfile eexec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%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 10 /Omega put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 69 /E put
dup 72 /H put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 120 /x put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueXX 5000791 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMTT10
%!PS-AdobeFont-1.1: CMTT10 1.00B
%%CreationDate: 1992 Apr 26 10:42:42

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 33 /exclam put
dup 34 /quotedbl put
dup 35 /numbersign put
dup 36 /dollar put
dup 37 /percent put
dup 38 /ampersand put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 42 /asterisk put
dup 43 /plus put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 47 /slash put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 59 /semicolon put
dup 60 /less put
dup 61 /equal put
dup 62 /greater put
dup 63 /question put
dup 65 /A put
dup 66 /B put
dup 68 /D put
dup 70 /F put
dup 71 /G put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 82 /R put
dup 91 /bracketleft put
dup 92 /backslash put
dup 93 /bracketright put
dup 94 /asciicircum put
dup 96 /quoteleft put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 119 /w put
dup 120 /x put
dup 123 /braceleft put
dup 124 /bar put
dup 125 /braceright put
dup 126 /asciitilde put
readonly def
/FontBBox{-4 -235 731 800}readonly def
/UniqueXX 5000832 def
currentdict end
currentfile eexec
8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8
97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc
8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011
b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415
50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e
784c6bf7389803d9450b0c21756a017306462c563d51ecefaacd079732f12c29
315e4b9623a5752c6f1d8145869e120d910b2644887cea7e30b15676a92537c2
9d3aa80dc30082aba94b40990b82fb1a877e805e0c8c48f61e9f2edac05b944e
e4d8084ec1d5cc517aaeec5b3ea379dd011eeb454cecab3ad2443c887c582789
72355673e503affe0394fc7db31de364e4f56c24033c7df2265c56445ec63a1d
5695a6041ea1b94407e1cdb7c5635603a4fd047e6edcaeb2d0da6c9e0e9396d5
1a4a58e8fdc1578730f992435560a6e2d3e3687703ee2f78f5896389ac8470bc
806169eb01762e89b6dc9adf857ead656620e2589aae722c37a2ed7a2941c360
b067ee34d8d5ca3bf68db725614d936bcb207781f4d4ec2ac67b13a5ad161f3f
059add7b5e3d904831e31c20c04546fae83ca93a35989e65c201756888f727b9
7e5313c9870ec96e4cec3901ea03a5c744754485e7d169bbc98bf872d0796e18
9d43b712950c3786257d8be06ab6080b9d9392313298327549a8a41c00a2cea3
690b4a333e45aa815a64facae1c2b44640860b8b8687afbefdbd5b4a541a7251
4ee7d3e0752af3e96a88c95d31fa16d34ba2f02fdb0088b165844f02611c734a
dbcc2037bf741fdea7e8bcdc130c70c33772f777d8bcddf4611db99001ccff14
d7af2bb05defc3480bdab312a0eff2f375afb4e0df2f803c594b7c93f71ca4af
861ae1711932fef19bfe2f9b7adb69d68e5a70ea4d1a3d5bd85231b16993f65e
fb37a0a823f0542f8c2340a073dc1be4a8347f9d3068a6435cd8278b0db1809e
8af9664c2c989b9f16873c009449b6284b85da4dee11c96a05ba83292f0edbd9
2cf674aad6ac1a5ff966ffc37e88a66048deed053565b55ef6bdb8c2f2f67832
f8939a32442ea54f13d003494d6350a2864b427a9126a9c26a031909fe53fa84
63b80c3ffd536c9a5dd29ceb03148dbc24c0033aec95c7fb4031f65dbd67a2d7
0e6b0a1bd772f5450fa1ddf2a6424fa2a5efcf7fcf2260372574d7050bbf7d9a
c6340019109b8f57540141b0b96f948d83e0f470248390518a6e881ff4511933
1c274c98a1f8af8eaed5a700c5aa8785011be8b2a6cfdd0e35ddfa96332ccf95
f9ff25733ebed67bbfd4ab734fd429f2e389124ab79d94802f2ccd41741ef75f
566ad0184b43477ab5a074aaa7d5032d9e54a27050363bd1cc0fa3d7dd0c1330
618008fab04093d1b242b34a624b4fb22b90e1bf1f3e030052312cbd45e7437e
b00f079e154138a510686c845e231d38f64b591b0a6871dbfd0a5f6eec3326a5
b6e559e4955e6ec14bd4c08907d10bd3337e77f3e95ab24c7ede5326b7470523
1664c3fb73ca3b1cd256101b8f0e3082bb42cbc707bccb31c18ffe2ba5248204
baaeaa4827364a0ac244a673cefe846f1eb7efbaf56df912429af20762a52538
567a6f6616a568eb6894304a0f9263c78e398f8ab8840d94e0fc643c8a271bf8
70fe17c528baecd305c2a5952aa77763238b2f109df034e2e96a28b4878cd8fb
5949fb0cbe9b6eca9f5f55e3b3145a5de189b3cff1ae547da592695eef837015
67eed3990ebbaab899647e74a0561620086c55e5c1c9685409c1332ca8db71e2
6972a0af5b222612c0ea26dba1affdab5b381767b6fa39c2013ad8d48eea1a49
8be479a07fca391d405f80f2b8b5b42c52b4388be091bc37243f35c96d648dcc
bf2e0c6d86f410f11098f7ca5ec3f70e1285d754724d566a63aa077e58da1b32
79ae0927738ce2460f4e9b3daedb44a26e7b7c823fb24959a2a34f7a3d919e0f
7829ad5043519b470b1b043a7d0408729f3a725dc26675bce1f64dd426d4debd
93f21cea1b5f166269ab6a4a6f48218f0c833d212a4e620214db91b6518b0506
992ee728fcf361dd33844e75742a06d66d429d7fa44122283cab368737871911
1f334a0b12fff88f4e834102148418f797e9de45fb61eb806d118f00ad69e623
1e3c2d814ae638a49098654cf6f4b39ac76156210b68fccc4680118188d865fe
4de702950858d4c88187023bbcd5684a96a8a601d235b66d3f93a43a69a5af81
c8928106115d95616cabc661684c2e0d9e817a3708b83592638747c79d6ac49b
e613dfdc02b0ed5d40abb4f10edb26f73a2eb38df78f1f4379f78097804fb114
47d61b2fd649e8976e5432750d7c84e4838f64eef2484bf8a096f42a336e3ae8
e394a213d65151915cc7df1b8d15c5635768e5143c07bdad89007e81407821c8
c5382dc822e6b0665861b87616bfb785ad525c58d877d01c1184005ad5dadf15
53ee57f543487b31dc5ab82221d3f0754284bb2b0e398e3a3b8ca7a5065c40f3
2bdab0a5936452346e86c6990811847663eaa59e07979cd10fae2511a00795c3
df752732373a13c69ab33f1b23c9f7eceeaf1542272d252ef368015817b3191e
cd28903738431bba7a0c9a0077a3db0992f50d09a90e9cace5ed9e8accc37f9a
0085a9c53d1cca39be57c2b8c289e32332a355ef70afcfa0fc429b9c383ea3c2
bb85d2f9edea061bff3f6b369d8e5b6e66c45ee6bdcf0eb34cd9e22549c09b35
776815b68a4a9920abb8d8a102f6bd91d264b3b974d0cb9bf146c780f3191401
65798a6a4d5151690a86637db1dc72d4efb671b4c6bd65c2b43329757983e3d6
3d9b00994e15f1dc0f6cc88a3e46cf53c7d59be8e68a9f0513b51d130e4bc504
08d12746c39666bb7ededd76e80f3e6adc7a7da01537fba98703a00d32a99394
5d4552a91e4f2ccd823cba2e4f0f0d176f3863cce50608787a6231d6768c8b26
453f09172c0569c0339fa4f15adece4ad8d33570f54bc8ca33b29ed51015f19d
5142b82ae6d8c80ac0d6015b81af094d01d74e611fa6f1f8525fa4f43cf41f1d
f5d55e1d2d2d5d42f10aa5b7718514b605f7d004ca0f2ae11cabd7ecfe1a6836
e759f7ca5907ab2d157d010a0ecdb715deb14c52190d6800c693429c100aa516
25de2a1b2a584c19972a41be35bf676810db61b00bb2ffb65e0151c674cab3e4
532593a492b54db5151c0c7fd11dda76a80cbd925adfbafb4f8bc4e85850613e
d369b58464c68177a72b87317921183efb1ae64ad50a5e32633c66510d542928
60c553c0b7e06d23d06c5fe05f15c8a24409f38fd7c57a441d4f3710c61b7f92
2e3090e0a6563a34118adf0b4bdb5aa2e9d8aa08f05e98d7636c1b14e1c212d1
fbeab21bbf558bf2f61e1b4cc35d93378b0197d7fb0632baa04727373e230bf7
7ab2fb8c84ed6ddb74476aaa7ddca09643b17872483c78cfee032c5459335f6a
3f96eeb5077f6f517619225a187ba0d0e796382f7403e87b52bbcbad570ab441
a0518bd4563c669b8bc5389cc7fb32464e7a65a35f49429048e359509cd75832
c7124d7cbca5b04c9665bdb1a7ac527e2066b2e1c44013ed9fc36c4c02d22e39
d64922b5e603e50e5cea50370028c7ee4266f90920792191a03e07ff66d519d0
f7939a03e0f71915bc40e1bd322de60a61a06769b10123a010d651af87c398c5
ebb890abd8035ab57cc52c863d1d1a7f63099f83da61275cb7ab8be2e87068ef
973b02bd74873bc0b323df7f18e3948a0770f1e6712feaf2a66f6a27bc3460b6
c74423ef52f747832e882c1483c13cc99118bdf2d4e5c6cd53de0548ca20b037
304f19c23ae27092433ed9a818e2d45fc8c33f54cd75103784548b637dce62e0
97ca863bda4b875d9043064fbbb95f48455ee290f4dfd2d340f0dffd94735521
a96c48aee78d7aaeb74a66c2efdbf4459d49e910a72abade079e234dc1803234
424c4230840b33856a12f579c5357506e5bad60858bb7e1dcfee432fadf71dda
6a30d1b67f32be95796a937a7c7c43ff2add537129ee7053bc4fc6410075f20b
a849c75ee75afa31ff1327a66efb1ac66fb91b0d80f3a7a65f131acbb6a59014
898c8058e5eca7c5741fb595827b2ca5c024ff82ba08ea3233d50aebff6563b1
40ce84b4d426827e505462acd82affa40513b69a47e6482496e13430c7b8acf9
2c3609d46964a399600df8f0043d896e1395468721b71e50985ff0e6fb6792ec
49fc0645dd00a086bb3c422734bfd3f2d630d952e1a4984ff2c81def3d222d26
87bbe8926a836a12886b528d991beb9ba7028797d12daba1fa87acd4692ab221
daf2d73602ad328a345440b5b0a3723d3916a13e1d43ddc45bffa6763b97e730
85817d7e7e1499bb139abc8abe8555b41b1dae5fa625afa46d7e7c93d02d7805
129a388ad94f436c57cb59e2af06bf6512f36fbd4066d25628336f74f030ae94
9fe619f0295c6fc91030ebd30cf5a1e7adc93fd0caf5a32391ab8ba99079136c
aac454bc8eb2d611ea736709af9a5a410d85275698a15886f68ab302df56dce7
2ff01bb84e1b6aa5a72a1025a5ce21d4c59781c624b9c4bd64f9e4d4446ccf83
f35c80adfdf95e5965c05c0219839b8f0398680027a9f3f6b619461018d5524b
d5613342a1b8e2895baf1d05eb1697e97a97c102c83423e97f56b6aa912af709
f815837f41fa3f64f11cc6e7205dbf96efaa6c20e34e1e753fc3a803f4037301
bc8bcf404a4c97447f68ef2b6c244bc49a22d7b6de5793cf5fbb6121ec01d9d7
99e99aadeb663fba3154c517685d6f220200ad92208821f7ad75dfa5cce1db93
1e78eadd88ea84a366a4f0dc629b07e671eee39dc0e8e4621d253e9da22d466b
c16c4875f6141101558c91cc57401b7404e730724edb271f8ce24abf7cf8210e
72a5927d55c4ddc75a988c6a53452d33d5dba09d188b83a30ef97fdc38b0fb29
f1beefa7f3192d22b23ad9e16cb2ba5bf62c4388d262bdf226c988b198ac894a
8b917a8c3248e961f480a1b76302711c62bc82e2f9cbef02dcdf40ea72830eb4
37b639825b8f96409ac16059ea9545c4038f855a644e87cd9129737072eadde4
7014d33aaeb5d1b9f2e62add730f1925d91f993e31a23e583d26c57a18cbd535
ad6217c9cb3277ea0b7ada57d53291c11bd6c0e51832f44e69a06ed640dc5885
e758a5721f3eb5243714a501d2bf857cd53ea15c4f3ec21088265105ac61cbdd
75c673539719ba9150e7ffe3792efdf83502aef0e8ebdc84c4fd645945b7de9e
5ebd2010adc246010d083966ff02f3c481654c952e1bf047f66b8dc70415aa56
1bb80edf628f7fb82625c62ea0bafb1c732a4f3c1dccf423ecb719cdeb6632f0
07fe2645d5e284d48f53605639a4a99f7612429b1a5516e36f9dba8447204f26
5c1fe1328b9c4ec4ec849c9af77c3541a7050ce8b76fee27d393e231bf50e2cc
5e3d06d4329f62a203c8398b4e2b8e1daeb33524aa55ff888a833a796c9eaaa4
019dc2e217bd70975648029379a40948a166b796d492ed8d5a6c3635866fe197
124781640ed1f8e0b632712a73eac02c892da22d8da0a9d288be10235b58b5ac
46b4c27458b637fb19c0b61d41682e861d75230c202b586a7bf3327528b4707e
77a5d91f80d4371f4704a0e550004cd35f4ce75167fe8a2fe00486b6cdf41ff4
75445d9a885b6732ad0f14f1331e2e635b251fcdbe339be5ee52b736ff3f8fb3
59f9c918f68534fa89ea811e4aecdddb7c6af6177afe02ed40237b09c2fedd03
1a1d25ce4a98bd2ecbab0b67a637f1cb0de53d88e70bfaa9b4fcb80d84508daf
2894500984e12a3389235cbffd4f5a3e3b141d13f8b7f755ba78fa2c83f3e4fc
df970ad40d1c98766ef7f6620848c681de17f861d2f89dde5b6f07cc213dd25d
a790015a2730373604b37bb27228c200608e8a32e53838f50d3cfa17e0c8b652
fe05adbb8eb4c41eb3526b439e29196deeacb4cf4076794e841c5083471b2c2f
ae72b4e6dea3aaf7ae5381fa47648be5206f7916be989b1580b60622afd9753b
32a37d6511a30c7a84ab2ba2c1add1a779a4ffd570e438b978432d3e1476c852
7dbfaa838def3d01f40e40eee4ff938a2d378d00ed1a661c03b37c2ad411095c
6f04fc201944d5380c5657af8a8a25077ede67ae8ea77a5e6f37b1cd34b8951a
2beb639c9c4bfe0a79addecb2fc2c676cf965d424d267b17e69e0db7d3760be3
b0b81e313d798aedf11549a93e4b6b4172af54ebaa585c7acb51ef142a525dac
fd634c3d441dfc4d8c30ffddc709b7722dce9c1f7a482a86713a0349b6b77047
6bae1c334d0f08ae953b01323767d2f9f7be0d798afb4c7e5eb2508ae37f5bc6
732bcb98d74fe47f7cc85e5da4d3915b7899a6732052cb7a58b5f4791f5b395b
7f9252992efedd71b7aba97cfec32057ede8530cbbd1b6549c00c60f4e33c874
4e0c926c0420ce545954e3169a3280c1c0ab6c12d2282503b00040b4363d3e42
c47be5752e33fa33b04d7ff5822f178c0016a95e7e9b6d321cd45557533c81e3
8f6518121ba8ba48cd607ca0ef08a0266287133b73bcbb4200c38f56061178ab
44331f4cdaa41f285446b46626ae15f6c464eb5d3452a907609ea1341d4e636f
3e7b34cf1c198c2fd6bac972ad183ac2c8a7ea4c29b3a3011e0bf36d01a16ab9
8f8e5ec28cd5c2872a317db6b10fa1e762c7a66ce9366119f565727493a7b92c
05bafc4459c90412802699a8b0ddcd444db0a83f0761eb5081dfae012d3614b4
2fa05949800f1d85d79984ad3591db13e83d4ade80c4f1a4dc5e39858005924d
c60bec55b62991499f19b8c9bede6fee233fbb73ff11ca67b2669a8d804c4991
30d8b565f085c3175ce41b4705b5e0097ce3b041b1a19fa51c0bb1d863dbde23
4a15c0b293bd6a57790841153eef874d04008ce5c79f0b044cffe805beb49e72
8df296c507016e4a193340c46a7199e1df9d082a77abc3aa78476cccd8c21570
8aa7ddf1b4c33cd89f2aacb9c7a555b063362c9b15ca11bfc99b2432cbe4437d
79aff0cf21b2e2e4b9610e2ab5e322efbcb8c305c1047f9da4168a7a902dbf4f
7349644e2b6a6786be23a9bc0366be38ebd7aebd46edf5ca6c12a51da866d394
8a427325c82f1ed51764c845b7d21a3921f0ad5c680af199749dffd1c1bc7d47
d034bfea94a22bea81f983b32e805e4cfded33a4561b520ae30e0f1d1e04dc1c
6e175f1a0a89dd63852ec275a8a253585afe3f32f5d8c1370f4930c01153125d
7a527f765f5b1ba5ac87a0b3ab4e02e1bdbd44a5ea10554f81c28fdea20452a2
15f0ebede394556f01dc214aa5befef3d6ae17e99a2d39fe8ee2ab756c5f4f1e
c8c34f909ff9327b32ce3d652f3d28539ceb4b3e1f64d2e8fab402709d9eaf79
2314f8309edda4a1116b04f32a3b7ea4b32a35fd20c44a116bf243d9efcc51f3
48c1af9886f56c896670916baaeff4af41fa455e2f3e5acb4e683985b3cd210a
6cf615c27a08247b322e34cfaf773f541a62073285a529c2162b23a53fe93755
6833d0c65205c12f508a94aef24ccf5bb0c6cd3c48cc976155c3d03ccf16a0d9
e2b4ee360616d9f3e7d92a2078e419240ee99bbe9cbd7fd02fd53fc92d886764
9108934d7ddbcedc46e581dde3bc76095f97ba100b802475a815538376e7e380
163d416c43088ce795287543a5a2cca3698a65f835704d150af3a9545f19cda2
3540216773de1346fcc83b2133137e6833f5e9dae7f8f35200588947eb3dc9f4
bfadd0b913177747d047559756caff04a2f9cd3276e387c7ed1d83e378cc019f
faf8a806792334724f7e6780da8bd196ba01e4a33146df9a9876b1a2dc9b37c6
311b1e59613de983c199412e554e31cafaeea271dbe13822db903156fd95f0e5
b06ad0f3fdde13aa4e00c19b02987d055bd60e4233c23e193fe35b67245cc58a
de71666f10de4560d2f6a56152002575310fffe3f07f955f71d76ec8bfb2c41a
87fb31ea44be4512e9fcad241ef4e0d1a140f96588584cc0158d685f0960354b
248eb3561a84fa84901b7e74410514e584158c5d464741c215fb0929fad6bc75
3cd849a180ca030161949505c2635c682f134953edc7825c5b7d1472dc998d89
42b46d54e7b77a5b68fd5c708a5ed9333421af8b7cfcd0ae17eaec4c816e023a
5c0752c9038693ad089900661aea8683e7e4352e870c68bf73ed9c1057585487
aa1d9d7a49d80e912d88997b852c058de017ac39a308d7e2a466edbe25734aa8
f1e5f71c107129e17a371b31e0c212862a070bbb57c279b47aea2c417fc646c0
8a2688032ea0a297b8cd3530ce1013d61d966d4730b71569c97299c7758768a2
a5818745bfb07b39018daef2b7e78461ff7bc976b2d729f20895d77a58891b48
2000353bf252c19425d2abd980f4ed1d90fd80a02d391981e9b708189b01f1c0
c8bf3b35a98cfad8d5ddec4c0e0845ef7fe52c2308ec3bd9cb7a69390773d871
6b8affd683f827ed9df0e7397d2342d6931c6a950c3817f450389b8854975325
07df03ca871ef52739cb7a5be89e90225982843c642d66314930c594f5f1ca34
2db441a34bcf8c41b9bda1d9896089786d0aa5534878ac9d34f095bed18a9967
ed119ab1f1f1d36e5c918761fb05e48d4f162cbfdc1cbd898b6ec29e012a6086
77fc2bc6e1bfd66c8785e983178c7244a64f50d6313132d6a636fbb722ebf352
58a20c693bf7b1dbb3eb21b1e059e51662be8908a97a7df8a90d0fcbf546487f
a66618bd70611c9f319bc6dfb4c8e42a4363eea360359c63050218e8c9250779
c8702150f9e7588a09d7671b75e867b29c690dd4f24133fefa70ced0171889a5
58393c6c874c641004bc38dc302c4852a3b3f7b8297e1f1c819d4bf51ca0f4fa
c8e72b68b739d2fdf501a9fa09bb3740116d8232024ab94d4f3e991cf6032fb5
8793a412a62724b7540c0f88fcd0946a0e731474c160cca3bf2fd39d81622bd1
68fab63b02b54f6107e0348d3217354a8745d71ebf25025052ac538adce3bfa3
eb5974a09618348bf8cd35e8cdaef609341a74828df002df34456221354b1d0b
7cf90f5b890feb0c2b0f107d10a863f49eb8aa0b3c582a5efac9d4e96778e90d
1d5b8e3d539eff436230606fcf152d97c3ac244a5baf7f3a85fe520d8a91e9ef
cf2aa14d65bc7652aa06546f1b6ac73a7e2799c5f14efd48f4a2c696cdf27117
abf059bc04fdf2740d66c302537a41118a4e1b6a2ce8f5cf8fce388a1c609de4
93c0bb62724c65553f39f7467f78fb3ce1231a0351a4b3347b4bf25cdfe44597
9b96678aae3595d4584e1125c3b17870c45988e465cd9391287e512e13ea78f6
e4b643111de22e0b0bd29073736441e551f961afdab9dee89c11417e881b802d
48f85f2e2fbb47b64223442da111f4d504fd6edc7c7ccae0c136c0534e9b634a
bbf304d3ecf446ee470efb3aadcea9d198365ecf59bb983c0d1e9d05b00b92d4
b6ecc86fa9d86ab5fca321ea4a65483f1a6b747865a653a9c00581930c303506
5fa325735c1aa83e47c96d24f473270d4ad2f6d89f1e536d2ec0dc9423f45d4f
1482f1e12feb5810dfeeb0709cc300fb57603b10f60aa54242ff34aafa5b272e
c3b2c20771814ddb5b5122a91a925d3622adf56d6b015978d4ccbd9de7da772e
6df1470158b3f56d2cc51eb640970c4f6bc0cc1933130d2dae93ebd663b89829
024b076d37fb7f47da2b63f2dbe82e5eeadd39e45e12de10ffcec4aeae8334e2
d2548d0d7801038fb2167d6f469d11d4bfe83af868b1e3f25a8a164549b77adb
40370cadf2ad1115ece979990c6bb5e992f0d069bc5e79cd1704df17b088dc01
16c6d39de5f84b99b73c61e969f7678d281e0d782a3f7854af6a02abc864c997
a5db643aef8f88a4c1785deeae55c02c7b2c863b63ac39631e009230f24f487e
413d8079355ba1b74fd01f11cd2ba91c5eae14ea1ca6877d028fd4e3deee9e29
3c37997f02b0bdfc017e2e7078a9e27076a2b4d16967ff8c7df7229374869875
3e0db25cc389e5986b1e5cb6750517312bff97688e0a7c9daa944b3e26954b14
e11a36a1028a60552cdfc41e72910ae5d76898b6cb6ca7b2b89e07c8f373d178
9420ced04da3a41758d4ef33659e52dcddfd4a80d072610e24c6f4cdad0d67b9
da92dc3a7eacc4261245418ffea63d5c414ecf6ce95e823094c91b168b886775
4bfe09ee9af94ff3b89479e579827fe51f5583fb41b8ec87268a625aa8754ffc
434bcbb3166558072ddf5067aac19ca6543a8d71fc1e2ba6edd22ea8e23e9639
0353ad00940fb37ab4c746c80ef040bd415dde1a4bfde33ecf12e507a0d8c823
11f4bec57a3d67dc303dd6f6acfa065d0f3934b6508aa67bb65543c5e079b3ae
3cd1b1eee2cbb4334926083fee36edb0c6417eb64bdd75037839bfc3cc5e5dd3
178ef1c8c9f587fa9af6a0e22e5a0d9c0f3025e18bbcae8065e779148fafc594
19c77b8506c1ab3f280e63abe46c30a1d4f8fcccb192a3def1fd7e27b8f1870e
635e2d09bd1f839d8ed15defa8cd0e78a57fdc3e8c2f63076c8f7bb449bf6c08
0e5867dba4772486b488c6f8022532f4f1ea1e1a7ca9888108d1981ab16e3183
489ec86dd8e7a5bce08144c19bd088743e74af49c4981f47a689fc29c5d7374b
e159910f076b0e82aef6ec1b462e020a6fb94d7d07d7f17f40f6a22662725f5e
ae78821eec72b1736f5a3abb9d8f77be674efcaefad5dd03f8aa576962922a72
38385fe55fe7febcaffec98eb31beb2b57aeab044ccb6990742ffb17f02d43ec
5d84b4a5beb5e31c7c7a14e925fe2a235b11f19e4d8c99d5a216e05abbe6c734
6776843d91b408370069b46206365b7b144f26735ef728f64dd3805736a2a51d
11ac604e4d108b40b001adaf54008f5351141340ca7a3c114d8a837550e06806
aeed4bcad7f242dc40ac1d8c9397f18224f843f1d2f47b74c86481611fa52c2d
2f90a88bdcf9c89d11ac5c3cc9a4cd901fdf39e1954afb15bdfe5b58c04506f2
4f5980b37dc18438111f4f0677fdef746d65a0e6c672b24b9d80e6a8d56361bd
cf088ba7c3724db8af48a40b72ca481160f0d12ec138cd892640999517bf20bd
cfb0e88b518dc143cefbc2183e3be179edab8d4ee35b39b3c764860295235464
0c9367fbd04a1dee3d2d7e9b5a47ebccc4ce5470af915a206b5a7a548e9c0884
81f689e556177c675b8cf6ec31e61f31bbbbcf33b86548caa724cbffb44fa071
25437099ac25f82ede5ed5078d6ca958e6cee6ba28e1d6b4144ec3805dc591f8
8d9075d95138a69e8a20e6e0b1f1ae76c0b6b6a98669193d19da67898de677f1
62b281296ba7a417704c61219fdb8125d45616a94ce870432d96885f1b09305c
04df5abd05fafdcfc27b1a89a04c6425c3993540cdb37cdd00c0c039d858374d
f55e1d185d6e1e66ccf29270fee345002e32b8977199f9c0719b062e14be8796
dbeab21b50ebb79507f95d9f26c841b9c62643500c50ea32ee7d7dcd64d381c7
ea64bd435683ec96138c10317b3b08ca2c4b17b5b52b40a2386e34c7408dee89
ed2e5751648ad264eb45e333922ac1c5090b3f4fbd7f647b8faaf5e2e31e05e6
68bbac03c9ceca04418165c277fe19e2536b6ade4ffed2f5e1373455e9f174a8
c21080891262bdd71aedc07082c64e0f48566d007738e0a8af4d39e1d7ce0eaf
0e11d37251230778a1325977d5c3aba3b634c51ca541f54a5d340aa446e53775
d42fcee631bf4eae749654c29ed6436ad19c7ed2820e5d06413181afb7e2ea9c
bfa08f5e3da98ead19b07a0a6020295126b904c74ba2360595ab8fc8fb389a94
d5657a2634b209e14c180adaa50ca136ce5ee44e2206cc390ee93f6b127dc98a
0caf2b6083a44fbc5895bbf6e831afee514d1d792598f1177737e8fb98e9228c
5bfe7718fb223a5f89d72c2af6659a6dc41c9c1979934a9188cf6e9bf8510d10
eac1b9da6b02d5fb65d71a8a57d32f7e4e6041981788e56478c503faee34e664
5056c2d7d9bc17d7ea1e07efeede6c59dfb4394c9d60f221a64b9897fd3916dc
8cf01937988e0bb6224fc5e7d0b2b45942b88f61f39faebec8a35cc61d7496b2
df5e4fe1392aa949fef9498c4695d81ccd513592dd256ffb40c6d499090d003f
489abff6a2fece75b5aa035890c1cc45bd632d59df29429bcde82ec5aa1c4cfa
85c51ca653bf51e150484b3a44b28c223151844743baf15636eeda3c519c50f3
5b0fd6041f3237d9e31690863b5e6aa8f3fdb5c7b42d4ac173acfa30a1e9de17
29a460273e73559cd57f02dcc8aa74044f8a3d7b0c4e25ba4fcd786ddb426cf7
6564dbfe9807b506800cde2574479d9a14bdf40a03a5703d487c5c7518c0e2c4
9cb6ee37a4669c1a9534b9311858ab917374c6a5181ea672489a295886753d2a
5fdbfc1ab5bd84226cecae91e93c1b67be7caf63a3007f9545c76af40b8097bb
888edcd916ef6167d2f159f18886ccb0945985ed541bc7fca22c8425634ca345
5e5e0e6461f754d4378463b096a9ea1cda50f792cf0216ce1a156642aafc2139
fb6321792e262e0f1c22575de8bd2c2da2cf1bd3bb18a38ba55291b4335cbfeb
c87c94b5df51b1d1c96c50c5262e2b4e49ed39e48e6900e255180ccd406144e1
2739ec3fb41866098d0bafd99dcfcdb5b541fb4047e7e212ce66555075e2673c
0cca9292bfc5ac95d1efd2261755a0574d78dfe17d8bc3136b4e8610ed5c1b8e
5501ae9b3b241a6f799545d7c060e5c344652855b9659895428bf4b8030ee267
e8ff6eb8f3157e4db50e27eef88baaa51ec399da91a87bfa970823069c130c81
cd28379465460dbc750b8bc277f4cfbe6b5ff594a64f66210f725d4fa199ac82
7ec091aeb06682a4724a4758afee0b0c0567fa653bfccd807c2153aec4af7f32
fd59e1572791226e290c2c9cbf78368b3e6c4a3138b7cd4e942e414a17ec2471
8b160a8b2ddfa3c6107737edf21869030c067be8527e79906c65ba1e12d652c9
c6204748b3f5c6c4cab1aadd72bfd3ab206c67d730b42326239b6e60268cbd62
84a7d84e2f28f3d7c0aa879e15c174a912e9e563adb97071721c8f231b4d01e0
aac77769fc6ba13ba469a20adae6f59eaee9736cf2934f65dc8d90bd890c1db2
423c5b350febe60e0411cd774146ef64b2283934305ce10c883a2089f56f4118
0f341bec509eb590450380d78331fbb4200515987e07e3efbadf1910b9cbf0d7
8d52c206dcc85197760c3d47f49ac457c43d4964a34ca01bcbb84eefe405a4fd
dbf771e9deeaf5b73f77b7b5f0201a8f23e23821031a8ff4f3caf8700d886bcd
b0797a3052f15a6cb22e9527e87d589d2eb811f078ee0d2c3c53902122f72929
b9389e102e08dd521835c19c10088d8268df23efe8846e164d28f5ac9c83b46f
8bf452afc44b91008feb255be38f444e36eb1c9cebb78a8179e1b5cf9bc46ac0
103d389129261c1d0f75e8639bc6afe7d02a07ff0ba9c23714f85e3a33986859
28801e5a5bf76962693a737ba73a835a292383f36b2131818f6d8e6eea9d7288
7a5d165240afa23dd58ad3dbebd61f8d4468671cbc58479b38e891e134b453dc
9322a75645fa81b89cdf53c3b32ad0d0a3f3f451846c575d8ac2d958248dbf1b
7f500fbdec15938ad1c7b625aa5687c0fd8c191020112bf0223933c1292117d7
d567b4514d7af0469cbd4bc70cbb14e6746c294234f65cf77f4b8648a007f340
58b97f0622ba32d1d6263f44513c8bbf05de61971cc58d67d98ba1deaf4338bb
018af24d18b32fd597fa2ffbbf5e9927d3a02f7c3102bf7aa88d4ad50b2fd33d
7d681c261b3d3d89c90e766b3412bfc15e2bcf304121a24e36e993b4b830aa2b
70b41904709de8c4dde151b4008e236d180cb67d0a66c5a3814c540dcad97c23
d094771a74cd763eb855abd0d8b58b6150ea608e7700413c57aeae4e84a21c10
d04fc93167d251f54765a6a15a25a04be4d382f6e393d364ec917a2bf7495e69
0468fa8d220e00ef296cab1c0ac19e99307a93923b909443ec97281533b8623e
24aa8470c4af45081151b1b3836e41e855ded39ce6b3e7dca1e96edf846539a1
6a78fc78e0cc9b7cfc72774ae811d3d17d3ad651294d50aa5b082f7fda55089b
b459c4fc55101a5a0c062150990ad164e3a17333c7c1273d7dba796f196c158a
603584309280d6f926f6f01e40d28838d6565c2fde5b479da704239648d30cce
6634dc9f163ed80732c532b181fb3ffe4faded5dfca0bacaf9749ce83176660c
fe857f3b968c251d94fbf543d9a03f7a2e797164910c5d310374a1c316dfcd19
f4505883cfa30fe8f32f4e3a1a30367a798d8a86620f3dd753bdfa9db8486076
898b1a44c1d756d905f73ac91733a65dffca0b969a353de66bd9775f271eb74b
345e65b9dfe6e33d640c17124fb9d098057bd346174a0b14f59b6b80eeb4d5ec
16f29d2900c975f8ffa610e7c60e64a4375b1619551fb23f4586c7aa4cf0fbea
32398444fbdd37a4f3cf8da8f6ae629c66e07e42204291faf68d4923dbe42d02
b6fc9553bcaaf85bd87b309a54be08ef77616e5b776542b3dd16a46ebae98c2e
d6f17c30a36286796e0bc58a601d7538cb9e8dad0ee29a3e5e9a5c5d5384df09
50e4b239e1a77f654daef4357e56ca390b1cb4d1652bce7ffa1b33b550b066c3
6e8bfe9aef3b558d0478c6e766349a30248b4c44385c093b9ccc603ea9e504a7
7d557f5962005e78645689f1ec442cfcc0b7d9b254c186ab362ebc843950d49d
0fba3a46dc394f5d2b680d64ea0c1ea309bf4ed12dd08c447ae0b06b74aff4a0
a1ebc1b541937c8bba0bb073a858dfa8b8ff0e3ff43fd6c61a07a89a35099e74
19c88e9c421bf2ec16208c80f0fc647c2cdb797750620ae14ed2810035e2707c
c3adb2d9bf31fee7e4dffbe0efe88983e4149119d7f750525324f21c9028f40c
ffdbfc1f07f280272f0e526c8501fd11f81b3d9e4803ce2f8f58abb78347fba8
0db6cc37d0e69198fe3f6c0cf4ad24751b024138f79ae7292279e3ff5a617012
3ef4c4158d7b975f75cc00145e8d17bbbfed56037b45b73b4a9a90a81ad32802
d4299b7a66cf8f982ff15241c0a5a7b79eabe71f2066b07c27cc061f38f549f4
55c442ddb64848d90f3bb4d9994a14b65c48f5fff067ef9dd27805278b70c131
843705b6dd28f36b2490b6c2db20282cd85343b5c5ef44f7e36bc8b4cd761898
387d5023e4643a2bd5eb64cac995f8ae2db3dd0d159182df04610d7cea086332
07799a8825d498be9f172fc5a9673d2d5268cc0f928d5641a5d3a927123ad02c
e27815128b2ca8fcedb458a94dab363ad8aea554fe88cea8836e04590acfb95b
69ae891b561b5c3edb89924bb413c781d87c111fc2edc89bef1e9999f56d2d88
343445b72ecae9a4e8d0cf2a587ac383eea39079b119c0bdeaca745cfd9b42a8
f9b8ce33e20fb3c63b26a10f0e45dfec78f825891892222d51b7b66d1c715c5f
20d419e6d8017cd407a70686be8a5e26548c40b88b891f134ad4defb17e838a9
fd3ebb66f30e91ef492fc6226ffa130755a93dc99274905a0adab9df6c4eb9ae
644f1ab789a5df2e02849f5864070fb0cd5c46d7983fa32488e2e51260b02b19
6608e8e04e069fedd52561d590fb9540682d395a5bb0c48c3eac439167509124
75cd371e2ac894d5e0bc578a527b74a7788c3ac343f156da35e05a4af8c9269b
9c80cd0532959d4aa4140c4baec656df2ce4c1b2d2379003d80d2867fe87149d
b81034da56263a537ef11f57f4ff99c201b7e3c4ab6ff736029b3f37824ab9e4
575c6ce71306d5f142653dc47c75c8823a88cf27b7e72333621a27e5ea685dae
4003a9a0eb21eb8fa0a28ee7422dafeeebb2116ce138ce4e5aa3b12bbd7adbf6
7b6e7abb8fc696157fe14bc27c19097adb0161ce55c12fc41606a71271e4b62e
63904b215b10e1b7b7aea6098597635923a5f4cb6ea88639ac9d391e5144f4d2
63f392f197620a0d90119a44108ef3b9a1383e9a09ecd6c233dc578726b77ef1
62fa360721b1a00f068d8f3c92f898fa2c426a89794383f2eee244f7224a1da1
7b497ce7a54c1e268f7c55fc13caaf75f466d228717114c42aaf7445f4d500e0
1fa3d9c40c7ee2c41035f7c80970d483038223a78f9e70e25f617f1ec68999b1
0c9e252bbf7f42bb949b4763aa5f913f1df928881b9a38be030ceab3cbc99490
f766e179a69ec6d587d4f0d5bf2c83de038f0e29cce2474d2d8637209d106e9c
40701a4a02fa7a5ea3337cbf5824986a35cc612befcd523b2ed8dadc1f8b14eb
ccb661e17712f97b2edc5f16200fdf902041f64f647f62df027995f31a37e353
7225b2c87dc9cd6a35667f0a8a855a250fb27ed3a416e05c24675cec24eae43e
030d3dfe57b3f75a3521ee949091ebb8addd2217f38e35d3db8d8a11acfafce6
0314190c782726b221d7ee3c9f3fbfc4941e2709dfdf70cd5302e08a3ffab961
fe2bf082078932761afe517acd606ac77430ab6de858eb373601e9f31b82bf21
216615bf880db0928caa415b340e13b830fccc44ac5612ebb95fe0148f0157a9
d955d6d45b120f64b95da02531b0e976cf84d50428be2fd5378db7c15d2292c4
0abb267fa92da83e4b93a549c9f9296afde98d0320ca4cf82999a1485890dbbd
87c78251e767877fdd9d82cacbd76c646a69159a6508e126f5e6662ea02be37f
d3fec239cfda3f288b54b51b81f67b9c4a4f24d32d4ce132d8d0b649e3f34a90
27db26979ee1be388d9e722b4c5cf67f73fde35c5a9a58c0a92a452c10eefabc
50e0725a0c6d2fd69ee7ad3bbd186b926abd168f92cf344cc848c62750688301
d96084651ebd71f449ce379817910f40fbcf3666ccd6a40edb3abb1a2702151b
7b4418f8de0292681ac8d12b133708eb693461c09026ce4c3cef28abe72667be
fc530611781c2104b467dade2177a2d7b0a875ff9e3ec3c02c28c3eedd66967d
88ddeea8c4658ae8c2a59bd453623461a6a23c98aef2cac87d1ea756a7c1738a
9147d68fd03d11b5606e01bff3b4bd6b4d969044ca19cc4fdbe4b24ba0cc0c28
0271d16e55666c353797c8509743416fdd093c12dbbc988d3dcde0302853fc1f
2961b3dbc9d65a8568cd93d8c0b8f9e258b393a63167aa2c5424af722eed65f4
fb8c9a6f56109a9fa2f8afc658b3f8c913a96c8cd8985af4c2b87a0e9d40e69a
78f062941687cccd682930d6c73e7d1777670ef5c66b48f328e15068afaef652
60365e16f6f45ad971b24d73e8e5c54be3eeaf23a0
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /fi put
dup 39 /quoteright put
dup 46 /period put
dup 47 /slash put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 73 /I put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 87 /W put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueXX 5000769 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 12 /beta put
dup 13 /gamma put
dup 14 /delta put
dup 15 /epsilon1 put
dup 22 /mu put
dup 25 /pi put
dup 27 /sigma put
dup 28 /tau put
dup 34 /epsilon put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 88 /X put
dup 89 /Y put
dup 90 /Z put
dup 96 /lscript put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueXX 5087385 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 2 /multiply put
dup 3 /asteriskmath put
dup 8 /circleplus put
dup 12 /circledot put
dup 15 /bullet put
dup 17 /equivalence put
dup 18 /reflexsubset put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 26 /propersubset put
dup 32 /arrowleft put
dup 33 /arrowright put
dup 40 /arrowdblleft put
dup 41 /arrowdblright put
dup 50 /element put
dup 54 /negationslash put
dup 55 /mapsto put
dup 56 /universal put
dup 57 /existential put
dup 59 /emptyset put
dup 63 /perpendicular put
dup 67 /C put
dup 68 /D put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 79 /O put
dup 91 /union put
dup 92 /intersection put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 106 /bar put
dup 110 /backslash put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueXX 5000820 def
currentdict end
currentfile eexec
9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352
05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23
b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a
f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145
d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8
3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0
aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a
9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4
9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa
4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd
df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79
c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074
b64c43002d04ea14cd9bb1688e1f6ebfa918d6022436c68abe6269f9279784a0
d7a93f78a02c635d7bb34ce2275c60c9e017ae034a6b5c9cbbe6538d4a63054f
7e2c76393c4b45900e88421d57790d16de183f240fb2fa865c156fc0e0ff66f7
9638e2f269f1095a7e5626f012baf0e8a4a4e16de5644dd9d2cde378caaf66b5
406bd43e823b3e91115cd73ad1cf998fbd149610e2775cc27f17ad72c05eaf2b
7ccd054bd569047a37364372604df8b15680e71276e6b17ecd19be1f99d93671
98fc943f24d07675973a26b6585f6bfa2d76c55275d9ba374afc54b60b3830db
06fd6ec552dd641faeae0522821ac749e0f456daba79f1a86ea8087b422460e5
5d06b527fc9413f787905a298ec61f2d429fc9074ac9b2d978cb7c79abf2334a
acd4be69b61cda800f3f6c5c3bbbbe28571cc92999496c3d8a80b56ee6a0d2dd
fb4bc8d21cc8a9936a8fae75d0238447cb468ca3b4bf08294b7b309b52575b46
764488ae21227f859211f38c9f91e3e7bc7e25178e465e5ec93cf07e7b480cbb
ca7c88170308f451715b3f7863192199761f11c46c6927f65f0cbacd03f507d3
9b59553f0c4c55379afb630ddfc90df3c330538dcac91f14f0cb2472b4acd08e
c307bc26de8386e5efab03d1f56b5ad1ab544cce507ef20d339e3b5a896d354e
18fd1667f7bfe844688a1317d867ea54eb4106d17def98e20b5b1c90501f90e5
4770a4957a00a2d2f33fcc765ca0aabe441c543ea7743403004970823c495f01
fe501f1ced9a0a6f329cf92cd98ac922660e0c2fa440922201c5dc374534c549
67c3a69e9af93adfc9b575648cfa851703084f4a948c5c6635931a0db9afc012
204cea9f642042ea1908150a6fd17e75fe4d2a8b20b40adac3dbe3862ca08c7c
974aaf4c4779a467d8ced00926973711a76282ca297165a1691abab663dd9aea
05ad19f6e43dc072ab3256cee5c82704c90af4b66925675fc89987d5a99d4c32
78fec3243332a12fdf18d53326a07e07e64d459aaf0d7fa4be1c4d313df6ee33
4ded1698563e32b9d3e8550dfc3c580cf04418bc6e55835debb3bb163d099222
73da5cd8b2872b15753d34e8fcda528008a7c7757fb0db9a30371743f659e04b
0133fa8eb060cb1294ed3bdba57d299279b5c50dd197450e28dd6610e891bba2
26f9ae82aaec2a5c9f41e93a87df5d0d31198cea6209630c889905e37ce06040
26f80599a6f144a3b3e5d6ef47fd8d2bd55bdf4ec98ed464d481f38dbd506f5b
2f8a764d4aed6a269f6cf7a524279659e6308d17e30081f5cabc06398bb272c1
78ce3146a4eec07530fd7ceb7443d67c04c6eda4fbf931ea19a615bbad09b9bc
85a2222212789f72b71292abcfd54f9594a54d8d36830ed6c3f8f7c6cfa41286
4d58230a9dc5ec525a4c76ae2b66ff12a90d5bba119ae62e276d871b41aae3c2
cb0df0e5a84dcd92a3c7f0c5e01a4ff239d14eecc4120956d56b97a6a87e4cdc
dd8f4d2d8ca7289874ce46ad6d9c80de21e3df87867c2b98a91a8a90618d0a33
e616715e7600037ae1b6802b08a7598180e75d28e982017ad7aa2ccb219feaa2
bfc43d0b0b55b2fffc72c9573fd6f280c15f609054eff15ccd89b1f4e79bea98
63ddb5241bd17ab8ef0a78eec9fd092456339e0d25e1494277db0135a06e1cfc
f0b7d233fc66f571f43debeb040e0fc5469cacb54f99bc20f40586502fe1ba1c
78efdde52b480cd13047fcc8db50bcadf4e73c0b3fbe69afbf28e5bf8bf2192a
1be879a07a5290e77bab074b2c9a0ad2787da51a61eb2463172d4684a29735a4
2f3c007efb13f6053495c2f1f8b221bf42ff995f0def5f49621d77b4cfac864a
24371da0941fdd6bb2f561d2d47e066b81648e04189ea844b2e6c5e0185df8a6
feeff81a0c670f5684ae961f4b57cdb4dbb4d1d1f6b43f8b2bbfdb91e1c3ed2b
f46e396694b2decb0b86f9d1979440a641fb7f6b67b34caccf0ceec1b1b221f8
9ca7b402e625ba4e8eeeb3805bf3648c63e9e7390e5e247b6792f7db208ab9e5
7aa740823f2b42b2f06d8020c5d3d95c0da256e68340da9963fe09f3559f7066
d385d71154932ab5aaa6f92d68ee4180aebee3c1a7d561e72eba06c239868540
070d93a3c2b13467dd7b84290f4594e1deac2edd7e7325f1f2a9088075f1e7aa
3d40c97af5be381eeba69d8f9e52c621560ce61c1186fe3d938064b4caf75fba
92a1f6793ebf7944d7805504d31f03c72be6acb8128052d5ce9a49ca36db0a44
b60a494ee47b4816a57b3c70221ea3eb560a4479fced4ddc522ac3d451782ce7
02dbc8c05ad98cc5be753d2be87cdb70040ca5c675d7656c5e96370eb7857938
9dbf1ed457d164c64d167972d704e0c61613c55a3a395a98d7be5460ac2debea
6da517f0c43a9df69da82798923d4620e567dade01b084f44bd37f8ed3a246e0
d44a4c1b09ed7946598722a7fdceaa1f7e6b4da7b2c1854cea27c76cde6e3392
b6750674f0701d650698d4e18e3ef8a63f9a4234dc1103e905f80acc7444570b
66bfb632980d00897534d6b058d513087ccbdfaa137d677aec4898c6a29b6b94
5ab23f10b5bac213b15804fa679b501c82cff21bd6fe73ae21d58cca52f5393e
db68e8929cf04bd877eeaf183ecd8954b34b806516cb1ae778bd1c89fefdd26e
8c655dcbe33269030e50203b57d7bb4f72e0c7d460525463ba6f1ce423b8333f
41fd8e1ca4d941be631341d259fb3e62fce9861e1799e4247c9b0c0f7349d6fd
e06772625b937248f19c0c89940d6a419ce2edba289fa49e102032307a9396d5
e6fcb235fdac6b27ca90caa70b24ce98108a70c0fc1cb98fe1fe3fadd4d55c83
45da7663f63f6fc7461859b8ca989f1908bc8d485ef8f5fd2872e4afbdd30eb1
6dde8fd3b3b5ad6a34b9ed996c24c8fd0e0f576e2fa6f84fbfdecfe8ce527928
8b7d4b531542c4687ce8ced6d7c1dc454219b31e59cd610c3f4df4d54de7b0d3
cef96b9e95758a57c1bd9ef451f00124dfab25246dc506b3fec166dc3cefd884
99759508529f08ec2d926783ffed59284fe8e4761f5fdd558b023257a19d86a1
decf1147db404ded06481891d51d999b64cab2e7a9e9387d5dc1d6faf7df2433
4ad0a7527d5a30ff4935c46c9eb55ef5cdacec97cbe7cd05abd6dbcc5eaddf6d
762f0596a353789bb0c13f8806fc003f7a787ef119f3201325df22d04187527d
31410fa64ac5775be109924de5750e542da58ff9a548ea2414826cf9fc35152a
d13e96f137c050e3f04e3473ccf94319a06f178bb09029cb158285da5a0055f8
4a0836677b2c851c793a777e18ce402a922fdaa1fb3619f90c23addcd5575cfe
3ce92b056ccf27eb15dc2ad14be65dba1d0c6b129c30d5446f213d3aa7db57d1
e4930f36793caa47841ef8b4ebf96ddc9997ae8e6060757dcccedb57395dbb43
bf8f93934eb1cf71733dcf28b4a4ba9504f6fb7042b80b6590c0042bdb119182
781b9368111548fc392aab5a552052ffee8ef24b550763d116d70c36556c4cb0
2b2dc329d83a6f183f55820db4d855771c057f22d658854a20751a462bae3142
7a8e9b4c033569f91f9facdebea9d1c19d52c4ccd51391cfc31e560b28f9604f
48c2bbaed8fcbac892a732af405a9de1e732f3aa72465d52e0b389db18c04923
eaed8467ea6d6a6ca5047de907fdd173b5e2e715d470f77f5a274931a995ba36
518b1d3eed9500c90edbff7602a95dd075055dd281eea68b2effce7efbe2a1ec
0c19834fac0052a9b909317bc7537a57416465a00a95981842bd86a0153d148a
8b51870d0ba4fd03f49f7ee4b0de68461575a651109f0a2b71b17c708163d66f
c2e637379df138c5b5e319e3247e639aadbfe411c32a3d1bca637936d24a8c87
f99e5c873e32b9fda8042830cd95fca20711fd9c929fdadce6b486bd265c360f
ae83ad57cc2f0d3c80e12f5f51a87ef419930224605a857225bed6637ad81249
c8483f358ad0f65f0471e3d3c72e98d06398c10934da5b4d00956ae692f45ee0
02e49e42b31a1ffe80a8622353da7feba2614a5d319482a0a0f040268542a774
05620023eacdc90c78c9c590ba5270aa6c00932b81d538decb3592093fa30cd1
d8555e39a26c82e7476273e8f2f9913f3d9cbd8e361b80370e7cce254ac770d7
0cd7b832a9c6f4f4807237ed59b10cba22c28823b4bc0f5c0f512cbdd225d0f3
6f626b3be913f7e1138405cf6dbb06870f97942ab85692a29852254115bc798f
8782d0d39a6f6400db671ec3edd4a2e9233d428940746681346d05d2e9c9c307
c9472c54e38590db3e47ef9d2d66bc632eb242ca89395fd937cfdfab64b3d194
14ccf2e17621bb88ae87dcde802db0bb85e97ff996fa1449f69e12c0f9f35241
11b6866d980e863fdfd67b8659a897a092b909bf31391216c1975f2cbf0dd6b8
cf0e97667281d37e6088fbfbd87b1780af2d80e70c93a8e48c107768b5dc6e6d
e49dd3269d93c928108b35aa34e53c8d784770e1e5cb93498a570f91803e412e
218a6a0e5bc87a597d0578b97bb89c7e1460d3a7004739d576547a21572ba70a
2d16f6963abb48e48f4ef8032c997e09103904997084e15fdd31ece6d9d4a850
c2ec59b6aa72a2d839b514b83629be2b01b951a8e79001ad954668cd5f38f41f
217118a67a68d696e1d7bfb61653e2208e0d84f17fded95f82bd460d28923e67
4c43b9a829da62a13a84356cc317f85a96a057653635b05d007b862e58b76bc5
e1e906c846ec85ee34ea188f535a4f333aff3901d7df42796df3f029247ba523
1708e5e6bf307ba847a5daadb2f823ce1ce8c767a8d306a6c06f71ac919b239f
fa67fac2de8a65ebdee194629cb5b5779c2f37a6261689ccc48d3639ca879270
12ce5c4ed559a152e11c976b805833e60e07df36604e066f4dfd7a38fbd3580f
a4441beec295a6c211a362a92c140e7b700524861a6b96263dd7b9d311086f76
429c8371651b1399b5fb89338b4e74830bb81a14486a396efbe49151b8f014a8
cf34c340983aab6182c5ee4e62f5330f0c3a89764b0be9cb6552c2534c4e0300
098427313c4fba045cfa7a42872c8bf42c339c3c1d05e487e3492a9ba7dfe5f3
a8f9094254a7545244e9432eebda8245e3630f730587d5938edd81eb5f45f6cd
ce64580355fc5fc8bb5a4a3d563b03843f6a13d9e9a1a511b807ca6efa21eff6
c60e54ef62dc4454a5c93ceaa85bd313ffe7dbb2ada9ba163c047700ef81351e
ec345434bdd3635dbce6ade21bdf5cb0de283b8360a83ab78c4c22c06f24e725
57a4c0a5e35ded540cf3ff24e268e957ff4240413ab2f76add71e3ba62d0e495
dabf6148bf7da2ae1a259936e3cf814258fb3bc935a6eb2bdc02f3d459b74ee2
cf600ae6b0296f243428d13268bc8a93f376a8ec8996068da00befbecc96fab4
772c874830786b4531b504f35dca4dc1fca11fe3b43a93e8ffa01917dba6e794
1afb36aea467af4f74db2598ab16d8b62428a9133511f6b217f177748418605c
5f55da584c51ed224bdb45b96e8f56000e4e143bb42eb85fd3b639e9862feac4
3225ddfe8d22f03697d3b8a57be4823636052d92e16a74c6e9a35b9ac3b27601
0a5c58682411565f95a2bc4ecd3ae9f1adfa83725a7a9307690d1b5de1e1f2c2
d0cfcacf4923193a85bd8879a6adacaeb3cb0d1a41bd1c96bf78279036318c05
0e9d03b460254300572b5518547b34c47dcc4664f1d55bdfe68ad6abfd9b1ea9
699dd0e5c03fecd059d445f840d1355717bc2583930b80195fae9ccdb4eed0be
fea5e0b6c5909ecfdfb6cce2b1e56d9f120e31ddbde3fc204f4a34b0e43f7a0e
2ef19d4a0f5bb951ca5912c95bf810846cd8232513be87a850a6604bfb8648d0
94b07a3a8f35d631070c3fb7ae6598567843cd1c6ca88fa3320447880b37aa90
21a969266de8b27e78f09d45afaf293a5ebe90010a267145c1c1e3e7efd386e0
c02424c424692ff3d5976719305467d606d00cfc733393fbf7feb5b8712aefd6
df0f0ab110ba09be2ff9e8b7712ff6fcc7bbe57e3d3d1c36b8ae9609d421b08a
316da0f8400011c927af66ab2b3fe93c1f8d53b92b9ced16bcd345183dbdf1a0
5fd9c5f67c69051037c547f7c4f906c8091ad37c2c6fa8dba09d672e00650cda
c1819f66aa2b506a38ec66968797a1d99bc8b88977f89e4ead9a775ebb373613
9d5d826de5fc74ff1b6e25fe14214c8de4e3133a42146ad3b12d436c4aac7ddc
8e600b5374a72dfb3ac590ca1998755cc7e1f815cda5173c9875a4da1a82f991
6ccafce0c20058219f06cf7be07b4a15c817a1769a4253f87bcd43a231fdf21e
3e9bcc8f2d006e745f908f7fbfedab79e9b7b5f9d88a490661b2a31fcad9979c
4e7ccbb5f63ded70d03ad82559ed709e4df14afe76e889e7628707a0b1e0a034
d9eacee35e88c0c392d8f28371a113b2325ba087aca5fdde4aad4c22d45317de
aadc1873f83a2641d6d47f32edf2d37115418edcb20d9c8cc412f87380fd8ff9
f418c7bdb5c50ff0b3d27736417e481839151bb1bfc42dd87cfb4a05739855f8
136835e83ca6383b6c0646d4b09726d2f50f2793197d5e8372cd072267503081
814fe9a5687d290f5924a8e842bacfc441bb693ba4974d37867262430c2613e3
0bc311ab458657a9ce9b9453e1aa5beea8058caf0a6ad28d52c91a87dbf1416e
7d4fc7036ce65ac2eb8d13f075a26330d4f6833af421d6dfcdc6d38cf7364b2b
7b6d19287f9de5ebba78c8922f07038bff556484f95339a89ecb77675af66784
250c0a2935b04a28f7ec5d4cb49044adea5c6c0f8ee7cb2d56bd4ad375cb11e4
a76e4ba59a55a031f5fdcb41650e4c5b76773d9f152da4ae566c92951624ebdc
04e548019f0f4adf8e795d30460f422239dd16db9dcaaebc51f12e845d0c7682
c3a1f8fba4c112ed091921cbd42ea5de54daf93c646cb5d227fc14540cd1cd62
1ab94a044913a3e3f9a285f1d024203118bb991ec0371866555f2f08165d2320
3b77e2e3a0e3915ac337846fdee3da4c27d9f0970fccdd6f034fd4805d4493b5
0092e114a38bd7ae53f5601a6e0a0a155dcada1b74575940a2d3ac31b1dc2b1a
69a33397f0da8f3f4e3e7aa44f6535222fe08d9d09d83e9af7f78a02fdffcb55
94e371397651c8779c5cd4d7580901bef4b544ccecec9921f84118ae7490ae12
7c28707a7597167ad0bb08dcf6678c84a3bac9dc01f164d1d70bc80fd9adaf03
8c069cdc3c10c1ecbf0dd2187ea3ad7c3b1e7eae3aef24236b2aceb3c0128804
cacca92ed65047c4998911c42a68869f5fd1d5ee37eef05a941e5719a9beb036
896b0ac3449a047ed73421a323ee3805d19302c73afaab88a7cc87443b03edd3
5e1129a153f0ff86e5df5ee0cb34e2905535467de9fa5304d987183b89a5f66a
42fdd9d7eb1e490353f20ce488371e761526e71377c22c51f42bd32b47540a3a
76f1171f7a4fc9075187cd277a75fcf95959282b49e2592f71162f826553c20f
01342929c572f9837517d40b416c7b8a89a21bc5d4f81fd67b274b4bee0300f2
10b075b8a8efaf776386a4c2e376ad34ca9f2cb48b9de39dccbdc646f9a3f223
0f53b6fd6ef9e960985a24c016adffaf484afb2d060ed7f0229796a4fa246488
2cfc5770fee7342bac78951d422a39d25addf090478eb9027d2b7ef5b8bd00f1
33c9691ac5a330a1c14f97ab298069a58276e4a5aa5d676259b74c1cb1c16dda
3ce04240731e1a7866472b7f79084bee9e7d66a552f7f254556d86e0d8469e48
e3f836b7820adbeb7397c5cb73d6d64301
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMCSC10
%!PS-AdobeFont-1.1: CMCSC10 1.0
%%CreationDate: 1991 Aug 18 17:46:49

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMCSC10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMCSC10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 72 /H put
dup 80 /P put
dup 83 /S put
dup 84 /T put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 121 /y put
readonly def
/FontBBox{14 -250 1077 750}readonly def
/UniqueXX 5000772 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /ff put
dup 12 /fi put
dup 14 /ffi put
dup 34 /quotedblright put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 86 /V put
dup 87 /W put
dup 89 /Y put
dup 92 /quotedblleft put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
dup 124 /emdash put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueXX 5000828 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 2 /Theta put
dup 5 /Pi put
dup 10 /Omega put
dup 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 14 /ffi put
dup 23 /ring put
dup 34 /quotedblright put
dup 35 /numbersign put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 47 /slash put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 59 /semicolon put
dup 61 /equal put
dup 63 /question put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 89 /Y put
dup 90 /Z put
dup 91 /bracketleft put
dup 92 /quotedblleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
dup 124 /emdash put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueXX 5000793 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMTT9
%!PS-AdobeFont-1.1: CMTT9 1.0
%%CreationDate: 1991 Aug 20 16:46:24

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 44 /comma put
dup 46 /period put
dup 47 /slash put
dup 58 /colon put
dup 64 /at put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 121 /y put
dup 122 /z put
dup 126 /asciitilde put
readonly def
/FontBBox{-6 -233 542 698}readonly def
/UniqueXX 5000831 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMSY9
%!PS-AdobeFont-1.1: CMSY9 1.0
%%CreationDate: 1991 Aug 15 07:22:27

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 2 /multiply put
dup 50 /element put
dup 79 /O put
dup 102 /braceleft put
dup 103 /braceright put
readonly def
/FontBBox{-30 -958 1146 777}readonly def
/UniqueXX 5000819 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 82 /R put
dup 105 /i put
dup 110 /n put
dup 111 /o put
dup 116 /t put
dup 117 /u put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueXX 5000789 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMR9
%!PS-AdobeFont-1.1: CMR9 1.0
%%CreationDate: 1991 Aug 20 16:39:59

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 10 /Omega put
dup 12 /fi put
dup 14 /ffi put
dup 34 /quotedblright put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 47 /slash put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 59 /semicolon put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 87 /W put
dup 89 /Y put
dup 91 /bracketleft put
dup 92 /quotedblleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
readonly def
/FontBBox{-39 -250 1036 750}readonly def
/UniqueXX 5000792 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 3 /asteriskmath put
dup 48 /prime put
dup 107 /bardbl put
dup 120 /section put
dup 121 /dagger put
dup 122 /daggerdbl put
dup 123 /paragraph put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueXX 5000816 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /ff put
dup 12 /fi put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 87 /W put
dup 90 /Z put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 124 /emdash put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueXX 5000768 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMBX7
%!PS-AdobeFont-1.1: CMBX7 1.0
%%CreationDate: 1991 Aug 20 16:35:49

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 48 /zero put
readonly def
/FontBBox{-55 -250 1289 751}readonly def
/UniqueXX 5000765 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 3 /asteriskmath put
dup 32 /arrowleft put
dup 33 /arrowright put
dup 48 /prime put
dup 50 /element put
dup 55 /mapsto put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 106 /bar put
dup 107 /bardbl put
dup 110 /backslash put
dup 120 /section put
dup 121 /dagger put
dup 122 /daggerdbl put
dup 123 /paragraph put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueXX 5000818 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 44 /comma put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 65 /A put
dup 66 /B put
dup 71 /G put
dup 73 /I put
dup 75 /K put
dup 78 /N put
dup 79 /O put
dup 82 /R put
dup 83 /S put
dup 86 /V put
dup 89 /Y put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 122 /z put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueXX 5000794 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
%%BeginFont: CMR17
%!PS-AdobeFont-1.1: CMR17 1.0
%%CreationDate: 1991 Aug 20 16:38:24

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR17) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR17 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 40 /parenleft put
dup 41 /parenright put
dup 73 /I put
dup 79 /O put
dup 80 /P put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 121 /y put
readonly def
/FontBBox{-33 -250 945 749}readonly def
/UniqueXX 5000795 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark

%%EndFont 
TeXDict begin 40258431 52099146 1000 8000 8000 (obfuscate.dvi)
@start /Fa 169[705 2[534 7[730 75[{}3 885.568 /CMTI8
rf /Fb 205[906 50[{}1 1212.12 /LASY10 rf /Fc 135[464
3[317 1[385 3[511 2[439 8[364 22[681 3[633 11[252 44[376
1[464 534 11[{}12 664.176 /CMMI6 rf /Fd 173[566 82[{}1
664.176 /CMTI7 rf /Fe 177[640 78[{}1 885.568 /MSBM7 rf
/Ff 145[808 4[572 572 1751 1212 10[1347 7[1010 41[976
976 27[404 8[505 505 556 556{}14 1212.12 /CMEX10 rf /Fg
177[875 78[{}1 1212.12 /MSBM10 rf /Fh 207[542 48[{}1
885.568 /CMBX8 rf /Fi 148[533 16[697 6[626 5[989 7[753
844 731 1[768 5[284 46[577 656 11[{}11 996.264 /CMMI9
rf /Fj 133[438 465 530 3[340 1[425 420 474 1[571 832
281 490 1[320 1[448 452 3[403 1[384 5[638 545 4[548 572
709 1[597 1[747 904 640 793 3[733 595 1[772 671 5[470
1[261 261 43[416 485 528 604 11[{}37 885.568 /CMMI8 rf
/Fk 138[549 340 418 431 1[523 523 575 1[261 2[314 2[314
2[470 470 523 97[{}13 996.264 /CMTI9 rf /Fl 135[497 8[470
523 784 6[288 418 523 418 1[470 3[261 1[261 18[705 2[640
11[470 470 470 470 470 470 470 470 470 470 4[732 1[366
366 29[680 10[{}27 885.568 /CMR8 rf /Fm 129[636 636 636
636 2[636 636 1[636 636 636 636 1[636 636 636 636 636
636 1[636 636 636 636 636 636 636 636 636 636 1[636 636
636 636 8[636 2[636 636 636 636 636 636 636 1[636 636
1[636 1[636 636 1[636 636 636 636 636 636 636 636 636
636 636 636 636 636 636 636 636 636 636 636 636 636 636
636 636 636 636 636 636 636 636 33[{}74 1212.12 /CMTT10
rf /Fn 133[664 789 789 1079 789 830 581 589 610 1[830
747 830 1245 415 789 1[415 830 747 457 682 830 664 830
726 9[1544 2[1039 830 1115 1[1021 1122 1[1418 3[557 4[1145
2[1128 11[747 747 747 747 747 1[747 415 46[{}41 1328.35
/CMBX12 rf /Fo 133[564 594 693 3[438 568 547 541 610
1[728 1064 1[631 1[418 698 578 593 1[631 525 520 641
505 5[827 704 1004 1[707 828 708 743 2[778 925 974 1176
825 1029 2[1008 953 779 895 1004 866 919 909 2[943 606
943 337 337 23[565 5[530 693 1[691 2[730 6[492 539 628
686 775 11[{}56 1212.12 /CMMI10 rf /Fp 145[606 3[337
471 471 606 606 9[808 808 11[965 6[1024 721 871 1[935
638 3[943 3[606 1[673 673 0 0 3[808 8[1212 1212 6[1212
1212 5[943 4[943 943 1[943 943 1[606 2[943 3[943 5[943
337 943{}36 1212.12 /CMSY10 rf /Fq 134[743 3[743 716
555 730 2[770 743 905 622 2[366 1[777 649 676 1[716 703
743 12[953 741 2[899 7[987 3[1005 953 1[987 65[{}23 1212.12
/CMCSC10 rf /Fr 131[1239 620 496 589 562 805 558 651
403 496 511 558 620 620 681 991 310 558 372 372 620 558
372 558 620 558 558 620 4[624 2[901 1[1211 901 1[867
681 884 1[822 929 901 1087 760 932 636 467 1[938 791
822 915 867 853 901 6[372 620 620 620 620 620 620 620
620 620 620 1[372 434 372 2[496 496 372 4[624 19[1069
1[681 743 11[{}71 1212.12 /CMTI10 rf /Fs 133[797 946
946 1295 1[996 697 707 732 1[996 897 996 1494 498 946
1[498 996 897 548 818 996 797 996 872 9[1852 2[1247 996
1338 1[1225 1347 1402 1701 3[668 1[1410 1127 1177 1374
1295 1275 1354 7[897 897 897 897 897 897 897 897 897
2[498 6[498 26[996 12[{}51 1594.02 /CMBX12 rf /Ft 131[1212
606 539 640 640 875 640 673 471 478 475 640 673 606 673
1010 337 640 370 337 673 606 370 539 673 539 673 606
3[337 606 337 741 909 1[1246 909 909 875 673 892 1[825
943 909 1111 758 943 623 438 909 951 791 825 926 875
859 909 1[572 1[943 1[337 337 606 606 606 606 606 606
606 606 606 606 606 337 404 337 943 1[471 471 337 3[1010
606 10[909 8[1010 673 673 707 875 4[909 2[943 2[{}87
1212.12 /CMR10 rf /Fu 129[523 3[523 523 1[523 523 523
523 523 523 1[523 523 523 523 523 523 1[523 523 523 1[523
523 523 523 523 32[523 5[523 10[523 523 1[523 44[{}28
996.264 /CMTT9 rf /Fv 152[512 512 22[813 28[683 47[796
2[{}5 996.264 /CMSY9 rf /Fw 138[449 320 4[406 449 4[234
22[587 24[406 406 406 406 406 406 406 406 406 406 4[621
1[320 320 40[{}19 664.176 /CMR6 rf /Fx 132[512 455 540
540 740 540 569 398 404 401 540 569 512 569 853 284 540
1[284 569 512 313 455 569 455 569 512 3[284 512 284 1[768
1[1052 1[768 740 569 754 1[697 796 768 938 640 1[526
370 768 803 668 697 782 740 725 768 5[284 284 512 512
512 512 512 512 512 512 512 512 512 284 341 284 2[398
398 284 4[512 19[853 1[569 1[740 10[{}73 996.264 /CMR9
rf /Fy 132[510 381 381 381 12[424 58[243 44[424 2[640{}8
664.176 /CMSY6 rf /Fz 131[1394 1[620 736 736 1007 736
774 542 550 574 1[774 697 774 1162 387 736 426 387 774
697 426 639 774 620 774 678 6[852 2[1441 2[970 774 1045
1[953 1047 1091 1323 838 1093 1[529 1091 1096 877 916
1069 1007 992 1054 6[387 697 697 697 697 697 697 697
697 697 697 1[387 465 3[542 542 387 26[774 813 11[{}64
1212.12 /CMBX10 rf /FA 207[502 48[{}1 774.872 /CMBX7
rf /FB 153[542 21[711 20[307 33[631 25[{}4 1106.96 /CMMI10
rf /FC 152[553 553 22[881 8[796 19[738 16[1107 33[{}6
1106.96 /CMSY10 rf /FD 134[537 514 2[594 368 453 467
1[566 566 622 905 283 2[339 1[509 339 509 1[509 509 566
97[{}18 1106.96 /CMTI10 rf /FE 133[492 584 584 799 584
615 430 437 434 1[615 553 615 922 307 584 1[307 615 553
338 492 615 492 615 553 4[553 4[1138 2[799 1[815 2[861
1[1015 3[400 14[307 8[553 553 1[307 369 307 2[430 430
307 4[553 19[922 14[{}42 1106.96 /CMR10 rf /FF 139[495
502 524 14[566 707 619 12[886 16[919 1[962 65[{}9 1106.96
/CMBX10 rf /FG 132[575 418 418 418 9[470 2[470 261 366
366 470 470 29[801 559 672 14[0 4[627 1[255 14[941 941
28[470 2[732{}21 885.568 /CMSY8 rf /FH 133[578 3[686
723 506 513 506 1[723 650 723 1084 361 686 1[361 723
650 1[578 723 578 723 650 7[975 2[975 2[723 957 2[1012
975 2[1011 1[469 1[1021 4[921 975 13[650 650 650 650
3[361 44[{}36 1328.35 /CMR12 rf /FI 252[664 3[{}1 1328.35
/CMSY10 rf /FJ 134[927 3[977 677 687 677 1[977 877 977
1477 478 2[478 977 877 527 777 1[777 977 877 16[1202
1377 5[627 31[677 677 40[{}23 1912.83 /CMR17 rf end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 8000dpi
TeXDict begin
%%BeginPaperSize: Letter
letter
%%EndPaperSize

%%EndSetup
%%Page: 1 1
1 0 bop 7590 6077 a FJ(On)578 b(the)g(\(Im\)p)50 b(ossibilit)-50
b(y)578 b(of)g(Obfuscating)i(Programs)44409 5383 y FI(\003)3767
10119 y FH(Boaz)435 b(Barak)10414 9637 y FG(y)13571 10119
y FH(Oded)e(Goldreic)-36 b(h)22505 9637 y FG(y)25661
10119 y FH(Russell)434 b(Impagliazzo)36855 9637 y FG(z)40013
10119 y FH(Stev)-36 b(en)433 b(Rudic)-36 b(h)48233 9637
y FG(x)12121 12381 y FH(Amit)434 b(Sahai)18588 11899
y FG({)23164 12381 y FH(Salil)h(V)-108 b(adhan)30391
11899 y FG(k)34966 12381 y FH(Ke)434 b(Y)-108 b(ang)39879
11899 y FG(x)20581 15098 y FH(No)-36 b(v)g(em)g(b)36
b(er)433 b(13,)i(2001)23565 20902 y FF(Abstract)4691
22922 y FE(Informally)-92 b(,)433 b(an)418 b FD(obfusc)-57
b(ator)418 b FC(O)449 b FE(is)418 b(an)h(\(e\016cien)-31
b(t,)433 b(probabilistic\))421 b(\\compiler")f(that)g(tak)-31
b(es)419 b(as)f(input)h(a)3030 24250 y(program)413 b(\(or)e(circuit\))i
FB(P)565 b FE(and)412 b(pro)31 b(duces)410 b(a)i(new)g(program)g
FC(O)31 b FE(\()p FB(P)154 b FE(\))412 b(that)g(has)g(the)f(same)h
(functionalit)-31 b(y)416 b(as)3030 25579 y FB(P)594
b FE(y)-31 b(et)440 b(is)g(\\unin)-31 b(telligible")445
b(in)440 b(some)h(sense.)704 b(Obfuscators,)458 b(if)441
b(they)f(exist,)459 b(w)-31 b(ould)441 b(ha)-31 b(v)g(e)441
b(a)g(wide)f(v)-61 b(ariet)-31 b(y)3030 26907 y(of)536
b(cryptographic)h(and)e(complexit)-31 b(y-theoretic)539
b(applications,)580 b(ranging)536 b(from)g(soft)-31 b(w)g(are)537
b(protection)g(to)3030 28235 y(homomorphic)462 b(encryption)f(to)f
(complexit)-31 b(y-theoretic)464 b(analogues)e(of)e(Rice's)g(theorem.)
765 b(Most)459 b(of)i(these)3030 29564 y(applications)470
b(are)e(based)f(on)g(an)h(in)-31 b(terpretation)470 b(of)e(the)f
(\\unin)-31 b(telligibilit)g(y")474 b(condition)c(in)d(obfuscation)3030
30892 y(as)429 b(meaning)h(that)g FC(O)31 b FE(\()p FB(P)154
b FE(\))428 b(is)h(a)g(\\virtual)h(blac)-31 b(k)430 b(b)31
b(o)-31 b(x,")444 b(in)429 b(the)g(sense)e(that)j(an)-31
b(ything)431 b(one)e(can)f(e\016cien)-31 b(tly)3030 32220
y(compute)371 b(giv)-31 b(en)370 b FC(O)31 b FE(\()p
FB(P)154 b FE(\),)370 b(one)f(could)h(also)g(e\016cien)-31
b(tly)372 b(compute)e(giv)-31 b(en)370 b(oracle)g(access)f(to)h
FB(P)154 b FE(.)4691 33549 y(In)383 b(this)h(w)-31 b(ork,)388
b(w)-31 b(e)384 b(initiate)j(a)d(theoretical)h(in)-31
b(v)g(estigation)388 b(of)c(obfuscation.)538 b(Our)383
b(main)i(result)e(is)h(that,)3030 34877 y(ev)-31 b(en)323
b(under)f(v)-31 b(ery)323 b(w)-31 b(eak)323 b(formalizations)k(of)c
(the)g(ab)31 b(o)-31 b(v)g(e)323 b(in)-31 b(tuition,)336
b(obfuscation)324 b(is)f(imp)31 b(ossible.)478 b(W)-92
b(e)322 b(pro)-31 b(v)g(e)3030 36205 y(this)310 b(b)-31
b(y)310 b(constructing)h(a)f(family)h(of)f(functions)h
FC(F)419 b FE(that)311 b(are)e FD(unobfusc)-57 b(atable)310
b FE(in)g(the)g(follo)-31 b(wing)313 b(sense:)462 b(there)3030
37534 y(is)384 b(a)f(prop)31 b(ert)-31 b(y)383 b FB(\031)371
b FE(:)332 b FC(F)440 b(!)331 b(f)p FE(0)p FB(;)184 b
FE(1)p FC(g)386 b FE(suc)-31 b(h)383 b(that)h(\(a\))h(giv)-31
b(en)384 b FD(any)410 b(pr)-57 b(o)g(gr)g(am)384 b FE(that)g(computes)g
(a)g(function)h FB(f)450 b FC(2)331 b(F)110 b FE(,)3030
38862 y(the)353 b(v)-61 b(alue)354 b FB(\031)40 b FE(\()p
FB(f)119 b FE(\))353 b(can)h(b)31 b(e)352 b(e\016cien)-31
b(tly)355 b(computed,)j(y)-31 b(et)354 b(\(b\))f(giv)-31
b(en)355 b FD(or)-57 b(acle)381 b(ac)-57 b(c)g(ess)353
b FE(to)h(a)f(\(randomly)i(selected\))3030 40190 y(function)371
b FB(f)427 b FC(2)307 b(F)110 b FE(,)369 b(no)g(e\016cien)-31
b(t)371 b(algorithm)h(can)d(compute)i FB(\031)40 b FE(\()p
FB(f)119 b FE(\))369 b(m)-31 b(uc)g(h)370 b(b)31 b(etter)370
b(than)g(random)f(guessing.)4691 41519 y(W)-92 b(e)485
b(extend)i(our)f(imp)31 b(ossibilit)-31 b(y)489 b(result)d(in)h(a)f(n)
-31 b(um)g(b)31 b(er)486 b(of)h(w)-31 b(a)g(ys,)517 b(including)488
b(ev)-31 b(en)486 b(obfuscators)h(that)3030 42847 y(\(a\))520
b(are)d(not)i(necessarily)f(computable)j(in)d(p)31 b(olynomial)521
b(time,)557 b(\(b\))519 b(only)g FD(appr)-57 b(oximately)517
b FE(preserv)-31 b(e)517 b(the)3030 44176 y(functionalit)-31
b(y)-92 b(,)391 b(and)383 b(\(c\))h(only)g(need)f(to)g(w)-31
b(ork)384 b(for)f(v)-31 b(ery)383 b(restricted)g(mo)31
b(dels)384 b(of)f(computation)k(\()p FF(TC)45604 44342
y FA(0)46161 44176 y FE(\).)534 b(W)-92 b(e)3030 45504
y(also)512 b(rule)f(out)g(sev)-31 b(eral)511 b(p)31 b(oten)-31
b(tial)514 b(applications)g(of)d(obfuscators,)547 b(b)-31
b(y)511 b(constructing)i(\\unobfuscatable")3030 46832
y(signature)370 b(sc)-31 b(hemes,)370 b(encryption)g(sc)-31
b(hemes,)369 b(and)h(pseudorandom)g(function)g(families.)0
55146 y Fz(Keyw)-39 b(ords:)1107 b FE(cryptograph)-31
b(y)-92 b(,)363 b(complexit)-31 b(y)363 b(theory)-92
b(,)362 b(soft)-31 b(w)g(are)362 b(protection,)h(homomorphic)f
(encryption,)i(Rice's)c(The-)0 56651 y(orem,)370 b(soft)-31
b(w)g(are)371 b(w)-31 b(atermarking,)372 b(pseudorandom)e(functions,)h
(statistical)h(zero)d(kno)-31 b(wledge)p 0 57686 20800
45 v 1363 58403 a Fy(\003)1843 58826 y Fx(An)341 b(extended)i(abstract)
e(of)g(this)h(w)-28 b(ork)340 b(is)i(to)f(app)28 b(ear)341
b(in)h(CR)-85 b(YPTO)340 b(2001)h([BGI)33716 58403 y
Fw(+)34393 58826 y Fx(01].)1406 59653 y Fy(y)1843 60076
y Fx(Departmen)-28 b(t)1120 b(of)f(Computer)f(Science,)1316
b(W)-85 b(eizmann)1120 b(Institute)f(of)g(Science,)1316
b(Reho)-28 b(v)g(ot,)1313 b(ISRAEL.)1118 b(E-mail:)0
61293 y Fv(f)p Fu(boaz,oded)p Fv(g)p Fu(@wisdom.weizmann.ac.il)1406
62120 y Fy(z)1843 62543 y Fx(Departmen)-28 b(t)291 b(of)f(Computer)f
(Science)j(and)d(Engineering,)302 b(Univ)-28 b(ersit)g(y)290
b(of)g(California,)301 b(San)290 b(Diego,)302 b(La)290
b(Jolla,)301 b(CA)290 b(92093-0114.)0 63761 y(E-mail:)457
b Fu(russell@cs.ucsd.edu)1406 64587 y Fy(x)1843 65011
y Fx(Computer)407 b(Science)h(Departmen)-28 b(t,)425
b(Carnegie)407 b(Mellon)h(Univ)-28 b(ersit)g(y)-85 b(,)425
b(5000)407 b(F)-85 b(orb)28 b(es)407 b(Av)-28 b(e.)653
b(Pittsburgh,)423 b(P)-85 b(A)407 b(15213.)652 b(E-mail:)0
66228 y Fv(f)p Fu(rudich,yangke)p Fv(g)p Fu(@cs.cmu.edu)1277
67055 y Fy({)1843 67478 y Fx(Departmen)-28 b(t)672 b(of)f(Computer)f
(Science,)756 b(Princeton)671 b(Univ)-28 b(ersit)g(y)-85
b(,)754 b(35)671 b(Olden)h(St.)1446 b(Princeton,)754
b(NJ)670 b(08540.)1445 b(E-mail:)0 68696 y Fu(sahai@cs.princeton.edu)
1363 69559 y Fy(k)1843 69982 y Fx(Division)457 b(of)e(Engineering)h
(and)f(Applied)i(Sciences,)485 b(Harv)-57 b(ard)455 b(Univ)-28
b(ersit)g(y)-85 b(,)484 b(33)456 b(Oxford)f(Street,)485
b(Cam)-28 b(bridge,)484 b(MA)456 b(02138.)0 71200 y(E-mail:)h
Fu(salil@eecs.harvard.edu.)f Fx(URL:)341 b Fu
(http://eecs.harvard.edu/~)-160 b(salil)p Fx(.)25697
74521 y Ft(1)p eop
%%Page: 1 2
1 1 bop 0 1218 a Fs(Con)-50 b(ten)g(ts)0 3935 y Fz(1)1121
b(In)-39 b(tro)39 b(duction)41801 b(2)1818 5441 y Ft(1.1)1239
b(Some)404 b(Applications)h(of)g(Obfuscators)354 b(.)606
b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f
(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)1731 b(2)1818
6946 y(1.2)1239 b(Our)404 b(Results)647 b(.)606 b(.)g(.)f(.)h(.)g(.)f
(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)
g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)
1731 b(4)1818 8452 y(1.3)1239 b(Discussion)542 b(.)606
b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g
(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)
h(.)g(.)g(.)f(.)h(.)g(.)1731 b(5)1818 9957 y(1.4)1239
b(Additional)405 b(Related)f(W)-101 b(ork)1198 b(.)606
b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f
(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)1731
b(6)1818 11463 y(1.5)1239 b(Organization)404 b(of)h(the)f(P)-34
b(ap)34 b(er)690 b(.)606 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g
(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)
f(.)h(.)g(.)1731 b(6)0 14180 y Fz(2)1121 b(De\014nitions)42892
b(6)1818 15686 y Ft(2.1)1239 b(Preliminaries)985 b(.)606
b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h
(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)
g(.)f(.)h(.)g(.)1731 b(6)1818 17191 y(2.2)1239 b(Obfuscators)698
b(.)606 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g
(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)
f(.)h(.)g(.)g(.)f(.)h(.)g(.)1731 b(7)0 19909 y Fz(3)1121
b(The)465 b(Main)g(Imp)39 b(ossibilit)-39 b(y)465 b(Result)30746
b(9)1818 21414 y Ft(3.1)1239 b(Obfuscating)405 b(t)-34
b(w)g(o)406 b(TMs/circuits)454 b(.)606 b(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)
h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g
(.)g(.)f(.)h(.)g(.)1125 b(10)1818 22920 y(3.2)1239 b(Obfuscating)405
b(one)g(TM/circuit)535 b(.)605 b(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)
g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g
(.)f(.)h(.)g(.)1125 b(12)0 25637 y Fz(4)c(Extensions)42223
b(16)1818 27143 y Ft(4.1)1239 b(T)-101 b(otally)404 b(unobfuscatable)i
(functions)1005 b(.)606 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g
(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)1125
b(16)1818 28648 y(4.2)1239 b(Appro)-34 b(ximate)405 b(obfuscators)324
b(.)606 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h
(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)
1125 b(17)1818 30154 y(4.3)1239 b(Imp)34 b(ossibilit)-34
b(y)404 b(of)g(the)h(applications)357 b(.)606 b(.)g(.)f(.)h(.)g(.)g(.)f
(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)
g(.)g(.)f(.)h(.)g(.)1125 b(21)1818 31659 y(4.4)1239 b(Obfuscating)405
b(restricted)f(circuit)g(classes)803 b(.)606 b(.)g(.)g(.)f(.)h(.)g(.)g
(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)
h(.)g(.)1125 b(23)1818 33165 y(4.5)1239 b(Relativization)562
b(.)606 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g
(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)
h(.)g(.)g(.)f(.)h(.)g(.)1125 b(24)0 35882 y Fz(5)c(On)465
b(a)g(Complexit)-39 b(y)464 b(Analogue)h(of)f(Rice's)h(Theorem)20567
b(26)0 38600 y(6)1121 b(Obfuscating)465 b(Sampling)h(Algorithms)28131
b(28)0 41317 y(7)1121 b(W)-116 b(eak)-39 b(er)464 b(Notions)h(of)g
(Obfuscation)29712 b(30)0 44035 y(8)1121 b(W)-116 b(atermarking)464
b(and)h(Obfuscation)29598 b(32)0 46752 y(9)1121 b(Directions)465
b(for)g(F)-116 b(urther)465 b(W)-116 b(ork)31487 b(34)0
49470 y(A)764 b(Generalizing)465 b(Rice's)g(Theorem)f(to)h(Promise)f
(Problems.)17406 b(37)0 52188 y(B)826 b(Pseudorandom)465
b(Oracles)34782 b(40)25697 74521 y Ft(1)p eop
%%Page: 2 3
2 2 bop 0 1218 a Fs(1)1793 b(In)-50 b(tro)50 b(duction)0
3923 y Ft(The)321 b(past)g(t)-34 b(w)g(o)322 b(decades)e(of)h
(cryptograph)-34 b(y)321 b(researc)-34 b(h)320 b(has)h(had)g(amazing)f
(success)h(in)f(putting)i(most)f(of)g(the)f(clas-)0 5429
y(sical)475 b(cryptographic)h(problems)g(|)f(encryption,)494
b(authen)-34 b(tication,)495 b(proto)34 b(cols)476 b(|)f(on)i
(complexit)-34 b(y-theoretic)0 6934 y(foundations.)538
b(Ho)-34 b(w)g(ev)g(er,)399 b(there)d(still)h(remain)f(sev)-34
b(eral)396 b(imp)34 b(ortan)-34 b(t)398 b(problems)f(in)g(cryptograph)
-34 b(y)397 b(ab)34 b(out)398 b(whic)-34 b(h)0 8440 y(theory)388
b(has)h(had)f(little)g(or)g(nothing)h(to)g(sa)-34 b(y)-101
b(.)533 b(One)388 b(suc)-34 b(h)389 b(problem)f(is)g(that)h(of)f
Fr(pr)-62 b(o)g(gr)g(am)418 b(obfusc)-62 b(ation.)529
b Ft(Roughly)0 9945 y(sp)34 b(eaking,)380 b(the)c(goal)f(of)h
(\(program\))g(obfuscation)g(is)f(to)h(mak)-34 b(e)375
b(a)g(program)g(\\unin)-34 b(telligible")376 b(while)f(preserving)0
11451 y(its)385 b(functionalit)-34 b(y)-101 b(.)534 b(Ideally)-101
b(,)387 b(an)e(obfuscated)i(program)e(should)h(b)34 b(e)385
b(a)g(\\virtual)g(blac)-34 b(k)385 b(b)34 b(o)-34 b(x,")388
b(in)d(the)h(sense)f(that)0 12956 y(an)-34 b(ything)441
b(one)f(can)g(compute)h(from)f(it)f(one)h(could)g(also)g(compute)g
(from)g(the)h(input-output)i(b)34 b(eha)-34 b(vior)439
b(of)i(the)0 14462 y(program.)1882 15967 y(The)334 b(hop)34
b(e)333 b(that)h(some)f(form)h(of)f(obfuscation)i(is)e(p)34
b(ossible)333 b(arises)g(from)g(the)h(fact)f(that)i(analyzing)e
(programs)0 17473 y(expressed)487 b(in)f(ric)-34 b(h)487
b(enough)h(formalisms)f(is)g(hard.)787 b(Indeed,)508
b(an)-34 b(y)487 b(programmer)g(kno)-34 b(ws)487 b(that)i(total)e(unin)
-34 b(tel-)0 18978 y(ligibilit)g(y)480 b(is)h(the)g(natural)h(state)f
(of)h(computer)f(programs)g(\(and)h(one)f(m)-34 b(ust)483
b(w)-34 b(ork)481 b(hard)g(in)g(order)g(to)g(k)-34 b(eep)481
b(a)0 20484 y(program)548 b(from)f(deteriorating)h(in)-34
b(to)548 b(this)g(state\).)970 b(Theoretically)-101 b(,)582
b(results)548 b(suc)-34 b(h)548 b(as)g(Rice's)e(Theorem)i(and)0
21989 y(the)361 b(hardness)h(of)f(the)g Fq(Hal)-81 b(ting)414
b(Pr)-27 b(oblem)361 b Ft(and)h Fq(Sa)-81 b(tisfiability)388
b Ft(all)360 b(seem)h(to)g(imply)f(that)i(the)f(only)f(useful)0
23495 y(thing)321 b(that)f(one)g(can)g(do)g(with)g(a)g(program)g(or)f
(circuit)g(is)g(to)h(run)h(it)e(\(on)i(inputs)g(of)f(ones)f(c)-34
b(hoice\).)511 b(Ho)-34 b(w)g(ev)g(er,)336 b(this)0 25000
y(informal)458 b(statemen)-34 b(t)459 b(is,)471 b(of)458
b(course,)470 b(an)458 b(o)-34 b(v)g(er-generalization,)470
b(and)459 b(the)f(existence)f(of)h(obfuscators)h(requires)0
26505 y(its)404 b(o)-34 b(wn)405 b(in)-34 b(v)g(estigation.)1882
28011 y(T)-101 b(o)485 b(b)34 b(e)485 b(a)f(bit)i(more)e(clear)g
(\(though)i(still)f(informal\),)505 b(an)485 b Fr(obfusc)-62
b(ator)481 b Fp(O)519 b Ft(is)484 b(an)h(\(e\016cien)-34
b(t,)506 b(probabilistic\))0 29516 y(\\compiler")272
b(that)i(tak)-34 b(es)274 b(as)f(input)h(a)f(program)g(\(or)g
(circuit\))g Fo(P)442 b Ft(and)274 b(pro)34 b(duces)273
b(a)g(new)g(program)g Fp(O)34 b Ft(\()p Fo(P)168 b Ft(\))274
b(satisfying)0 31022 y(the)405 b(follo)-34 b(wing)404
b(t)-34 b(w)g(o)406 b(conditions:)1818 33524 y Fp(\017)606
b Ft(\(functionalit)-34 b(y\))406 b Fp(O)34 b Ft(\()p
Fo(P)168 b Ft(\))405 b(computes)g(the)g(same)f(function)h(as)g
Fo(P)168 b Ft(.)1818 36025 y Fp(\017)606 b Ft(\(\\virtual)351
b(blac)-34 b(k)350 b(b)34 b(o)-34 b(x")350 b(prop)34
b(ert)-34 b(y\))351 b(\\An)-34 b(ything)352 b(that)f(can)g(b)34
b(e)350 b(e\016cien)-34 b(tly)350 b(computed)h(from)g
Fp(O)34 b Ft(\()p Fo(P)168 b Ft(\))351 b(can)f(b)34 b(e)3030
37531 y(e\016cien)-34 b(tly)404 b(computed)h(giv)-34
b(en)404 b(oracle)f(access)h(to)g Fo(P)168 b Ft(.")0
40033 y(While)477 b(there)g(are)g(heuristic)g(approac)-34
b(hes)479 b(to)f(obfuscation)h(in)e(practice)g(\(cf.,)496
b(Figure)477 b(1)h(and)g([CT00]\),)496 b(there)0 41538
y(has)431 b(b)34 b(een)430 b(little)g(theoretical)g(w)-34
b(ork)431 b(on)g(this)g(problem.)617 b(This)431 b(is)g(unfortunate,)439
b(since)430 b(obfuscation,)438 b(if)431 b(it)f(w)-34
b(ere)0 43043 y(p)34 b(ossible,)404 b(w)-34 b(ould)405
b(ha)-34 b(v)g(e)405 b(a)f(wide)g(v)-67 b(ariet)-34 b(y)403
b(of)i(cryptographic)f(and)h(complexit)-34 b(y-theoretic)403
b(applications.)1882 44549 y(In)385 b(this)h(w)-34 b(ork,)390
b(w)-34 b(e)386 b(initiate)f(a)h(theoretical)f(in)-34
b(v)g(estigation)386 b(of)g(obfuscation.)534 b(W)-101
b(e)385 b(examine)g(v)-67 b(arious)385 b(formal-)0 46054
y(izations)395 b(of)h(the)g(notion,)i(in)d(an)h(attempt)g(to)g
(understand)h(what)g(w)-34 b(e)396 b(can)f(and)h(cannot)h(hop)34
b(e)395 b(to)h(ac)-34 b(hiev)g(e.)535 b(Our)0 47560 y(main)357
b(result)h(is)e(a)i(negativ)-34 b(e)357 b(one,)366 b(sho)-34
b(wing)359 b(that)f(obfuscation)h(\(as)e(it)h(is)e(t)-34
b(ypically)357 b(understo)34 b(o)g(d\))358 b(is)f Fr(imp)-62
b(ossible)p Ft(.)0 49065 y(Before)322 b(describing)g(this)i(result)e
(and)i(others)f(in)f(more)h(detail,)338 b(w)-34 b(e)323
b(outline)g(some)g(of)g(the)g(p)34 b(oten)-34 b(tial)323
b(applications)0 50571 y(of)404 b(obfuscators,)i(b)34
b(oth)405 b(for)f(motiv)-67 b(ation)404 b(and)h(to)g(clarify)e(the)h
(notion.)0 53817 y Fn(1.1)1495 b(Some)499 b(Applications)i(of)d
(Obfuscators)0 56105 y Fz(Soft)-39 b(w)g(are)443 b(Protection.)1212
b Ft(The)386 b(most)f(direct)g(applications)g(of)g(obfuscators)i(are)d
(for)h(v)-67 b(arious)384 b(forms)i(of)f(soft-)0 57610
y(w)-34 b(are)416 b(protection.)575 b(By)415 b(de\014nition,)k
(obfuscating)f(a)e(program)g(protects)g(it)g(against)h(rev)-34
b(erse)414 b(engineering.)574 b(F)-101 b(or)0 59116 y(example,)331
b(if)313 b(one)g(part)-34 b(y)-101 b(,)332 b(Alice,)e(disco)-34
b(v)g(ers)313 b(a)g(more)g(e\016cien)-34 b(t)314 b(algorithm)f(for)g
(factoring)h(in)-34 b(tegers,)331 b(she)314 b(ma)-34
b(y)313 b(wish)0 60621 y(to)367 b(sell)e(another)i(part)-34
b(y)-101 b(,)374 b(Bob,)g(a)366 b(program)h(for)f(apparen)-34
b(tly)367 b(w)-34 b(eak)g(er)367 b(tasks)f(\(suc)-34
b(h)368 b(as)f(breaking)f(the)g(RSA)h(cryp-)0 62127 y(tosystem\))373
b(that)f(use)g(the)g(factoring)g(algorithm)g(as)g(a)f(subroutine)i
(without)h(actually)d(giving)g(Bob)h(a)f(factoring)0
63632 y(algorithm.)539 b(Alice)403 b(could)h(hop)34 b(e)404
b(to)h(ac)-34 b(hiev)g(e)403 b(this)i(b)-34 b(y)405 b(obfuscating)g
(the)g(program)f(she)g(giv)-34 b(es)404 b(to)h(Bob.)1882
65138 y(In)-34 b(tuitiv)g(ely)-101 b(,)421 b(obfuscators)e(w)-34
b(ould)419 b(also)f(b)34 b(e)417 b(useful)h(in)g Fr(watermarking)e
Ft(soft)-34 b(w)g(are)420 b(\(cf.,)g([CT00)q(,)d(NSS99)r(]\).)579
b(A)0 66643 y(soft)-34 b(w)g(are)390 b(v)-34 b(endor)389
b(could)f(mo)34 b(dify)389 b(a)f(program's)g(b)34 b(eha)-34
b(vior)388 b(in)h(a)f(w)-34 b(a)g(y)389 b(that)h(uniquely)e(iden)-34
b(ti\014es)389 b(the)g(p)34 b(erson)389 b(to)0 68149
y(whom)d(it)e(is)h(sold,)j(and)e(then)f(obfuscate)h(the)f(program)g(to)
g(guaran)-34 b(tee)386 b(that)g(this)f(\\w)-34 b(atermark")385
b(is)g(di\016cult)g(to)0 69654 y(remo)-34 b(v)g(e.)25697
74521 y(2)p eop
%%Page: 3 4
3 3 bop 12002 44 27996 45 v 11980 23078 45 23034 v 13939
1550 a Fm(#include<stdio.h>)643 b(#include<string.h>)13939
3055 y(main\(\){char*O,l[999]="'`acgo\\177~|xp)650 b(.)13939
4561 y(-\\0R^8\)NJ6\045K4O+A2M\(*0ID57$3G1FBL";)13939
6066 y(while\(O=fgets\(l+45,954,stdin\)\){*l=O[)13939
7572 y(strlen\(O\)[O-1]=0,strspn\(O,l+11\)];)13939 9077
y(while\(*O\)switch\(\(*l&&isalnum\(*O\)\)-!*l\))13939
10583 y({case-1:{char*I=\(O+=strspn\(O,l+12\))13939 12088
y(+1\)-2,O=34;while\(*I&3&&\(O=\(O-16<<1\)+)13939 13593
y(*I---'-'\)<80\);putchar\(O&93?*I)13939 15099 y(&8||!\()1275
b(I=memchr\()640 b(l)c(,)h(O)g(,)g(44)g(\))f(\))h(?'?':)13939
16604 y(I-l+47:32\);)k(break;)d(case)g(1:)f(;}*l=)13939
18110 y(\(*O&31\)[l-15+\(*O>61\)*32];while\(putchar)13939
19615 y(\(45+*l\0452\),\(*l=*l+32>>1\)>35\);)647 b(case)637
b(0:)13939 21121 y(putchar\(\(++O)k(,32\)\);}putchar\(10\);}})p
39975 23078 V 12002 23122 27996 45 v 0 26577 a Ft(Figure)308
b(1:)491 b(The)309 b(winning)g(en)-34 b(try)309 b(of)g(the)f(1998)h
Fr(International)345 b(Obfusc)-62 b(ate)g(d)342 b(C)k(Co)-62
b(de)344 b(Contest)p Ft(,)326 b(an)309 b(ASCI)34 b(I/Morse)0
28083 y(co)g(de)404 b(translator)g(b)-34 b(y)405 b(F)-101
b(rans)404 b(v)-67 b(an)404 b(Dorsselaer)f([vD98])g(\(adapted)j(for)f
(this)f(pap)34 b(er\).)0 31817 y Fz(Homomorphic)462 b(Encryption.)1213
b Ft(A)403 b(long-standing)h(op)34 b(en)402 b(problem)h(in)f
(cryptograph)-34 b(y)403 b(is)f(whether)i Fr(homo-)0
33323 y(morphic)416 b Ft(encryption)i(sc)-34 b(hemes)418
b(exist)g(\(cf.,)j([RAD78,)d(FM91,)g(DDN00,)g(BL96)o(,)g(SYY99]\).)580
b(That)419 b(is,)i(w)-34 b(e)419 b(seek)0 34828 y(a)360
b(secure)g(public-k)-34 b(ey)360 b(cryptosystem)g(for)h(whic)-34
b(h,)370 b(giv)-34 b(en)360 b(encryptions)g(of)h(t)-34
b(w)g(o)362 b(bits)e(\(and)i(the)f(public)f(k)-34 b(ey\),)369
b(one)0 36334 y(can)345 b(compute)h(an)g(encryption)f(of)h(an)-34
b(y)346 b(binary)f(Bo)34 b(olean)345 b(op)34 b(eration)345
b(of)h(those)f(bits.)520 b(Obfuscators)346 b(w)-34 b(ould)347
b(allo)-34 b(w)0 37839 y(one)478 b(to)h(con)-34 b(v)g(ert)478
b(an)-34 b(y)479 b(public-k)-34 b(ey)478 b(cryptosystem)g(in)-34
b(to)479 b(a)f(homomorphic)h(one:)687 b(use)478 b(the)h(secret)f(k)-34
b(ey)477 b(to)i(con-)0 39345 y(struct)352 b(an)g(algorithm)g(that)h(p)
34 b(erforms)351 b(the)h(required)f(computations)i(\(b)-34
b(y)353 b(decrypting,)362 b(applying)352 b(the)g(Bo)34
b(olean)0 40850 y(op)g(eration,)445 b(and)437 b(encrypting)g(the)h
(result\),)445 b(and)437 b(publish)h(an)f(obfuscation)i(of)e(this)g
(algorithm)g(together)g(with)0 42356 y(the)405 b(public)f(k)-34
b(ey)-101 b(.)7745 41916 y Fl(1)0 45557 y Fz(Remo)-39
b(ving)447 b(Random)g(Oracles.)1211 b Ft(The)389 b Fr(R)-62
b(andom)418 b(Or)-62 b(acle)418 b(Mo)-62 b(del)387 b
Ft([BR93)o(])h(is)g(an)h(idealized)e(cryptographic)0
47063 y(setting)k(in)g(whic)-34 b(h)391 b(all)f(parties)h(ha)-34
b(v)g(e)391 b(access)e(to)i(a)g(truly)f(random)h(function.)536
b(It)390 b(is)g(\(heuristically\))h(hop)34 b(ed)391 b(that)0
48568 y(proto)34 b(cols)425 b(designed)g(in)f(this)i(mo)34
b(del)424 b(will)g(remain)g(secure)g(when)i(implemen)-34
b(ted)425 b(using)h(an)f(e\016cien)-34 b(t,)429 b(publicly)0
50074 y(computable)275 b(cryptographic)e(hash)i(function)g(in)f(place)f
(of)h(the)g(random)h(function.)496 b(While)273 b(it)h(is)g(kno)-34
b(wn)275 b(that)f(this)0 51579 y(is)408 b(not)i(true)f(in)g(general)f
([CGH98],)h(it)g(is)g(unkno)-34 b(wn)410 b(whether)g(there)e(exist)h
(e\016cien)-34 b(tly)408 b(computable)h(functions)0 53085
y(with)322 b(strong)f(enough)h(prop)34 b(erties)320 b(to)h(b)34
b(e)321 b(securely)e(used)j(in)f(place)f(of)h(the)g(random)h(function)g
(in)f(v)-67 b(arious)320 b Fr(sp)-62 b(e)g(ci\014c)0
54590 y Ft(proto)34 b(cols)517 b(\(e.g.,)545 b(in)517
b(Fiat-Shamir)h(t)-34 b(yp)34 b(e)517 b(sc)-34 b(hemes)517
b([FS87)q(]\).)877 b(One)517 b(migh)-34 b(t)518 b(hop)34
b(e)518 b(to)f(obtain)h(suc)-34 b(h)518 b(functions)0
56095 y(b)-34 b(y)453 b(obfuscating)i(a)d(family)h(of)g(pseudorandom)h
(functions)h([GGM86],)464 b(whose)453 b(input-output)k(b)34
b(eha)-34 b(vior)452 b(is)h(b)-34 b(y)0 57601 y(de\014nition)405
b(indistinguishable)h(from)e(that)h(of)g(a)f(truly)g(random)h
(function.)0 60803 y Fz(T)-116 b(ransforming)668 b(Priv)-77
b(ate-Key)666 b(Encryption)k(in)-39 b(to)669 b(Public-Key)f
(Encryption.)1213 b Ft(Obfuscation)583 b(can)0 62308
y(also)412 b(b)34 b(e)411 b(used)i(to)f(create)f(new)i(public-k)-34
b(ey)411 b(encryption)h(sc)-34 b(hemes)412 b(b)-34 b(y)413
b(obfuscating)g(a)f(priv)-67 b(ate-k)-34 b(ey)411 b(encryption)0
63813 y(sc)-34 b(heme.)560 b(Giv)-34 b(en)412 b(a)f(secret)f(k)-34
b(ey)411 b Fo(K)499 b Ft(of)411 b(a)h(priv)-67 b(ate-k)-34
b(ey)410 b(encryption)h(sc)-34 b(heme,)413 b(one)f(can)f(publish)i(an)e
(obfuscation)p 0 64868 20800 45 v 1382 65585 a Fw(1)1843
66008 y Fx(There)357 b(is)g(a)h(subtlet)-28 b(y)358 b(here,)j(caused)d
(b)-28 b(y)357 b(the)g(fact)h(that)f(encryption)h(algorithms)f(m)-28
b(ust)357 b(b)28 b(e)358 b Fk(pr)-52 b(ob)g(abilistic)357
b Fx(to)g(b)28 b(e)358 b(seman)-28 b(tically)0 67226
y(secure)412 b(in)g(the)h(usual)e(sense)i([GM84)q(].)668
b(Ho)-28 b(w)g(ev)g(er,)428 b(b)28 b(oth)413 b(the)f(\\functionalit)-28
b(y")413 b(and)e(\\virtual)i(blac)-28 b(k)412 b(b)28
b(o)-28 b(x")412 b(prop)28 b(erties)413 b(of)e(obfus-)0
68444 y(cators)406 b(b)28 b(ecome)409 b(more)e(complex)h(for)e
(probabilistic)i(algorithms,)424 b(so)407 b(in)g(this)g(w)-28
b(ork,)423 b(w)-28 b(e)406 b(restrict)h(our)g(atten)-28
b(tion)407 b(to)g(obfuscating)0 69661 y(deterministic)343
b(algorithms\(except)h(in)d(Section)i(6\).)455 b(This)341
b(restriction)h(only)g(mak)-28 b(es)342 b(our)e(main)i(\(imp)28
b(ossibilit)-28 b(y\))344 b(result)e(stronger.)25697
74521 y Ft(3)p eop
%%Page: 4 5
4 4 bop 0 1218 a Ft(of)445 b(the)g(encryption)g(algorithm)g(Enc)17223
1406 y Fj(K)18135 1218 y Ft(.)18472 778 y Fl(2)19658
1218 y Ft(This)g(allo)-34 b(ws)445 b(ev)-34 b(ery)g(one)444
b(to)h(encrypt,)455 b(y)-34 b(et)445 b(only)f(one)h(p)34
b(ossessing)445 b(the)0 2723 y(secret)403 b(k)-34 b(ey)404
b Fo(K)491 b Ft(should)405 b(b)34 b(e)404 b(able)g(to)h(decrypt.)1882
4229 y(In)-34 b(terestingly)-101 b(,)576 b(in)543 b(the)f(original)g
(pap)34 b(er)542 b(of)h(Di\016e)f(and)h(Hellman)f([DH76],)576
b(the)542 b(ab)34 b(o)-34 b(v)g(e)543 b(w)-34 b(as)543
b(the)g(reason)0 5734 y(giv)-34 b(en)451 b(to)g(b)34
b(eliev)-34 b(e)450 b(that)i(public-k)-34 b(ey)451 b(cryptosystems)g
(migh)-34 b(t)452 b(exist)e(ev)-34 b(en)451 b(though)i(there)e(w)-34
b(ere)451 b(no)g(candidates)0 7240 y(kno)-34 b(wn)379
b(y)-34 b(et.)530 b(That)379 b(is,)k(they)378 b(suggested)h(that)g(it)f
(migh)-34 b(t)378 b(b)34 b(e)378 b(p)34 b(ossible)378
b(to)g(obfuscate)h(a)f(priv)-67 b(ate-k)-34 b(ey)377
b(encryption)0 8745 y(sc)-34 b(heme.)4081 8305 y Fl(3)0
11991 y Fn(1.2)1495 b(Our)499 b(Results)0 14279 y Fz(The)453
b(Basic)f(Imp)39 b(ossibilit)-39 b(y)453 b(Result.)1213
b Ft(Most)395 b(of)f(the)h(ab)34 b(o)-34 b(v)g(e)394
b(applications)g(rely)f(on)i(the)f(in)-34 b(tuition)395
b(that)h(an)0 15785 y(obfuscated)458 b(program)f(is)g(a)f(\\virtual)h
(blac)-34 b(k)456 b(b)34 b(o)-34 b(x.")697 b(That)458
b(is,)469 b(an)-34 b(ything)458 b(one)f(can)g(e\016cien)-34
b(tly)456 b(compute)i(from)0 17290 y(the)453 b(obfuscated)i(program,)
465 b(one)453 b(should)h(b)34 b(e)452 b(able)h(to)g(e\016cien)-34
b(tly)453 b(compute)g(giv)-34 b(en)453 b(just)h(oracle)e(access)g(to)h
(the)0 18796 y(program.)1882 20301 y(Our)361 b(main)g(result)g(sho)-34
b(ws)363 b(that)f(it)f(is)g(imp)34 b(ossible)360 b(to)i(ac)-34
b(hiev)g(e)361 b(this)g(notion)h(of)g(obfuscation.)526
b(W)-101 b(e)360 b(pro)-34 b(v)g(e)361 b(this)0 21806
y(b)-34 b(y)361 b(constructing)h(\(from)g(an)-34 b(y)361
b(one-w)-34 b(a)g(y)362 b(function\))h(a)e(family)f Fp(F)481
b Ft(of)362 b(functions)g(whic)-34 b(h)362 b(is)f Fr(unobfusc)-62
b(atable)356 b Ft(in)k(the)0 23312 y(sense)404 b(that)h(there)f(is)g
(some)g(prop)34 b(ert)-34 b(y)405 b Fo(\031)380 b Ft(:)336
b Fp(F)457 b(!)337 b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)404
b Ft(suc)-34 b(h)405 b(that:)1818 25814 y Fp(\017)606
b Ft(Giv)-34 b(en)327 b Fr(any)f Ft(program)h(\(circuit\))g(that)h
(computes)f(a)g(function)h Fo(f)467 b Fp(2)337 b(F)120
b Ft(,)343 b(the)327 b(v)-67 b(alue)326 b Fo(\031)43
b Ft(\()p Fo(f)130 b Ft(\))328 b(can)f(b)34 b(e)326 b(e\016cien)-34
b(tly)3030 27319 y(computed;)1818 29821 y Fp(\017)606
b Ft(Y)-101 b(et,)448 b(giv)-34 b(en)439 b(oracle)f(access)h(to)h(a)f
(\(randomly)g(selected\))g(function)i Fo(f)526 b Fp(2)395
b(F)120 b Ft(,)449 b(no)439 b(e\016cien)-34 b(t)439 b(algorithm)h(can)
3030 31326 y(compute)405 b Fo(\031)43 b Ft(\()p Fo(f)130
b Ft(\))406 b(m)-34 b(uc)g(h)406 b(b)34 b(etter)404 b(than)h(b)-34
b(y)404 b(random)h(guessing.)0 33828 y(Th)-34 b(us,)494
b(there)476 b(is)f(no)g(w)-34 b(a)g(y)477 b(of)e(obfuscating)i(the)f
(programs)f(that)i(compute)f(these)g(functions,)494 b(ev)-34
b(en)475 b(if)g(\(a\))h(the)0 35334 y(obfuscation)360
b(is)e(mean)-34 b(t)360 b(to)f(hide)f(only)g(one)h(bit)g(of)g
(information)g(ab)34 b(out)360 b(the)f(function)g(\(namely)g
Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\)\),)370 b(and)359
b(\(b\))0 36839 y(the)405 b(obfuscator)g(itself)f(has)h(un)-34
b(b)34 b(ounded)406 b(computation)g(time.)1882 38344
y(W)-101 b(e)345 b(b)34 b(eliev)-34 b(e)345 b(that)j(the)e(existence)g
(of)g(suc)-34 b(h)347 b(functions)h(sho)-34 b(ws)348
b(that)f(the)g(\\virtual)f(blac)-34 b(k)346 b(b)34 b(o)-34
b(x")346 b(paradigm)h(for)0 39850 y(obfuscators)437 b(is)e(inheren)-34
b(tly)436 b(\015a)-34 b(w)g(ed.)635 b(An)-34 b(y)436
b(hop)34 b(e)436 b(for)f(p)34 b(ositiv)-34 b(e)436 b(results)f(ab)34
b(out)437 b(obfuscator-lik)-34 b(e)436 b(ob)67 b(jects)437
b(m)-34 b(ust)0 41355 y(abandon)406 b(this)e(viewp)34
b(oin)-34 b(t,)405 b(or)f(at)g(least)g(b)34 b(e)404 b(reconciled)f
(with)i(the)f(existence)g(of)g(functions)i(as)e(ab)34
b(o)-34 b(v)g(e.)0 44557 y Fz(Appro)-39 b(ximate)601
b(Obfuscators.)1212 b Ft(The)523 b(basic)g(imp)34 b(ossibilit)-34
b(y)523 b(result)f(as)h(describ)34 b(ed)523 b(ab)34 b(o)-34
b(v)g(e)523 b(applies)f(to)h(ob-)0 46062 y(fuscators)485
b Fp(O)518 b Ft(for)485 b(whic)-34 b(h)485 b(w)-34 b(e)485
b(require)e(that)i(the)g(obfuscated)h(program)e Fp(O)34
b Ft(\()p Fo(P)168 b Ft(\))486 b(computes)f(exactly)e(the)i(same)0
47568 y(function)444 b(as)e(the)g(original)g(program)g
Fo(P)168 b Ft(.)653 b(Ho)-34 b(w)g(ev)g(er,)452 b(for)443
b(some)f(applications)g(it)h(ma)-34 b(y)442 b(su\016ce)g(that,)453
b(for)442 b(ev)-34 b(ery)0 49073 y(input)494 b Fo(x)p
Ft(,)515 b Fp(O)34 b Ft(\()p Fo(P)168 b Ft(\))493 b(and)h
Fo(P)662 b Ft(agree)492 b(on)h Fo(x)g Ft(with)g(high)h(probabilit)-34
b(y)493 b(\(o)-34 b(v)g(er)493 b(the)h(coin)e(tosses)i(of)f
Fp(O)34 b Ft(\).)805 b(Using)493 b(some)0 50579 y(additional)405
b(ideas,)f(our)g(imp)34 b(ossibilit)-34 b(y)404 b(result)g(extends)g
(to)h(suc)-34 b(h)405 b Fr(appr)-62 b(oximate)431 b(obfusc)-62
b(ators)p Ft(.)p 0 51634 20800 45 v 1382 52351 a Fw(2)1843
52774 y Fx(This)585 b(application)i(in)-28 b(v)g(olv)g(es)586
b(the)g(same)g(subtlet)-28 b(y)586 b(p)28 b(oin)-28 b(ted)586
b(out)g(in)g(F)-85 b(o)28 b(otnote)586 b(1.)1188 b(Th)-28
b(us,)646 b(our)585 b(results)g(regarding)g(the)0 53992
y(\(un\)obfuscatabilit)-28 b(y)341 b(of)f(priv)-57 b(ate-k)-28
b(ey)343 b(encryption)d(sc)-28 b(hemes)341 b(\(describ)28
b(ed)341 b(later\))g(refer)f(to)h(a)f(relaxed)h(notion)g(of)f(securit)
-28 b(y)340 b(in)h(whic)-28 b(h)0 55209 y(m)g(ultiple)354
b(encryptions)e(of)h(the)f(same)h(message)g(are)g(not)f(allo)-28
b(w)g(ed)352 b(\(whic)-28 b(h)352 b(is)g(consisten)-28
b(t)353 b(with)f(a)g(deterministic)j(encryption)d(algo-)0
56427 y(rithm\).)1382 57221 y Fw(3)1843 57644 y Fx(F)-85
b(rom)304 b([DH76)q(]:)437 b(\\A)305 b(more)f(practical)i(approac)-28
b(h)303 b(to)i(\014nding)f(a)h(pair)f(of)g(easily)j(computed)e(in)-28
b(v)g(erse)304 b(algorithms)i Fi(E)361 b Fx(and)305 b
Fi(D)28 b Fx(;)318 b(suc)-28 b(h)0 58862 y(that)313 b
Fi(D)341 b Fx(is)313 b(hard)e(to)i(infer)f(from)g Fi(E)57
b Fx(,)319 b(mak)-28 b(es)313 b(use)g(of)f(the)h(di\016cult)-28
b(y)314 b(of)e(analyzing)h(programs)f(in)g(lo)-28 b(w)312
b(lev)-28 b(el)315 b(languages.)446 b(An)-28 b(y)g(one)312
b(who)0 60080 y(has)380 b(tried)g(to)g(determine)i(what)e(op)28
b(eration)380 b(is)h(accomplished)g(b)-28 b(y)380 b(someone)h(else's)g
(mac)-28 b(hine)381 b(language)f(program)f(kno)-28 b(ws)380
b(that)g Fi(E)0 61297 y Fx(itself)338 b(\(i.e.)456 b(what)336
b Fi(E)394 b Fx(do)28 b(es\))338 b(can)f(b)28 b(e)337
b(hard)g(to)g(infer)g(from)f(an)h(algorithm)g(for)g Fi(E)57
b Fx(.)454 b(If)336 b(the)i(program)e(w)-28 b(ere)336
b(to)h(b)28 b(e)338 b(made)f(purp)28 b(osefully)0 62515
y(confusing)281 b(through)f(the)i(addition)f(of)g(unneeded)h(v)-57
b(ariables)282 b(and)f(statemen)-28 b(ts,)294 b(then)282
b(determining)g(an)f(in)-28 b(v)g(erse)281 b(algorithm)h(could)f(b)28
b(e)0 63733 y(made)342 b(v)-28 b(ery)342 b(di\016cult.)456
b(Of)342 b(course,)f Fi(E)399 b Fx(m)-28 b(ust)341 b(b)28
b(e)342 b(complicated)h(enough)e(to)g(prev)-28 b(en)g(t)342
b(its)f(iden)-28 b(ti\014cation)343 b(from)e(input-output)g(pairs.)1024
64950 y(Essen)-28 b(tially)421 b(what)e(is)h(required)h(is)f(a)f(one-w)
-28 b(a)g(y)420 b(compiler:)613 b(one)420 b(whic)-28
b(h)419 b(tak)-28 b(es)420 b(an)g(easily)h(understo)28
b(o)g(d)420 b(program)f(written)h(in)g(a)0 66168 y(high)365
b(lev)-28 b(el)366 b(language)f(and)f(translates)h(it)g(in)-28
b(to)364 b(an)h(incomprehensible)h(program)d(in)i(some)g(mac)-28
b(hine)365 b(language.)526 b(The)364 b(compiler)i(is)0
67386 y(one-w)-28 b(a)g(y)375 b(b)28 b(ecause)377 b(it)f(m)-28
b(ust)375 b(b)28 b(e)376 b(feasible)h(to)f(do)f(the)h(compilation,)386
b(but)376 b(infeasible)h(to)f(rev)-28 b(erse)376 b(the)g(pro)28
b(cess.)558 b(Since)376 b(e\016ciency)i(in)0 68603 y(size)h(of)f
(program)f(and)g(run)g(time)j(are)e(not)g(crucial)g(in)g(this)g
(application,)389 b(suc)-28 b(h)377 b(compilers)i(ma)-28
b(y)378 b(b)28 b(e)379 b(p)28 b(ossible)379 b(if)f(the)h(structure)e
(of)0 69821 y(the)342 b(mac)-28 b(hine)342 b(language)f(can)h(b)28
b(e)342 b(optimized)g(to)g(assist)g(in)f(the)h(confusion.")25697
74521 y Ft(4)p eop
%%Page: 5 6
5 5 bop 0 1218 a Fz(Imp)39 b(ossibilit)-39 b(y)605 b(of)f
(Applications.)1213 b Ft(T)-101 b(o)526 b(giv)-34 b(e)525
b(further)h(evidence)f(that)i(our)f(imp)34 b(ossibilit)-34
b(y)525 b(result)h(is)f(not)0 2723 y(an)599 b(artifact)g(of)g
(de\014nitional)g(c)-34 b(hoices,)647 b(but)600 b(rather)e(that)i
(there)e(is)h(something)g(inheren)-34 b(tly)599 b(\015a)-34
b(w)g(ed)600 b(in)e(the)0 4229 y(\\virtual)481 b(blac)-34
b(k)481 b(b)34 b(o)-34 b(x")482 b(idea,)500 b(w)-34 b(e)482
b(also)f(demonstrate)h(that)h(sev)-34 b(eral)480 b(of)i(the)g
(applications)g(of)g(obfuscators)g(are)0 5734 y(also)366
b(imp)34 b(ossible.)526 b(W)-101 b(e)366 b(do)h(this)g(b)-34
b(y)367 b(constructing)h Fr(unobfusc)-62 b(atable)361
b Ft(signature)367 b(sc)-34 b(hemes,)374 b(encryption)367
b(sc)-34 b(hemes,)0 7240 y(and)579 b(pseudorandom)h(functions.)1062
b(These)578 b(are)g(ob)67 b(jects)579 b(satisfying)g(the)g(standard)g
(de\014nitions)h(of)e(securit)-34 b(y)0 8745 y(\(except)400
b(for)h(the)f(subtlet)-34 b(y)402 b(noted)f(in)f(F)-101
b(o)34 b(otnote)401 b(2\),)g(but)h(for)e(whic)-34 b(h)401
b(one)g(can)f(e\016cien)-34 b(tly)400 b(compute)h(the)g(secret)0
10250 y(k)-34 b(ey)478 b Fo(K)566 b Ft(from)479 b Fr(any)502
b(pr)-62 b(o)g(gr)g(am)477 b Ft(that)j(signs)f(\(or)g(encrypts)g(or)f
(ev)-67 b(aluates)479 b(the)g(pseudorandom)h(function,)499
b(resp.\))0 11756 y(relativ)-34 b(e)318 b(to)g Fo(K)87
b Ft(.)510 b(\(Hence)319 b(handing)h(out)f(\\obfuscated)h(forms")f(of)g
(these)g(k)-34 b(ey)g(ed-algorithms)319 b(is)f(highly)g(insecure.\))
1882 13261 y(In)269 b(particular,)296 b(w)-34 b(e)270
b(complemen)-34 b(t)270 b(Canetti)g(et.)g(al.'s)e(critique)h(of)h(the)f
(Random)i(Oracle)d(Metho)34 b(dology)270 b([CGH98].)0
14767 y(They)456 b(sho)-34 b(w)457 b(that)g(there)f(exist)f(\(con)-34
b(triv)g(ed\))457 b(proto)34 b(cols)456 b(that)h(are)e(secure)g(in)h
(the)g(idealized)f(Random)i(Oracle)0 16272 y(Mo)34 b(del)386
b(\(of)h([BR93]\),)j(but)e(are)e Fr(inse)-62 b(cur)g(e)475
b Ft(when)388 b(the)f(random)g(oracle)f(is)g(replaced)g(with)h
Fr(any)494 b Ft(\(e\016cien)-34 b(tly)386 b(com-)0 17778
y(putable\))535 b(function.)928 b(Our)533 b(results)h(imply)f(that)h
(for)g(ev)-34 b(en)533 b(for)h Fr(natur)-62 b(al)657
b Ft(proto)34 b(cols)534 b(that)g(are)f(secure)g(in)g(the)0
19283 y(random)466 b(oracle)f(mo)34 b(del)465 b(\(e.g.,)480
b(Fiat-Shamir)467 b(t)-34 b(yp)34 b(e)465 b(sc)-34 b(hemes)466
b([FS87]\),)481 b(there)466 b(exist)f(\(con)-34 b(triv)g(ed\))467
b(pseudoran-)0 20789 y(dom)372 b(functions,)378 b(suc)-34
b(h)372 b(that)h(these)e(proto)34 b(cols)371 b(are)f(insecure)h(when)h
(the)f(random)h(oracle)e(is)h(replaced)g(with)g Fr(any)0
22294 y(pr)-62 b(o)g(gr)g(am)496 b Ft(that)405 b(computes)g(the)g(con)
-34 b(triv)g(ed)404 b(pseudorandom)i(function.)0 25496
y Fz(Obfuscating)310 b(restricted)g(complexit)-39 b(y)309
b(classes.)1211 b Ft(Ev)-34 b(en)270 b(though)h(obfuscation)g(of)f
(general)f(programs/circuits)0 27001 y(is)346 b(imp)34
b(ossible,)358 b(one)346 b(ma)-34 b(y)347 b(hop)34 b(e)347
b(that)g(it)g(is)f(p)34 b(ossible)347 b(to)g(obfuscate)g(more)f
(restricted)h(classes)f(of)h(computations.)0 28507 y(Ho)-34
b(w)g(ev)g(er,)452 b(using)443 b(the)g(pseudorandom)h(functions)g(of)f
([NR97])f(in)h(our)f(construction,)453 b(w)-34 b(e)443
b(can)f(sho)-34 b(w)444 b(that)f(the)0 30012 y(imp)34
b(ossibilit)-34 b(y)323 b(result)g(holds)h(ev)-34 b(en)323
b(when)h(the)g(input)h(program)e Fo(P)492 b Ft(is)323
b(a)g(constan)-34 b(t-depth)326 b(threshold)f(circuit)d(\(i.e.,)0
31518 y(is)404 b(in)g Fz(TC)4610 31700 y Fh(0)5207 31518
y Ft(\),)g(under)h(widely)f(b)34 b(eliev)-34 b(ed)403
b(complexit)-34 b(y)403 b(assumptions)j(\(e.g.,)e(the)g(hardness)h(of)g
(factoring\).)0 34719 y Fz(Obfuscating)523 b(Sampling)g(Algorithms.)
1212 b Ft(Another)455 b(w)-34 b(a)g(y)455 b(in)f(whic)-34
b(h)455 b(the)g(notion)g(of)f(obfuscators)i(can)e(b)34
b(e)0 36225 y(w)-34 b(eak)g(ened)432 b(is)f(b)-34 b(y)432
b(c)-34 b(hanging)432 b(the)g(functionalit)-34 b(y)432
b(requiremen)-34 b(t.)620 b(Up)432 b(to)f(no)-34 b(w,)439
b(w)-34 b(e)432 b(ha)-34 b(v)g(e)432 b(considered)f(programs)0
37730 y(in)365 b(terms)g(of)g(the)g(functions)i(they)e(compute,)373
b(but)366 b(sometimes)f(one)g(is)g(in)-34 b(terested)365
b(in)g(other)g(kinds)g(of)g(b)34 b(eha)-34 b(vior.)0
39236 y(F)-101 b(or)472 b(example,)488 b(one)472 b(sometimes)g
(considers)f Fr(sampling)495 b(algorithms)p Ft(,)486
b(i.e.)741 b(probabilistic)472 b(programs)g(that)h(tak)-34
b(e)0 40741 y(no)446 b(input)h(\(other)g(than,)457 b(sa)-34
b(y)-101 b(,)456 b(a)446 b(length)g(parameter\))g(and)h(pro)34
b(duce)446 b(an)g(output)i(according)d(to)h(some)g(desired)0
42247 y(distribution.)522 b(W)-101 b(e)350 b(consider)g(t)-34
b(w)g(o)352 b(natural)g(de\014nitions)f(of)g(obfuscators)h(for)f
(sampling)g(algorithms,)361 b(and)352 b(pro)-34 b(v)g(e)0
43752 y(that)299 b(the)f(stronger)f(de\014nition)i(is)e(imp)34
b(ossible)297 b(to)h(meet.)503 b(W)-101 b(e)296 b(also)i(observ)-34
b(e)297 b(that)i(the)f(w)-34 b(eak)g(er)297 b(de\014nition)i(implies)0
45257 y(the)405 b(non)-34 b(trivialit)g(y)404 b(of)g(statistical)h
(zero)e(kno)-34 b(wledge.)0 48459 y Fz(Soft)-39 b(w)g(are)474
b(W)-116 b(atermarking.)1212 b Ft(As)412 b(men)-34 b(tioned)414
b(earlier,)f(there)f(app)34 b(ears)413 b(to)f(b)34 b(e)413
b(some)f(connection)h(b)34 b(et)-34 b(w)g(een)0 49965
y(the)276 b(problems)f(of)h(soft)-34 b(w)g(are)276 b(w)-34
b(atermarking)276 b(and)g(co)34 b(de)275 b(obfuscation.)497
b(W)-101 b(e)275 b(consider)g(a)g(couple)g(of)h(formalizations)0
51470 y(of)404 b(the)h(w)-34 b(atermarking)405 b(problem)f(and)h
(explore)e(their)h(relationship)g(to)h(our)f(results)g(on)h
(obfuscation.)0 54716 y Fn(1.3)1495 b(Discussion)0 57004
y Ft(Our)491 b(w)-34 b(ork)491 b(rules)g(out)g(the)h(standard,)514
b(\\virtual)490 b(blac)-34 b(k)491 b(b)34 b(o)-34 b(x")491
b(notion)h(of)f(obfuscators)i(as)e(imp)34 b(ossible,)512
b(along)0 58510 y(with)521 b(sev)-34 b(eral)520 b(of)h(its)f
(applications.)889 b(Ho)-34 b(w)g(ev)g(er,)550 b(it)520
b(do)34 b(es)521 b(not)g(mean)g(that)g(there)g(is)f(no)h(metho)34
b(d)521 b(of)g(making)0 60015 y(programs)458 b(\\unin)-34
b(telligible")457 b(in)h(some)f(meaningful)i(and)f(precise)f(sense.)698
b(Suc)-34 b(h)459 b(a)f(metho)34 b(d)458 b(could)g(still)f(pro)-34
b(v)g(e)0 61521 y(useful)405 b(for)f(soft)-34 b(w)g(are)406
b(protection.)1882 63026 y(Th)-34 b(us,)559 b(w)-34 b(e)527
b(consider)g(it)g(to)g(b)34 b(e)527 b(b)34 b(oth)528
b(imp)34 b(ortan)-34 b(t)528 b(and)g(in)-34 b(teresting)528
b(to)f(understand)i(whether)f(there)e(are)0 64532 y(alternativ)-34
b(e)532 b(senses)h(\(or)g(mo)34 b(dels\))532 b(in)h(whic)-34
b(h)533 b(some)f(form)h(of)g(obfuscation)h(is)e(p)34
b(ossible.)924 b(T)-101 b(o)-34 b(w)g(ard)534 b(this)f(end,)0
66037 y(w)-34 b(e)380 b(suggest)h(t)-34 b(w)g(o)382 b(w)-34
b(eak)g(er)380 b(de\014nitions)h(of)g(obfuscators)g(that)g(a)-34
b(v)g(oid)381 b(the)f(\\virtual)g(blac)-34 b(k)380 b(b)34
b(o)-34 b(x")380 b(paradigm)h(\(and)0 67542 y(hence)332
b(are)f(not)i(ruled)e(out)i(b)-34 b(y)332 b(our)g(imp)34
b(ossibilit)-34 b(y)332 b(pro)34 b(of)94 b(\).)515 b(These)332
b(de\014nitions)i(could)e(b)34 b(e)331 b(the)i(sub)67
b(ject)333 b(of)f(future)0 69048 y(in)-34 b(v)g(estigations,)405
b(but)g(w)-34 b(e)405 b(hop)34 b(e)404 b(that)h(other)g(alternativ)-34
b(es)404 b(will)f(also)h(b)34 b(e)404 b(prop)34 b(osed)405
b(and)g(examined.)25697 74521 y(5)p eop
%%Page: 6 7
6 6 bop 1882 1218 a Ft(As)482 b(is)f(usually)h(the)g(case)f(with)h(imp)
34 b(ossibilit)-34 b(y)482 b(results)f(and)i(lo)-34 b(w)g(er)482
b(b)34 b(ounds,)502 b(w)-34 b(e)482 b(sho)-34 b(w)483
b(that)g(obfuscators)0 2723 y(\(in)356 b(the)h(\\virtual)f(blac)-34
b(k)355 b(b)34 b(o)-34 b(x")357 b(sense\))f(do)g(not)h(exist)f(b)-34
b(y)356 b(presen)-34 b(ting)357 b(a)f(somewhat)h(con)-34
b(triv)g(ed)357 b(coun)-34 b(terexample)0 4229 y(of)403
b(a)h(function)g(ensem)-34 b(ble)403 b(that)i(cannot)f(b)34
b(e)402 b(obfuscated.)540 b(It)403 b(is)g(in)-34 b(teresting)404
b(whether)g(obfuscation)g(is)f(p)34 b(ossible)0 5734
y(for)543 b(a)g(restricted)f(class)h(of)g(algorithms,)577
b(whic)-34 b(h)544 b(nonetheless)f(con)-34 b(tains)544
b(some)f(\\useful")h(algorithms.)954 b(This)0 7240 y(restriction)411
b(should)h(not)g(b)34 b(e)411 b(con\014ned)h(to)g(the)g(computational)g
(complexit)-34 b(y)411 b(of)h(the)g(algorithms:)553 b(if)411
b(w)-34 b(e)412 b(try)f(to)0 8745 y(restrict)367 b(the)g(algorithms)h
(b)-34 b(y)368 b(their)f(computational)h(complexit)-34
b(y)-101 b(,)374 b(then)368 b(there's)f(not)h(m)-34 b(uc)g(h)369
b(hop)34 b(e)367 b(for)h(obfusca-)0 10250 y(tion.)680
b(Indeed,)463 b(as)452 b(men)-34 b(tioned)452 b(ab)34
b(o)-34 b(v)g(e,)463 b(w)-34 b(e)452 b(sho)-34 b(w)452
b(that)h(\(under)f(widely)f(b)34 b(eliev)-34 b(ed)450
b(complexit)-34 b(y)451 b(assumptions\))0 11756 y(our)486
b(coun)-34 b(terexample)485 b(can)h(b)34 b(e)485 b(placed)g(in)h
Fz(TC)22371 11938 y Fh(0)22968 11756 y Ft(.)782 b(In)486
b(general,)504 b(the)486 b(complexit)-34 b(y)485 b(of)h(our)g(coun)-34
b(terexample)485 b(is)0 13261 y(essen)-34 b(tially)359
b(the)g(same)g(as)g(the)g(complexit)-34 b(y)359 b(of)g(pseudorandom)i
(functions,)369 b(and)360 b(so)f(a)g(complexit)-34 b(y)358
b(class)h(whic)-34 b(h)0 14767 y(do)34 b(es)404 b(not)h(con)-34
b(tain)405 b(our)f(example)g(will)f(also)h(not)h(con)-34
b(tain)405 b(man)-34 b(y)405 b(cryptographically)e(useful)i
(algorithms.)0 18013 y Fn(1.4)1495 b(Additional)500 b(Related)f(W)-125
b(ork)0 20301 y Ft(There)456 b(are)g(a)h(n)-34 b(um)g(b)34
b(er)458 b(of)e(heuristic)h(approac)-34 b(hes)457 b(to)g(obfuscation)h
(and)g(soft)-34 b(w)g(are)458 b(w)-34 b(atermarking)457
b(in)f(the)h(lit-)0 21806 y(erature,)517 b(as)494 b(describ)34
b(ed)495 b(in)f(the)h(surv)-34 b(ey)495 b(of)g(Collb)34
b(erg)494 b(and)h(Thom)-34 b(b)34 b(orson)497 b([CT00].)810
b(A)495 b(theoretical)f(study)i(of)0 23312 y(soft)-34
b(w)g(are)372 b(protection)e(w)-34 b(as)371 b(previously)f(conducted)h
(b)-34 b(y)370 b(Goldreic)-34 b(h)370 b(and)h(Ostro)-34
b(vsky)370 b([GO96],)376 b(who)371 b(considered)0 24817
y Fr(har)-62 b(dwar)g(e-b)g(ase)g(d)400 b Ft(solutions.)1882
26323 y(Hada)342 b([Had00])f(ga)-34 b(v)g(e)342 b(some)f(de\014nitions)
i(for)f(co)34 b(de)340 b(obfuscators)j(whic)-34 b(h)343
b(are)e(stronger)g(than)i(the)f(de\014nitions)0 27828
y(w)-34 b(e)364 b(consider)g(in)g(this)h(pap)34 b(er,)372
b(and)365 b(sho)-34 b(w)g(ed)365 b(some)g(implications)f(of)g(the)h
(existence)e(of)h(suc)-34 b(h)365 b(obfuscators.)527
b(\(Our)0 29334 y(result)404 b(rules)g(out)h(also)f(the)g(existence)f
(of)i(obfuscators)g(according)f(to)h(the)f(de\014nitions)i(of)e([Had00)
q(].\))1882 30839 y(Canetti,)413 b(Goldreic)-34 b(h)410
b(and)h(Halevi)e([CGH98)q(])h(sho)-34 b(w)g(ed)412 b(another)f(setting)
g(in)g(cryptograph)-34 b(y)411 b(where)f(getting)0 32345
y(a)524 b(function's)h(description)f(is)g(pro)-34 b(v)-67
b(ably)523 b(more)h(p)34 b(o)-34 b(w)g(erful)525 b(than)g(blac)-34
b(k-b)34 b(o)-34 b(x)524 b(access.)897 b(As)524 b(men)-34
b(tioned)525 b(ab)34 b(o)-34 b(v)g(e,)0 33850 y(they)481
b(ha)-34 b(v)g(e)481 b(sho)-34 b(wn)482 b(that)g(there)e(exist)g(proto)
34 b(cols)481 b(that)h(are)e(secure)g(when)h(executed)f(with)i(blac)-34
b(k-b)34 b(o)-34 b(x)481 b(access)0 35356 y(to)388 b(a)f(random)h
(function,)k(but)c(insecure)f(when)h(instead)g(the)g(parties)f(are)g
(giv)-34 b(en)387 b(a)g(description)h(of)f(an)-34 b(y)388
b(explicit)0 36861 y(function.)0 40107 y Fn(1.5)1495
b(Organization)501 b(of)d(the)g(P)-42 b(ap)42 b(er)0
42395 y Ft(In)453 b(Section)g(2,)465 b(w)-34 b(e)453
b(giv)-34 b(e)453 b(some)f(basic)h(de\014nitions)h(along)g(with)f(\(v)
-34 b(ery)453 b(w)-34 b(eak\))453 b(de\014nitions)h(of)g(obfuscators.)
686 b(In)0 43901 y(Section)520 b(3,)547 b(w)-34 b(e)520
b(pro)-34 b(v)g(e)520 b(the)f(imp)34 b(ossibilit)-34
b(y)519 b(of)h(obfuscators)h(b)-34 b(y)519 b(constructing)h(an)g
(unobfuscatable)i(function)0 45406 y(ensem)-34 b(ble.)1107
b(In)594 b(Section)g(4,)641 b(w)-34 b(e)594 b(giv)-34
b(e)593 b(a)h(n)-34 b(um)g(b)34 b(er)594 b(of)g(extensions)g(of)g(our)g
(imp)34 b(ossibilit)-34 b(y)594 b(result,)641 b(including)0
46912 y(imp)34 b(ossibilit)-34 b(y)480 b(results)g(for)f(obfuscators)j
(whic)-34 b(h)480 b(only)g(need)g(to)g(appro)-34 b(ximately)480
b(preserv)-34 b(e)479 b(functionalit)-34 b(y)-101 b(,)500
b(for)0 48417 y(obfuscators)584 b(computable)f(in)g(lo)-34
b(w)583 b(circuit)f(classes,)626 b(and)583 b(for)g(some)f(of)h(the)g
(applications)g(of)g(obfuscators.)0 49923 y(W)-101 b(e)580
b(also)g(sho)-34 b(w)582 b(that)f(our)g(main)f(imp)34
b(ossibilit)-34 b(y)580 b(result)h(do)34 b(es)580 b(not)h(relativize.)
1065 b(In)581 b(Section)g(5,)624 b(w)-34 b(e)580 b(discuss)0
51428 y(some)i(conjectural)h(complexit)-34 b(y-theoretic)581
b(analogues)i(of)f(Rice's)g(Theorem,)626 b(and)583 b(use)f(our)h(tec)
-34 b(hniques)583 b(to)0 52934 y(sho)-34 b(w)575 b(that)f(one)g(of)g
(these)g(is)f(false.)1046 b(In)574 b(Section)g(6,)615
b(w)-34 b(e)574 b(examine)f(notions)h(of)g(obfuscators)h(for)f
Fr(sampling)0 54439 y Ft(algorithms.)633 b(In)435 b(Section)h(7,)443
b(w)-34 b(e)436 b(prop)34 b(ose)436 b(w)-34 b(eak)g(er)435
b(notions)i(of)e(obfuscation)i(that)g(are)e(not)h(ruled)g(out)g(b)-34
b(y)436 b(our)0 55944 y(imp)34 b(ossibilit)-34 b(y)319
b(results.)511 b(In)319 b(Section)h(8,)336 b(w)-34 b(e)320
b(discuss)g(the)g(problem)g(of)f(soft)-34 b(w)g(are)321
b(w)-34 b(atermarking)320 b(and)h(its)e(relation)0 57450
y(to)404 b(obfuscation.)541 b(Finally)-101 b(,)403 b(in)h(Section)g(9,)
g(w)-34 b(e)405 b(men)-34 b(tion)405 b(some)f(directions)g(for)g
(further)h(w)-34 b(ork)404 b(in)g(this)h(area.)0 61269
y Fs(2)1793 b(De\014nitions)0 64019 y Fn(2.1)1495 b(Preliminaries)0
66307 y Fz(Standard)442 b(Notations.)1240 b Fr(TM)384
b Ft(is)f(shorthand)j(for)e(T)-101 b(uring)385 b(mac)-34
b(hine.)532 b Fr(PPT)384 b Ft(is)g(shorthand)i(for)e(probabilistic)0
67813 y(p)34 b(olynomial-time)398 b(T)-101 b(uring)399
b(mac)-34 b(hine.)537 b(By)398 b Fr(cir)-62 b(cuit)511
b Ft(w)-34 b(e)399 b(refer)f(to)h(a)g(standard)h(b)34
b(o)g(olean)398 b(circuit)g(with)h(AND,OR)0 69318 y(and)438
b(NOT)f(gates.)637 b(If)437 b Fo(C)524 b Ft(is)436 b(a)h(circuit)f
(with)i Fo(n)e Ft(inputs)i(and)g Fo(m)f Ft(outputs,)447
b(and)438 b Fo(x)391 b Fp(2)g(f)p Ft(0)p Fo(;)202 b Ft(1)p
Fp(g)42021 68879 y Fj(n)43084 69318 y Ft(then)438 b(b)-34
b(y)437 b Fo(C)87 b Ft(\()p Fo(x)p Ft(\))438 b(w)-34
b(e)0 70824 y(denote)280 b(the)f(result)g(of)h(applying)f
Fo(C)366 b Ft(on)280 b(input)g Fo(x)p Ft(.)496 b(W)-101
b(e)279 b(sa)-34 b(y)279 b(that)h Fo(C)366 b Ft(computes)280
b(a)f(function)i Fo(f)467 b Ft(:)337 b Fp(f)p Ft(0)p
Fo(;)202 b Ft(1)p Fp(g)45638 70384 y Fj(n)46601 70824
y Fp(!)336 b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)51112
70384 y Fj(m)25697 74521 y Ft(6)p eop
%%Page: 7 8
7 7 bop 0 1218 a Ft(if)405 b(for)g(an)-34 b(y)406 b Fo(x)337
b Fp(2)i(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)10399 778
y Fj(n)11025 1218 y Ft(,)405 b Fo(C)87 b Ft(\()p Fo(x)p
Ft(\))339 b(=)e Fo(f)130 b Ft(\()p Fo(x)p Ft(\).)543
b(F)-101 b(or)405 b(algorithms)g Fo(A)g Ft(and)h Fo(M)537
b Ft(and)406 b(a)f(string)g Fo(x)p Ft(,)g(w)-34 b(e)405
b(denote)h(b)-34 b(y)405 b Fo(A)49311 778 y Fj(M)50364
1218 y Ft(\()p Fo(x)p Ft(\))0 2723 y(the)517 b(output)i(of)e
Fo(A)g Ft(when)h(executed)e(on)h(input)h Fo(x)f Ft(and)h(oracle)d
(access)h(to)i Fo(M)132 b Ft(.)876 b(When)517 b Fo(M)649
b Ft(is)517 b(a)f(circuit,)544 b(this)0 4229 y(carries)420
b(the)i(standard)h(meaning)e(\(on)i(answ)-34 b(er)422
b(to)f(oracle)g(query)f Fo(x)p Ft(,)425 b Fo(A)d Ft(receiv)-34
b(es)419 b Fo(M)132 b Ft(\()p Fo(x)p Ft(\)\).)591 b(When)422
b Fo(M)553 b Ft(is)421 b(a)h(TM,)0 5734 y(this)483 b(means)f(that)i
Fo(A)e Ft(can)h(mak)-34 b(e)482 b(oracle)f(queries)h(of)g(the)h(form)g
(\()p Fo(x;)202 b Ft(1)32493 5294 y Fj(t)32888 5734 y
Ft(\))482 b(and)h(receiv)-34 b(e)481 b(in)h(resp)34 b(onse)483
b(either)e(the)0 7240 y(output)363 b(of)e Fo(M)493 b
Ft(on)361 b(input)h Fo(x)f Ft(\(if)g Fo(M)493 b Ft(halts)361
b(within)h Fo(t)e Ft(steps)i(on)f Fo(x)p Ft(\),)369 b(or)361
b Fp(?)f Ft(\(if)h Fo(M)493 b Ft(do)34 b(es)361 b(not)h(halt)f(within)h
Fo(t)e Ft(steps)i(on)0 8745 y Fo(x)p Ft(\).)1501 8305
y Fl(4)2638 8745 y Ft(If)429 b Fo(A)f Ft(is)h(a)f(probabilistic)h(T)
-101 b(uring)429 b(mac)-34 b(hine)429 b(then)g(b)-34
b(y)429 b Fo(A)p Ft(\()p Fo(x)p Ft(;)202 b Fo(r)34 b
Ft(\))428 b(w)-34 b(e)429 b(refer)e(to)i(the)g(result)g(of)g(running)g
Fo(A)g Ft(on)0 10250 y(input)396 b Fo(x)e Ft(and)h(random)h(tap)34
b(e)394 b Fo(r)34 b Ft(.)535 b(By)394 b Fo(A)p Ft(\()p
Fo(x)p Ft(\))h(w)-34 b(e)395 b(refer)f(to)h(the)g(distribution)h
(induced)f(b)-34 b(y)395 b(c)-34 b(ho)34 b(osing)395
b Fo(r)428 b Ft(uniformly)0 11926 y(and)461 b(running)h
Fo(A)p Ft(\()p Fo(x)p Ft(;)202 b Fo(r)34 b Ft(\).)708
b(If)461 b Fo(D)493 b Ft(is)461 b(a)g(distribution)g(then)h(b)-34
b(y)461 b Fo(x)29339 11239 y Fw(R)29026 11926 y Fp( )307
b Fo(D)494 b Ft(w)-34 b(e)461 b(mean)g(that)g Fo(x)g
Ft(is)f(a)h(random)g(v)-67 b(ariable)0 13602 y(distributed)359
b(according)f(to)h Fo(D)34 b Ft(.)522 b(If)358 b Fo(S)428
b Ft(is)358 b(a)g(set)g(then)h(b)-34 b(y)359 b Fo(x)26760
12915 y Fw(R)26447 13602 y Fp( )177 b Fo(S)429 b Ft(w)-34
b(e)358 b(mean)g(that)i Fo(x)d Ft(is)h(a)g(random)h(v)-67
b(ariable)357 b(that)i(is)0 15108 y(distributed)432 b(uniformly)f(o)-34
b(v)g(er)431 b(the)g(elemen)-34 b(ts)431 b(of)g Fo(S)70
b Ft(.)619 b(Supp)r(\()p Fo(D)34 b Ft(\))430 b(denotes)i(the)f
Fr(supp)-62 b(ort)428 b Ft(of)k(distribution)g Fo(D)34
b Ft(,)436 b(i.e.)0 16613 y(the)342 b(set)g(of)g(p)34
b(oin)-34 b(ts)343 b(that)g(ha)-34 b(v)g(e)342 b(nonzero)g(probabilit)
-34 b(y)343 b(under)f Fo(D)34 b Ft(.)517 b(A)342 b(function)h
Fo(\026)337 b Ft(:)f Fg(N)h Fp(!)g Fg(N)342 b Ft(is)g(called)f
Fr(ne)-62 b(gligible)430 b Ft(if)0 18119 y(it)399 b(gro)-34
b(ws)400 b(slo)-34 b(w)g(er)400 b(than)g(the)g(in)-34
b(v)g(erse)398 b(of)i(an)-34 b(y)400 b(p)34 b(olynomial.)536
b(That)400 b(is,)g(for)f(an)-34 b(y)400 b(p)34 b(ositiv)-34
b(e)399 b(p)34 b(olynomial)399 b Fo(p)p Ft(\()p Fp(\001)p
Ft(\))g(there)0 19624 y(exists)473 b Fo(N)584 b Fp(2)451
b Fg(N)474 b Ft(suc)-34 b(h)474 b(that)g Fo(\026)p Ft(\()p
Fo(n)p Ft(\))452 b Fo(<)g Ft(1)p Fo(=p)p Ft(\()p Fo(n)p
Ft(\))473 b(for)g(an)-34 b(y)474 b Fo(n)451 b(>)g(N)132
b Ft(.)746 b(W)-101 b(e'll)472 b(sometimes)h(use)g(neg)17
b(\()p Fp(\001)p Ft(\))473 b(to)h(denote)g(an)0 21130
y(unsp)34 b(eci\014ed)406 b(negligible)e(function.)544
b(W)-101 b(e)405 b(will)f(iden)-34 b(tify)406 b(T)-101
b(uring)406 b(mac)-34 b(hines)406 b(and)g(circuits)f(with)h(their)f
(canonical)0 22635 y(represen)-34 b(tations)405 b(as)f(strings)h(in)f
Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)18192 22195 y FG(\003)18718
22635 y Ft(.)0 25803 y Fz(Nonstandard)331 b(Notations.)1212
b Ft(If)287 b Fo(M)419 b Ft(is)287 b(a)g(TM)g(then)h(w)-34
b(e)287 b(denote)h(b)-34 b(y)287 b Fp(h)p Fo(M)132 b
Fp(i)288 b Ft(the)f(function)i Fp(h)p Fo(M)132 b Fp(i)337
b Ft(:)g(1)45424 25363 y FG(\003)45985 25803 y Fp(\002)35
b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)49926 25363 y FG(\003)50788
25803 y Fp(!)0 27309 y(f)p Ft(0)p Fo(;)g Ft(1)p Fp(g)2963
26869 y FG(\003)3893 27309 y Ft(giv)-34 b(en)404 b(b)-34
b(y:)8072 30237 y Fp(h)p Fo(M)132 b Fp(i)p Ft(\(1)11399
29737 y Fj(t)11795 30237 y Fo(;)202 b(x)p Ft(\))13767
29550 y Fl(def)13945 30237 y Ft(=)15335 28892 y Ff(n)16345
29636 y Fo(y)1561 b(M)132 b Ft(\()p Fo(x)p Ft(\))405
b(halts)g(with)f(output)j Fo(y)447 b Ft(after)405 b(at)f(most)h
Fo(t)f Ft(steps)16345 31141 y Fp(?)1212 b Ft(otherwise)0
33166 y(If)431 b Fo(C)517 b Ft(is)431 b(a)f(circuit)g(then)i(w)-34
b(e)431 b(denote)g(b)-34 b(y)431 b([)p Fo(C)87 b Ft(])430
b(the)h(function)h(it)f(computes.)619 b(Similarly)430
b(if)g Fo(M)563 b Ft(is)430 b(a)h(TM)g(then)h(w)-34 b(e)0
34671 y(denote)405 b(b)-34 b(y)404 b([)p Fo(M)132 b Ft(])404
b(the)g(\(p)34 b(ossibly)405 b(partial\))f(function)i(it)e(computes.)0
37884 y Fn(2.2)1495 b(Obfuscators)0 40172 y Ft(In)520
b(this)h(section,)549 b(w)-34 b(e)521 b(aim)f(to)h(formalize)e(the)i
(notion)h(of)e(obfuscators)i(based)f(on)g(the)f(\\virtual)h(blac)-34
b(k)520 b(b)34 b(o)-34 b(x")0 41677 y(prop)34 b(ert)-34
b(y)445 b(as)h(describ)34 b(ed)444 b(in)i(the)f(in)-34
b(tro)34 b(duction.)663 b(Recall)444 b(that)j(this)e(prop)34
b(ert)-34 b(y)446 b(requires)e(that)i(\\an)-34 b(ything)447
b(that)0 43183 y(an)399 b(adv)-34 b(ersary)398 b(can)h(compute)g(from)g
(an)g(obfuscation)h Fp(O)34 b Ft(\()p Fo(P)168 b Ft(\))400
b(of)f(a)f(program)h Fo(P)168 b Ft(,)400 b(it)e(could)h(also)g(compute)
g(giv)-34 b(en)0 44688 y(just)422 b(oracle)e(access)h(to)g
Fo(P)168 b Ft(.")590 b(W)-101 b(e)421 b(shall)g(de\014ne)h(what)g(it)f
(means)g(for)h(the)f(adv)-34 b(ersary)421 b(to)g(successfully)g
(compute)0 46194 y(something)405 b(in)f(this)h(setting,)f(and)h(there)f
(are)g(sev)-34 b(eral)403 b(c)-34 b(hoices)404 b(for)g(this)h(\(in)f
(decreasing)g(order)g(of)g(generalit)-34 b(y\):)1818
48518 y Fp(\017)606 b Ft(\(computational)431 b(indistinguishabilit)-34
b(y\))432 b(The)d(most)h(general)f(c)-34 b(hoice)429
b(is)g(not)h(to)g(restrict)f(the)h(nature)g(of)3030 50023
y(what)489 b(the)g(adv)-34 b(ersary)487 b(is)h(trying)g(to)g(compute,)
510 b(and)488 b(merely)f(require)g(that)i(it)f(is)f(p)34
b(ossible,)509 b(giv)-34 b(en)488 b(just)3030 51529 y(oracle)286
b(access)g(to)h Fo(P)168 b Ft(,)310 b(to)287 b(pro)34
b(duce)286 b(an)h(output)i(distribution)e(that)h(is)e(computationally)i
(indistinguishable)3030 53034 y(from)405 b(what)g(the)f(adv)-34
b(ersary)404 b(computes)h(when)g(giv)-34 b(en)404 b Fp(O)34
b Ft(\()p Fo(P)168 b Ft(\).)1818 55465 y Fp(\017)606
b Ft(\(satisfying)344 b(a)e(relation\))g(An)h(alternativ)-34
b(e)343 b(is)f(to)g(consider)h(the)f(adv)-34 b(ersary)342
b(as)h(trying)f(to)h(pro)34 b(duce)342 b(an)h(out-)3030
56970 y(put)411 b(that)f(satis\014es)g(an)g(arbitrary)f(\(p)34
b(ossibly)409 b(p)34 b(olynomial-time\))409 b(relation)g(with)h(the)g
(original)f(program)3030 58476 y Fo(P)168 b Ft(,)429
b(and)c(require)d(that)j(it)f(is)g(p)34 b(ossible,)428
b(giv)-34 b(en)423 b(just)i(oracle)e(access)g(to)h Fo(P)168
b Ft(,)429 b(to)c(succeed)e(with)i(roughly)e(the)3030
59981 y(same)404 b(probabilit)-34 b(y)405 b(as)f(the)h(adv)-34
b(ersary)403 b(do)34 b(es)404 b(when)h(giv)-34 b(en)404
b Fp(O)34 b Ft(\()p Fo(P)168 b Ft(\).)1818 62412 y Fp(\017)606
b Ft(\(computing)465 b(a)f(function\))h(A)e(w)-34 b(eak)g(er)464
b(requiremen)-34 b(t)463 b(is)g(to)h(restrict)f(the)g(previous)h
(requiremen)-34 b(t)463 b(to)g(re-)3030 63918 y(lations)415
b(whic)-34 b(h)415 b(are)f(functions;)421 b(that)415
b(is,)i(the)e(adv)-34 b(ersary)413 b(is)i(trying)f(to)h(compute)g(some)
f(function)i(of)e(the)3030 65423 y(original)404 b(program.)p
0 66407 20800 45 v 1382 67124 a Fw(4)1843 67547 y Fx(In)489
b(t)-28 b(ypical)490 b(cases)f(\(i.e.,)529 b(when)489
b(the)h(running)e(time)j(is)e Fk(a)502 b(priori)488 b
Fx(b)28 b(ounded\),)527 b(this)490 b(con)-28 b(v)g(en)g(tion)489
b(mak)-28 b(es)490 b(our)e(de\014nitions)i(of)0 68765
y(obfuscator)332 b(ev)-28 b(en)334 b(w)-28 b(eak)g(er)332
b(since)i(it)g(allo)-28 b(ws)332 b Fi(A)h Fx(to)g(learn)h(the)f(actual)
h(running-time)g(of)f Fi(M)440 b Fx(on)333 b(particular)g(inputs.)453
b(This)333 b(seems)g(the)0 69982 y(natural)391 b(c)-28
b(hoice)392 b(b)28 b(ecause)393 b(a)e(mac)-28 b(hine)392
b(giv)-28 b(en)392 b(the)g(co)28 b(de)392 b(of)f Fi(M)500
b Fx(can)391 b(de\014nitely)i(learn)f(its)g(actual)g(running-time)g(on)
g(inputs)f(of)g(its)0 71200 y(o)-28 b(wn)340 b(c)-28
b(hoice.)25697 74521 y Ft(7)p eop
%%Page: 8 9
8 8 bop 1818 1218 a Fp(\017)606 b Ft(\(computing)406
b(a)e(predicate\))g(The)h(w)-34 b(eak)g(est)405 b(is)f(to)g(restrict)g
(the)g(previous)g(requiremen)-34 b(t)404 b(to)g Fp(f)p
Ft(0)p Fo(;)202 b Ft(1)p Fp(g)p Ft(-v)-67 b(alued)3030
2723 y(functions;)406 b(that)f(is,)e(the)i(adv)-34 b(ersary)404
b(is)g(trying)g(to)g(decide)g(some)g(prop)34 b(ert)-34
b(y)404 b(of)h(the)f(original)g(program.)1882 5179 y(Since)560
b(w)-34 b(e)562 b(will)e(b)34 b(e)560 b(pro)-34 b(ving)562
b(imp)34 b(ossibilit)-34 b(y)560 b(results,)600 b(our)561
b(results)g(are)f(strongest)i(when)f(w)-34 b(e)561 b(adopt)i(the)0
6685 y(w)-34 b(eak)g(est)293 b(requiremen)-34 b(t)292
b(\(i.e.,)314 b(the)293 b(last)f(one\).)502 b(This)293
b(yields)f(t)-34 b(w)g(o)294 b(de\014nitions)f(for)g(obfuscators,)316
b(one)292 b(for)h(programs)0 8190 y(de\014ned)405 b(b)-34
b(y)405 b(T)-101 b(uring)404 b(mac)-34 b(hines)405 b(and)g(one)f(for)g
(programs)h(de\014ned)g(b)-34 b(y)404 b(circuits.)0 10970
y Fz(De\014nition)466 b(2.1)f(\(TM)g(obfuscator\))606
b Fr(A)335 b(pr)-62 b(ob)g(abilistic)331 b(algorithm)j
Fp(O)369 b Fr(is)335 b(a)g Ft(TM)298 b(obfuscator)337
b Fr(if)e(the)g(fol)62 b(lowing)0 12475 y(thr)-62 b(e)g(e)432
b(c)-62 b(onditions)432 b(hold:)1818 14932 y Fp(\017)606
b Fr(\(functionality\))489 b(F)-93 b(or)492 b(every)f(TM)i
Fo(M)132 b Fr(,)507 b(the)491 b(string)g Fp(O)34 b Ft(\()p
Fo(M)132 b Ft(\))493 b Fr(describ)-62 b(es)490 b(a)i(TM)g(that)f(c)-62
b(omputes)490 b(the)h(same)3030 16437 y(function)433
b(as)f Fo(M)132 b Fr(.)1818 18921 y Fp(\017)606 b Fr(\(p)-62
b(olynomial)466 b(slowdown\))h(The)g(description)f(length)g(and)i
(running)g(time)g(of)f Fp(O)34 b Ft(\()p Fo(M)132 b Ft(\))469
b Fr(ar)-62 b(e)467 b(at)h(most)f(p)-62 b(oly-)3030 20426
y(nomial)62 b(ly)436 b(lar)-62 b(ger)435 b(than)i(that)e(of)h
Fo(M)132 b Fr(.)567 b(That)436 b(is,)h(ther)-62 b(e)436
b(is)g(a)g(p)-62 b(olynomial)435 b Fo(p)i Fr(such)e(that)g(for)i(every)
e(TM)i Fo(M)132 b Fr(,)3030 21932 y Fp(jO)34 b Ft(\()p
Fo(M)132 b Ft(\))p Fp(j)440 b(\024)g Fo(p)p Ft(\()p Fp(j)p
Fo(M)132 b Fp(j)p Ft(\))p Fr(,)505 b(and)490 b(if)h Fo(M)622
b Fr(halts)489 b(in)i Fo(t)g Fr(steps)e(on)h(some)g(input)g
Fo(x)p Fr(,)504 b(then)490 b Fp(O)34 b Ft(\()p Fo(M)132
b Ft(\))492 b Fr(halts)c(within)j Fo(p)p Ft(\()p Fo(t)p
Ft(\))3030 23437 y Fr(steps)432 b(on)i Fo(x)p Fr(.)1818
25921 y Fp(\017)606 b Fr(\(\\virtual)432 b(black)g(b)-62
b(ox")434 b(pr)-62 b(op)g(erty\))432 b(F)-93 b(or)434
b(any)g(PPT)i Fo(A)p Fr(,)e(ther)-62 b(e)433 b(is)h(a)g(PPT)h
Fo(S)505 b Fr(and)434 b(a)g(ne)-62 b(gligible)432 b(function)h
Fo(\013)3030 27426 y Fr(such)f(that)g(for)h(al)62 b(l)433
b(TMs)g Fo(M)13497 28936 y Ff(\014)13497 29663 y(\014)13497
30390 y(\014)13901 30330 y Ft(Pr)201 b([)p Fo(A)p Ft(\()p
Fp(O)34 b Ft(\()p Fo(M)132 b Ft(\)\))338 b(=)f(1])269
b Fp(\000)g Ft(Pr)26382 28984 y Ff(h)26955 30330 y Fo(S)27768
29829 y FG(h)p Fj(M)94 b FG(i)29553 30330 y Ft(\(1)30630
29829 y FG(j)p Fj(M)g FG(j)32207 30330 y Ft(\))337 b(=)f(1)34900
28984 y Ff(i)35473 28936 y(\014)35473 29663 y(\014)35473
30390 y(\014)36213 30330 y Fp(\024)h Fo(\013)t Ft(\()p
Fp(j)p Fo(M)132 b Fp(j)p Ft(\))p Fo(:)0 33722 y Fr(We)433
b(say)f(that)g Fp(O)468 b Fr(is)433 b Ft(e\016cien)-34
b(t)434 b Fr(if)f(it)g(runs)g(in)h(p)-62 b(olynomial)431
b(time.)0 36502 y Fz(De\014nition)466 b(2.2)f(\(circuit)g(obfuscator\))
606 b Fr(A)457 b(pr)-62 b(ob)g(abilistic)453 b(algorithm)i
Fp(O)491 b Fr(is)457 b(a)g Ft(\(circuit\))430 b(obfuscator)458
b Fr(if)f(the)0 38007 y(fol)62 b(lowing)432 b(thr)-62
b(e)g(e)432 b(c)-62 b(onditions)432 b(hold:)1818 40464
y Fp(\017)606 b Fr(\(functionality\))338 b(F)-93 b(or)342
b(every)f(cir)-62 b(cuit)339 b Fo(C)87 b Fr(,)360 b(the)341
b(string)g Fp(O)34 b Ft(\()p Fo(C)87 b Ft(\))343 b Fr(describ)-62
b(es)338 b(a)k(cir)-62 b(cuit)340 b(that)g(c)-62 b(omputes)339
b(the)i(same)3030 41969 y(function)433 b(as)f Fo(C)87
b Fr(.)1818 44453 y Fp(\017)606 b Fr(\(p)-62 b(olynomial)340
b(slowdown\))i(Ther)-62 b(e)342 b(is)g(a)g(p)-62 b(olynomial)341
b Fo(p)h Fr(such)f(that)h(for)g(every)f(cir)-62 b(cuit)341
b Fo(C)87 b Fr(,)360 b Fp(jO)34 b Ft(\()p Fo(C)87 b Ft(\))p
Fp(j)337 b(\024)f Fo(p)p Ft(\()p Fp(j)p Fo(C)87 b Fp(j)p
Ft(\))p Fr(.)1818 46936 y Fp(\017)606 b Fr(\(\\virtual)432
b(black)g(b)-62 b(ox")434 b(pr)-62 b(op)g(erty\))432
b(F)-93 b(or)434 b(any)g(PPT)i Fo(A)p Fr(,)e(ther)-62
b(e)433 b(is)h(a)g(PPT)h Fo(S)505 b Fr(and)434 b(a)g(ne)-62
b(gligible)432 b(function)h Fo(\013)3030 48442 y Fr(such)f(that)g(for)h
(al)62 b(l)433 b(cir)-62 b(cuits)431 b Fo(C)14482 49951
y Ff(\014)14482 50679 y(\014)14482 51406 y(\014)14886
51345 y Ft(Pr)202 b([)p Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p
Fo(C)87 b Ft(\)\))338 b(=)e(1])269 b Fp(\000)h Ft(Pr)27013
50000 y Ff(h)27586 51345 y Fo(S)28399 50845 y Fj(C)29187
51345 y Ft(\(1)30264 50845 y FG(j)p Fj(C)63 b FG(j)31576
51345 y Ft(\))337 b(=)f(1)34269 50000 y Ff(i)34842 49951
y(\014)34842 50679 y(\014)34842 51406 y(\014)35583 51345
y Fp(\024)g Fo(\013)t Ft(\()p Fp(j)p Fo(C)87 b Fp(j)p
Ft(\))p Fo(:)0 54738 y Fr(We)433 b(say)f(that)g Fp(O)468
b Fr(is)433 b Ft(e\016cien)-34 b(t)434 b Fr(if)f(it)g(runs)g(in)h(p)-62
b(olynomial)431 b(time.)1882 57517 y Ft(W)-101 b(e)483
b(call)f(the)i(\014rst)g(t)-34 b(w)g(o)484 b(requiremen)-34
b(ts)483 b(\(functionalit)-34 b(y)485 b(and)f(p)34 b(olynomial)483
b(slo)-34 b(wdo)g(wn\))486 b(the)e Fr(syntactic)504 b(r)-62
b(e-)0 59023 y(quir)g(ements)502 b Ft(of)404 b(obfuscation,)i(as)e
(they)g(do)h(not)f(address)h(the)g(issue)f(of)g(securit)-34
b(y)404 b(at)h(all.)1882 60528 y(There)559 b(are)f(a)h(couple)f(of)i
(other)f(natural)g(form)-34 b(ulations)560 b(of)f(the)h(\\virtual)e
(blac)-34 b(k)559 b(b)34 b(o)-34 b(x")559 b(prop)34 b(ert)-34
b(y)-101 b(.)1002 b(The)0 62034 y(\014rst,)390 b(whic)-34
b(h)387 b(more)f(closely)f(follo)-34 b(ws)387 b(the)g(informal)f
(discussion)h(ab)34 b(o)-34 b(v)g(e,)390 b(asks)c(that)i(for)e(ev)-34
b(ery)385 b(predicate)i Fo(\031)43 b Ft(,)389 b(the)0
63539 y(probabilit)-34 b(y)365 b(that)h Fo(A)p Ft(\()p
Fp(O)34 b Ft(\()p Fo(C)87 b Ft(\)\))338 b(=)e Fo(\031)43
b Ft(\()p Fo(C)87 b Ft(\))366 b(is)e(at)h(most)g(the)g(probabilit)-34
b(y)365 b(that)h Fo(S)35261 63099 y Fj(C)36049 63539
y Ft(\(1)37126 63099 y FG(j)p Fj(C)63 b FG(j)38438 63539
y Ft(\))337 b(=)g Fo(\031)43 b Ft(\()p Fo(C)87 b Ft(\))365
b(plus)g(a)g(negligible)0 65045 y(term.)644 b(It)439
b(is)g(easy)g(to)h(see)f(that)h(this)g(requiremen)-34
b(t)439 b(is)g(equiv)-67 b(alen)-34 b(t)439 b(to)g(the)h(one)f(ab)34
b(o)-34 b(v)g(e.)645 b(Another)440 b(form)-34 b(ulation)0
66550 y(refers)502 b(to)h(the)g(distinguishabilit)-34
b(y)504 b(b)34 b(et)-34 b(w)g(een)504 b(obfuscations)g(of)f(t)-34
b(w)g(o)504 b(TMs/circuits:)735 b(ask)503 b(that)h(for)e(ev)-34
b(ery)502 b Fo(C)51474 66732 y Fl(1)0 68055 y Ft(and)336
b Fo(C)3154 68237 y Fl(2)3681 68055 y Ft(,)348 b Fp(j)202
b Ft(Pr)g([)p Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(C)10460
68237 y Fl(1)10986 68055 y Ft(\)\))338 b(=)e(1])132 b
Fp(\000)g Ft(Pr)202 b([)p Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p
Fo(C)21250 68237 y Fl(2)21776 68055 y Ft(\)\)])203 b
Fp(j)335 b Ft(is)h(appro)-34 b(ximately)335 b(equal)g(to)h
Fp(j)202 b Ft(Pr)39579 67074 y Ff(\002)40084 68055 y
Fo(S)40897 67616 y Fj(C)41568 67739 y Fw(1)42084 68055
y Ft(\(1)43161 67616 y FG(j)p Fj(C)44093 67739 y Fw(1)44555
67616 y FG(j)44872 68055 y Fo(;)g Ft(1)46017 67616 y
FG(j)p Fj(C)46949 67739 y Fw(2)47410 67616 y FG(j)47726
68055 y Ft(\))337 b(=)g(1)50420 67074 y Ff(\003)51057
68055 y Fp(\000)0 69694 y Ft(Pr)1502 68713 y Ff(\002)2007
69694 y Fo(S)2820 69255 y Fj(C)3491 69378 y Fw(2)4007
69694 y Ft(\(1)5084 69255 y FG(j)p Fj(C)6016 69378 y
Fw(1)6478 69255 y FG(j)6794 69694 y Fo(;)202 b Ft(1)7939
69255 y FG(j)p Fj(C)8871 69378 y Fw(2)9388 69694 y Ft(\))9859
68713 y Ff(\003)10566 69694 y Fp(j)p Ft(.)700 b(This)459
b(de\014nition)g(app)34 b(ears)459 b(to)f(b)34 b(e)458
b(sligh)-34 b(tly)458 b(w)-34 b(eak)g(er)459 b(than)g(the)f(ones)h(ab)
34 b(o)-34 b(v)g(e,)472 b(but)459 b(our)0 71200 y(imp)34
b(ossibilit)-34 b(y)404 b(pro)34 b(of)404 b(also)g(rules)g(it)g(out.)
25697 74521 y(8)p eop
%%Page: 9 10
9 9 bop 1882 1218 a Ft(Note)519 b(that)h(in)g(b)34 b(oth)520
b(de\014nitions,)548 b(w)-34 b(e)520 b(ha)-34 b(v)g(e)519
b(c)-34 b(hosen)520 b(to)g(simplify)f(the)g(de\014nition)h(b)-34
b(y)520 b(using)g(the)f(size)g(of)0 2723 y(the)434 b(TM/circuit)f(to)g
(b)34 b(e)433 b(obfuscated)i(as)e(a)h(securit)-34 b(y)433
b(parameter.)625 b(One)433 b(can)h(alw)-34 b(a)g(ys)434
b(increase)e(this)i(length)g(b)-34 b(y)0 4229 y(padding)405
b(to)g(obtain)g(higher)f(securit)-34 b(y)-101 b(.)1882
5734 y(The)279 b(main)h(di\013erence)e(b)34 b(et)-34
b(w)g(een)280 b(the)g(circuit)e(and)i(TM)f(obfuscators)i(is)e(that)h(a)
f(circuit)g(computes)g(a)h(function)0 7240 y(with)508
b(\014nite)f(domain)g(\(all)g(the)g(inputs)h(of)g(a)e(particular)h
(length\))h(while)f(a)f(TM)h(computes)h(a)f(function)h(with)0
8745 y(in\014nite)343 b(domain.)519 b(Note)343 b(that)g(if)g(w)-34
b(e)343 b(had)h(not)f(restricted)f(the)h(size)f(of)h(the)g(obfuscated)h
(circuit)e Fp(O)34 b Ft(\()p Fo(C)87 b Ft(\),)355 b(then)343
b(the)0 10250 y(\(exp)34 b(onen)-34 b(tial)328 b(size\))f(list)h(of)g
(all)f(the)h(v)-67 b(alues)327 b(of)h(the)g(circuit)f(w)-34
b(ould)329 b(b)34 b(e)327 b(a)h(v)-67 b(alid)327 b(obfuscation)i(\(pro)
-34 b(vided)329 b(w)-34 b(e)328 b(allo)-34 b(w)0 11756
y Fo(S)522 b Ft(running)453 b(time)f(p)34 b(oly)16 b(\()p
Fp(jO)34 b Ft(\()p Fo(C)87 b Ft(\))p Fp(j)p Ft(\))452
b(rather)g(than)h(p)34 b(oly)17 b(\()p Fp(j)p Fo(C)87
b Fp(j)p Ft(\)\).)682 b(F)-101 b(or)452 b(T)-101 b(uring)453
b(mac)-34 b(hines,)464 b(it)451 b(is)h(not)h(clear)e(ho)-34
b(w)453 b(to)0 13261 y(construct)383 b(suc)-34 b(h)384
b(an)f(obfuscation,)388 b(ev)-34 b(en)382 b(if)g(w)-34
b(e)383 b(are)f(allo)-34 b(w)g(ed)383 b(an)g(exp)34 b(onen)-34
b(tial)383 b(slo)-34 b(wdo)g(wn.)533 b(Hence)382 b(obfuscating)0
14767 y(TMs)405 b(is)f(in)-34 b(tuitiv)g(ely)404 b(harder.)538
b(Indeed,)404 b(it)h(is)e(relativ)-34 b(ely)403 b(easy)h(to)g(pro)-34
b(v)g(e:)0 17236 y Fz(Prop)39 b(osition)465 b(2.3)607
b Fr(If)433 b(a)g(TM)h(obfusc)-62 b(ator)430 b(exists,)i(then)h(a)g
(cir)-62 b(cuit)432 b(obfusc)-62 b(ator)430 b(exists.)0
19705 y Ft(Th)-34 b(us,)467 b(when)454 b(w)-34 b(e)454
b(pro)-34 b(v)g(e)453 b(our)h(imp)34 b(ossibilit)-34
b(y)453 b(result)g(for)h(circuit)e(obfuscators,)467 b(the)454
b(imp)34 b(ossibilit)-34 b(y)453 b(of)h(TM)g(ob-)0 21210
y(fuscators)521 b(will)e(follo)-34 b(w.)887 b(Ho)-34
b(w)g(ev)g(er,)549 b(considering)520 b(TM)h(obfuscators)g(will)f(b)34
b(e)519 b(useful)i(as)f(motiv)-67 b(ation)520 b(for)g(the)0
22716 y(pro)34 b(of.)1882 24221 y(W)-101 b(e)348 b(note)h(that,)361
b(from)349 b(the)g(p)34 b(ersp)g(ectiv)-34 b(e)348 b(of)h
(applications,)360 b(De\014nitions)350 b(2.1)e(and)i(2.2)e(are)h
(already)f(to)34 b(o)349 b(w)-34 b(eak)0 25727 y(to)341
b(ha)-34 b(v)g(e)341 b(the)g(wide)f(applicabilit)-34
b(y)341 b(discussed)g(in)f(the)h(in)-34 b(tro)34 b(duction.)519
b(The)341 b(p)34 b(oin)-34 b(t)341 b(is)f(that)i(they)e(are)g(nev)-34
b(ertheless)0 27232 y Fr(imp)-62 b(ossible)401 b Ft(to)j(satisfy)h
(\(as)g(w)-34 b(e)404 b(will)g(pro)-34 b(v)g(e\).)0 30994
y Fs(3)1793 b(The)598 b(Main)g(Imp)50 b(ossibilit)-50
b(y)600 b(Result)0 33700 y Ft(T)-101 b(o)405 b(state)f(our)g(main)h
(result)f(w)-34 b(e)404 b(in)-34 b(tro)34 b(duce)405
b(the)g(notion)g(of)f(unobfuscatable)i(function)g(ensem)-34
b(ble.)0 36169 y Fz(De\014nition)466 b(3.1)606 b Fr(A)-31
b(n)480 b Ft(unobfuscatable)354 b(function)f(ensem)-34
b(ble)386 b Fr(is)f(an)h(ensemble)e Fp(fH)38697 36366
y Fj(k)39266 36169 y Fp(g)39872 36366 y Fj(k)24 b FG(2)p
Fe(N)42093 36169 y Fr(of)385 b(distributions)d Fp(H)51431
36366 y Fj(k)0 37674 y Fr(on)434 b(\014nite)f(functions)f(\(fr)-62
b(om,)433 b(say,)f Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)19334
37234 y Fj(l)19615 37377 y Fw(in)20353 37234 y Fl(\()p
Fj(k)24 b Fl(\))22087 37674 y Fr(to)433 b Fp(f)p Ft(0)p
Fo(;)202 b Ft(1)p Fp(g)26506 37234 y Fj(l)26787 37345
y Fw(out)28017 37234 y Fl(\()p Fj(k)24 b Fl(\))29317
37674 y Fr(\))434 b(satisfying:)1818 40039 y Fp(\017)606
b Fr(\(e\016cient)515 b(c)-62 b(omputability\))512 b(Every)j(function)h
Fo(f)26036 39352 y Fd(R)25713 40039 y Fp( )330 b(H)28279
40236 y Fj(k)29364 40039 y Fr(is)516 b(c)-62 b(omputable)513
b(by)j(a)g(cir)-62 b(cuit)515 b(of)h(size)f Ft(p)34 b(oly)16
b(\()p Fo(k)38 b Ft(\))p Fr(.)3030 41544 y(\(Mor)-62
b(e)g(over,)551 b(a)528 b(distribution)f(on)i(cir)-62
b(cuits)527 b(c)-62 b(onsistent)527 b(with)h Fp(H)32783
41741 y Fj(k)33881 41544 y Fr(c)-62 b(an)529 b(b)-62
b(e)528 b(sample)-62 b(d)528 b(uniformly)f(in)j(time)3030
43050 y Ft(p)34 b(oly)17 b(\()p Fo(k)38 b Ft(\))p Fr(.\))1818
45430 y Fp(\017)606 b Fr(\(unobfusc)-62 b(atability\))427
b(Ther)-62 b(e)433 b(exists)f(a)i(function)e Fo(\031)380
b Ft(:)27594 44521 y Ff(S)28604 45788 y Fj(k)24 b FG(2)p
Fe(N)30641 45430 y Ft(Supp)r(\()p Fp(H)34830 45627 y
Fj(k)35399 45430 y Ft(\))337 b Fp(!)g(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)433 b Fr(such)f(that)4100 47810 y(1.)605
b Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))436 b Fr(is)d(har)-62
b(d)432 b(to)h(c)-62 b(ompute)431 b(with)i(black-b)-62
b(ox)431 b(ac)-62 b(c)g(ess)431 b(to)i Fo(f)130 b Fr(:)558
b(F)-93 b(or)434 b(any)f(PPT)i Fo(S)20377 50678 y Ft(Pr)19627
51952 y Fj(f)20365 51441 y Fd(R)20177 51952 y FG( H)21919
52108 y Fc(k)22427 50678 y Ft([)p Fo(S)23577 50177 y
Fj(f)24182 50678 y Ft(\(1)25259 50177 y Fj(k)25828 50678
y Ft(\))337 b(=)f Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\)])339
b Fp(\024)32402 49858 y Ft(1)p 32402 50399 607 49 v 32402
51509 a(2)33411 50678 y(+)269 b(neg)17 b(\()p Fo(k)38
b Ft(\))4100 54279 y Fr(2.)605 b Fo(\031)43 b Ft(\()p
Fo(f)130 b Ft(\))419 b Fr(is)d(e)-62 b(asy)416 b(to)g(c)-62
b(ompute)415 b(with)h(ac)-62 b(c)g(ess)415 b(to)h(any)g(cir)-62
b(cuit)415 b(that)h(c)-62 b(omputes)414 b Fo(f)130 b
Fr(:)550 b(Ther)-62 b(e)416 b(exists)g(a)g(PPT)5697 55785
y Fo(A)434 b Fr(such)e(that)g(for)h(any)g Fo(f)468 b
Fp(2)18739 54875 y Ff(S)19749 56143 y Fj(k)24 b FG(2)p
Fe(N)21787 55785 y Ft(Supp)r(\()p Fp(H)25976 55982 y
Fj(k)26545 55785 y Ft(\))434 b Fr(and)f(for)g(any)g(cir)-62
b(cuit)432 b Fo(C)520 b Fr(that)433 b(c)-62 b(omputes)430
b Fo(f)25437 58143 y(A)p Ft(\()p Fo(C)87 b Ft(\))338
b(=)e Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))1882 60662
y(W)-101 b(e)525 b(pro)-34 b(v)g(e)526 b(in)g(Theorem)g(3.11)g(that,)
557 b(assuming)527 b(one-w)-34 b(a)g(y)526 b(functions)i(exist,)555
b(there)526 b(exists)g(an)g(unobfus-)0 62167 y(catable)455
b(function)i(ensem)-34 b(ble.)691 b(This)456 b(implies)f(that,)468
b(under)456 b(the)g(same)f(assumption,)469 b(there)455
b(is)g(no)h(obfuscator)0 63673 y(that)447 b(satis\014es)f(De\014nition)
g(2.2)f(\(actually)h(w)-34 b(e)446 b(pro)-34 b(v)g(e)445
b(the)h(latter)g(fact)g(directly)e(in)i(Theorem)f(3.8\).)663
b(Since)445 b(the)0 65178 y(existence)522 b(of)h(an)g
Fr(e\016cient)636 b Ft(obfuscator)525 b(implies)d(the)h(existence)f(of)
h(one-w)-34 b(a)g(y)524 b(functions)g(\(Lemma)f(3.9\),)553
b(w)-34 b(e)0 66684 y(conclude)404 b(that)h(e\016cien)-34
b(t)405 b(obfuscators)g(do)g(not)f(exist)g(\(unconditionally\).)1882
68189 y(Ho)-34 b(w)g(ev)g(er,)382 b(the)c(existence)e(of)h
(unobfuscatable)i(function)f(ensem)-34 b(ble)377 b(has)h(ev)-34
b(en)376 b(stronger)h(implications.)530 b(As)0 69694
y(men)-34 b(tioned)361 b(in)g(the)f(in)-34 b(tro)34 b(duction,)370
b(these)361 b(functions)g(can)g(not)g(b)34 b(e)360 b(obfuscated)h(ev)
-34 b(en)360 b(if)h(w)-34 b(e)360 b(allo)-34 b(w)361
b(the)f(follo)-34 b(wing)0 71200 y(relaxations)404 b(to)g(the)h
(obfuscator:)25697 74521 y(9)p eop
%%Page: 10 11
10 10 bop 1481 1218 a Ft(1.)606 b(As)352 b(men)-34 b(tioned)352
b(ab)34 b(o)-34 b(v)g(e,)362 b(the)351 b(obfuscator)i(do)34
b(es)351 b(not)h(ha)-34 b(v)g(e)352 b(to)f(run)h(in)f(p)34
b(olynomial)351 b(time)g(|)g(it)g(can)h(b)34 b(e)350
b(an)-34 b(y)3030 2723 y(random)405 b(pro)34 b(cess.)1481
5208 y(2.)606 b(The)545 b(obfuscator)g(has)f(only)g(to)g(w)-34
b(ork)544 b(for)g(functions)h(in)f(Supp)r(\()p Fp(H)34635
5405 y Fj(k)35204 5208 y Ft(\))g(and)g(only)g(for)g(a)g(non-negligible)
3030 6714 y(fraction)405 b(of)f(these)h(functions)g(under)g(the)g
(distributions)g Fp(H)30859 6911 y Fj(k)31428 6714 y
Ft(.)1481 9199 y(3.)606 b(The)372 b(obfuscator)g(has)f(only)g(to)g
(hide)g(an)g Fr(a)403 b(priori)369 b Ft(\014xed)i(prop)34
b(ert)-34 b(y)371 b Fo(\031)414 b Ft(from)371 b(an)g
Fr(a)403 b(priori)369 b Ft(\014xed)i(adv)-34 b(ersary)3030
10705 y Fo(A)p Ft(.)0 13898 y Fz(Structure)576 b(of)e(the)h(Pro)39
b(of)574 b(of)h(the)g(Main)g(Imp)39 b(ossibilit)-39 b(y)575
b(Result.)1213 b Ft(W)-101 b(e)499 b(shall)h(pro)-34
b(v)g(e)500 b(our)g(result)g(b)-34 b(y)0 15404 y(\014rst)488
b(de\014ning)h(obfuscators)g(that)g(are)e(secure)g(also)h(when)h
(applied)f(to)g(sev)-34 b(eral)487 b(\(e.g.,)508 b(t)-34
b(w)g(o\))489 b(algorithms)f(and)0 16909 y(pro)-34 b(ving)566
b(that)g(they)g(do)f(not)i(exist.)1021 b(Then)567 b(w)-34
b(e)565 b(shall)h(mo)34 b(dify)565 b(the)h(construction)g(in)f(this)h
(pro)34 b(of)566 b(to)g(pro)-34 b(v)g(e)0 18415 y(that)489
b(TM)g(obfuscators)g(in)f(the)h(sense)f(of)g(De\014nition)h(2.1)f(do)g
(not)h(exist.)790 b(After)488 b(that,)510 b(using)489
b(an)f(additional)0 19920 y(construction)399 b(\(whic)-34
b(h)400 b(requires)d(one-w)-34 b(a)g(y)399 b(functions\),)i(w)-34
b(e)399 b(will)f(pro)-34 b(v)g(e)399 b(that)g(a)f(circuit)g(obfuscator)
i(as)e(de\014ned)0 21426 y(in)508 b(De\014nition)h(2.2)f(do)34
b(es)508 b(not)h(exist)e(if)h(one-w)-34 b(a)g(y)510 b(functions)f
(exist.)850 b(W)-101 b(e)508 b(will)f(then)i(observ)-34
b(e)508 b(that)h(our)g(pro)34 b(of)0 22931 y(actually)404
b(yields)f(an)i(unobfuscatable)h(function)f(ensem)-34
b(ble)405 b(\(Theorem)f(3.11\).)0 26169 y Fn(3.1)1495
b(Obfuscating)500 b(t)-42 b(w)g(o)500 b(TMs/circuits)0
28458 y Ft(Obfuscators)587 b(as)g(de\014ned)g(in)g(the)g(previous)f
(section)g(pro)-34 b(vide)587 b(a)f(\\virtual)g(blac)-34
b(k)587 b(b)34 b(o)-34 b(x")586 b(prop)34 b(ert)-34 b(y)587
b(when)g(a)0 29963 y(single)475 b(program)g(is)f(obfuscated,)494
b(but)477 b(the)e(de\014nitions)h(do)g(not)f(sa)-34 b(y)475
b(an)-34 b(ything)477 b(ab)34 b(out)476 b(what)g(happ)34
b(ens)476 b(when)0 31469 y(the)524 b(adv)-34 b(ersary)523
b(can)g(insp)34 b(ect)524 b(more)f(than)i(one)e(obfuscated)i(program.)
897 b(In)523 b(this)h(section,)553 b(w)-34 b(e)524 b(will)f(consider)0
32974 y(extensions)396 b(of)f(those)h(de\014nitions)h(to)e(obfuscating)
i(t)-34 b(w)g(o)397 b(programs,)g(and)f(pro)-34 b(v)g(e)396
b(that)g(they)g(are)f(imp)34 b(ossible)395 b(to)0 34479
y(meet.)673 b(The)449 b(pro)34 b(ofs)450 b(will)e(pro)-34
b(vide)449 b(useful)h(motiv)-67 b(ation)449 b(for)g(the)h(imp)34
b(ossibilit)-34 b(y)448 b(of)i(the)f(original)g(one-program)0
35985 y(de\014nitions.)0 38769 y Fz(De\014nition)466
b(3.2)f(\(2-TM)f(obfuscator\))606 b Fr(A)582 b Ft(2-TM)566
b(obfuscator)583 b Fr(is)f(de\014ne)-62 b(d)581 b(in)h(the)f(same)g
(way)h(as)f(a)h(TM)0 40275 y(obfusc)-62 b(ator,)430 b(exc)-62
b(ept)432 b(that)g(the)g(\\virtual)f(black)h(b)-62 b(ox")433
b(pr)-62 b(op)g(erty)431 b(is)i(str)-62 b(engthene)g(d)431
b(as)i(fol)62 b(lows:)1818 42735 y Fp(\017)606 b Fr(\(\\virtual)432
b(black)g(b)-62 b(ox")434 b(pr)-62 b(op)g(erty\))432
b(F)-93 b(or)434 b(any)g(PPT)i Fo(A)p Fr(,)e(ther)-62
b(e)433 b(is)h(a)g(PPT)h Fo(S)505 b Fr(and)434 b(a)g(ne)-62
b(gligible)432 b(function)h Fo(\013)3030 44241 y Fr(such)f(that)g(for)h
(al)62 b(l)433 b(TMs)g Fo(M)65 b(;)202 b(N)7132 45755
y Ff(\014)7132 46483 y(\014)7132 47210 y(\014)7536 47149
y Ft(Pr)f([)p Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(M)132
b Ft(\))p Fo(;)202 b Fp(O)34 b Ft(\()p Fo(N)132 b Ft(\)\))338
b(=)f(1])269 b Fp(\000)g Ft(Pr)23604 45804 y Ff(h)24176
47149 y Fo(S)24989 46649 y FG(h)p Fj(M)94 b FG(i)p Fj(;)p
FG(h)p Fj(N)g FG(i)28609 47149 y Ft(\(1)29686 46649 y
FG(j)p Fj(M)g FG(j)p Fl(+)p FG(j)p Fj(N)g FG(j)33358
47149 y Ft(\))337 b(=)f(1)36051 45804 y Ff(i)36624 45755
y(\014)36624 46483 y(\014)36624 47210 y(\014)37365 47149
y Fp(\024)g Fo(\013)t Ft(\(min)r Fp(fj)p Fo(M)132 b Fp(j)p
Fo(;)202 b Fp(j)p Fo(N)132 b Fp(jg)p Ft(\))0 50615 y
Fz(De\014nition)466 b(3.3)f(\(2-circuit)f(obfuscator\))606
b Fr(A)537 b Ft(2-circuit)515 b(obfuscator)538 b Fr(is)e(de\014ne)-62
b(d)536 b(in)h(the)e(same)h(way)g(as)f(a)0 52121 y(cir)-62
b(cuit)432 b(obfusc)-62 b(ator,)429 b(exc)-62 b(ept)432
b(that)g(the)h(\\virtual)e(black)h(b)-62 b(ox")432 b(pr)-62
b(op)g(erty)432 b(is)h(r)-62 b(eplac)g(e)g(d)431 b(with)i(the)f(fol)62
b(lowing:)1818 54581 y Fp(\017)606 b Fr(\(\\virtual)432
b(black)g(b)-62 b(ox")434 b(pr)-62 b(op)g(erty\))432
b(F)-93 b(or)434 b(any)g(PPT)i Fo(A)p Fr(,)e(ther)-62
b(e)433 b(is)h(a)g(PPT)h Fo(S)505 b Fr(and)434 b(a)g(ne)-62
b(gligible)432 b(function)h Fo(\013)3030 56087 y Fr(such)f(that)g(for)h
(al)62 b(l)433 b(cir)-62 b(cuits)431 b Fo(C)19 b(;)202
b(D)8621 57601 y Ff(\014)8621 58329 y(\014)8621 59056
y(\014)9025 58995 y Ft(Pr)g([)p Fo(A)p Ft(\()p Fp(O)34
b Ft(\()p Fo(C)87 b Ft(\))p Fo(;)202 b Fp(O)34 b Ft(\()p
Fo(D)g Ft(\)\))337 b(=)g(1])269 b Fp(\000)g Ft(Pr)24669
57650 y Ff(h)25242 58995 y Fo(S)26055 58495 y Fj(C)10
b(;D)27851 58995 y Ft(\(1)28928 58495 y FG(j)p Fj(C)63
b FG(j)p Fl(+)p FG(j)p Fj(D)26 b FG(j)32292 58995 y Ft(\))337
b(=)g(1)34986 57650 y Ff(i)35558 57601 y(\014)35558 58329
y(\014)35558 59056 y(\014)36299 58995 y Fp(\024)f Fo(\013)t
Ft(\(min)r Fp(fj)p Fo(C)87 b Fp(j)p Fo(;)202 b Fp(j)p
Fo(D)34 b Fp(jg)p Ft(\))0 62394 y Fz(Prop)39 b(osition)465
b(3.4)607 b Fr(Neither)431 b(2-TM)i(nor)h(2-cir)-62 b(cuit)431
b(obfusc)-62 b(ators)430 b(exist.)0 65178 y Fz(Pro)39
b(of:)1956 b Ft(W)-101 b(e)517 b(b)34 b(egin)518 b(b)-34
b(y)518 b(sho)-34 b(wing)519 b(that)g(2-TM)g(obfuscators)g(do)f(not)g
(exist.)879 b(Supp)34 b(ose,)547 b(for)518 b(sak)-34
b(e)518 b(of)g(con-)0 66684 y(tradiction,)536 b(that)511
b(there)f(exists)f(a)h(2-TM)g(obfuscator)h Fp(O)34 b
Ft(.)855 b(The)510 b(essence)f(of)h(this)g(pro)34 b(of,)536
b(and)511 b(in)e(fact)h(of)h(all)0 68189 y(the)520 b(imp)34
b(ossibilit)-34 b(y)519 b(pro)34 b(ofs)520 b(in)f(this)h(pap)34
b(er,)548 b(is)519 b(that)i(there)e(is)g(a)h(fundamen)-34
b(tal)521 b(di\013erence)e(b)34 b(et)-34 b(w)g(een)520
b(getting)0 69694 y(blac)-34 b(k-b)34 b(o)-34 b(x)380
b(access)f(to)h(a)g(function)h(and)g(getting)f(a)g(program)g(that)h
(computes)f(it,)385 b(no)380 b(matter)g(ho)-34 b(w)381
b(obfuscated:)0 71200 y(A)464 b(program)g(is)f(a)h(succinct)g
(description)g(of)g(the)g(function,)479 b(on)465 b(whic)-34
b(h)464 b(one)g(can)g(p)34 b(erform)463 b(computations)j(\(or)25394
74521 y(10)p eop
%%Page: 11 12
11 11 bop 0 1218 a Ft(run)458 b(other)g(programs\).)699
b(Of)458 b(course,)470 b(if)458 b(the)g(function)h(is)e(\(exactly\))h
(learnable)f(via)g(oracle)f(queries)h(\(i.e.,)470 b(one)0
2723 y(can)507 b(acquire)e(a)i(program)f(that)i(computes)f(the)g
(function)h(b)-34 b(y)507 b(querying)e(it)i(at)g(a)f(few)h(lo)34
b(cations\),)532 b(then)507 b(this)0 4229 y(di\013erence)h(disapp)34
b(ears.)854 b(Hence,)534 b(to)509 b(get)h(our)f(coun)-34
b(terexample,)535 b(w)-34 b(e)509 b(will)g(use)g(a)g(function)h(that)g
(cannot)g(b)34 b(e)0 5734 y(exactly)348 b(learned)g(with)i(oracle)e
(queries.)519 b(A)349 b(v)-34 b(ery)348 b(simple)g(example)g(of)h(suc)
-34 b(h)350 b(an)f(unlearnable)g(function)i(follo)-34
b(ws.)0 7240 y(F)-101 b(or)404 b(strings)h Fo(\013)t(;)202
b(\014)401 b Fp(2)336 b(f)p Ft(0)p Fo(;)202 b Ft(1)p
Fp(g)12610 6800 y Fj(k)13179 7240 y Ft(,)404 b(de\014ne)h(the)f(T)-101
b(uring)405 b(mac)-34 b(hine)18822 10250 y Fo(C)19688
10447 y Fj(\013;\014)21182 10250 y Ft(\()p Fo(x)p Ft(\))23087
9563 y Fl(def)23265 10250 y Ft(=)24655 8905 y Ff(n)25665
9616 y Fo(\014)1701 b(x)337 b Ft(=)f Fo(\013)25665 11121
y Ft(0)26271 10681 y Fj(k)28052 11121 y Ft(otherwise)1882
12860 y(W)-101 b(e)430 b(assume)i(that)g(on)f(input)i
Fo(x)p Ft(,)k Fo(C)18168 13057 y Fj(\013;\014)20093 12860
y Ft(runs)432 b(in)f(10)287 b Fp(\001)g(j)p Fo(x)p Fp(j)431
b Ft(steps)g(\(the)h(constan)-34 b(t)433 b(10)e(is)g(arbitrary\).)619
b(No)-34 b(w)432 b(w)-34 b(e)0 14366 y(will)462 b(de\014ne)i(a)e(TM)i
Fo(D)10431 14563 y Fj(\013;\014)12387 14366 y Ft(that,)479
b(giv)-34 b(en)463 b(the)g(co)34 b(de)462 b(of)h(a)g(TM)g
Fo(C)87 b Ft(,)477 b(can)463 b(distinguish)i(b)34 b(et)-34
b(w)g(een)463 b(the)h(case)e(that)i Fo(C)0 15871 y Ft(computes)370
b(the)g(same)g(function)h(as)e Fo(C)17447 16068 y Fj(\013;\014)19312
15871 y Ft(from)g(the)h(case)f(that)i Fo(C)457 b Ft(computes)370
b(the)g(same)f(function)i(as)f Fo(C)48088 16075 y Fj(\013)48692
15823 y Fy(0)48991 16075 y Fj(;\014)49826 15823 y Fy(0)50549
15871 y Ft(for)0 17377 y(an)-34 b(y)405 b(\()p Fo(\013)3540
16937 y FG(0)3851 17377 y Fo(;)202 b(\014)5140 16937
y FG(0)5450 17377 y Ft(\))337 b Fp(6)p Ft(=)g(\()p Fo(\013)t(;)202
b(\014)64 b Ft(\).)18883 20503 y Fo(D)19887 20700 y Fj(\013;\014)21381
20503 y Ft(\()p Fo(C)87 b Ft(\))23546 19816 y Fl(def)23724
20503 y Ft(=)25114 19158 y Ff(n)26124 19902 y Ft(1)1212
b Fo(C)87 b Ft(\()p Fo(\013)t Ft(\))339 b(=)d Fo(\014)26124
21407 y Ft(0)1212 b(otherwise)0 23147 y(\(Actually)-101
b(,)473 b(this)460 b(function)g(is)f(uncomputable.)706
b(Ho)-34 b(w)g(ev)g(er,)473 b(as)460 b(w)-34 b(e)460
b(shall)f(see)g(b)34 b(elo)-34 b(w,)473 b(w)-34 b(e)459
b(can)h(use)f(a)h(mo)34 b(di\014ed)0 24652 y(v)-34 b(ersion)346
b(of)g Fo(D)6386 24849 y Fj(\013;\014)8227 24652 y Ft(that)h(only)f
(considers)g(the)g(execution)g(of)h Fo(C)87 b Ft(\()p
Fo(\013)t Ft(\))348 b(for)e(p)34 b(oly)16 b(\()p Fo(k)38
b Ft(\))347 b(steps,)358 b(and)347 b(outputs)i(0)d(if)g
Fo(C)433 b Ft(do)34 b(es)0 26158 y(not)459 b(halt)g(within)g(that)h
(man)-34 b(y)459 b(steps,)472 b(for)458 b(some)h(\014xed)f(p)34
b(olynomial)458 b(p)34 b(oly)16 b(\()p Fp(\001)p Ft(\).)702
b(W)-101 b(e)458 b(will)g(ignore)g(this)g(issue)h(for)0
27663 y(no)-34 b(w,)405 b(and)g(elab)34 b(orate)403 b(on)i(it)f
(later.\))538 b(Note)405 b(that)g Fo(C)23600 27860 y
Fj(\013;\014)25499 27663 y Ft(and)g Fo(D)28860 27860
y Fj(\013;\014)30758 27663 y Ft(ha)-34 b(v)g(e)405 b(description)f
(size)f(\002\()p Fo(k)38 b Ft(\).)1882 29169 y(Consider)396
b(an)f(adv)-34 b(ersary)395 b Fo(A)p Ft(,)i(whic)-34
b(h,)398 b(giv)-34 b(en)395 b(t)-34 b(w)g(o)397 b(\(obfuscated\))g(TMs)
f(as)g(input,)i(simply)d(runs)h(the)g(second)0 30674
y(TM)459 b(on)g(the)g(\014rst)g(one.)702 b(That)459 b(is,)472
b Fo(A)p Ft(\()p Fo(C)19 b(;)202 b(D)34 b Ft(\))428 b(=)f
Fo(D)34 b Ft(\()p Fo(C)87 b Ft(\).)701 b(\(Actually)-101
b(,)472 b(lik)-34 b(e)458 b(w)-34 b(e)459 b(mo)34 b(di\014ed)459
b Fo(D)42343 30871 y Fj(\013;\014)44295 30674 y Ft(ab)34
b(o)-34 b(v)g(e,)472 b(w)-34 b(e)459 b(also)0 32180 y(will)394
b(mo)34 b(dify)394 b Fo(A)h Ft(to)g(only)f(run)h Fo(D)428
b Ft(on)395 b Fo(C)481 b Ft(for)395 b(p)34 b(oly)16 b(\()p
Fp(j)p Fo(C)87 b Fp(j)p Fo(;)202 b Fp(j)p Fo(D)34 b Fp(j)p
Ft(\))393 b(steps,)k(and)e(output)i(0)d(if)h Fo(D)428
b Ft(do)34 b(es)394 b(not)h(halt)g(in)g(that)0 33685
y(time.\))539 b(Th)-34 b(us,)405 b(for)f(an)-34 b(y)405
b Fo(\013)t(;)202 b(\014)401 b Fp(2)337 b(f)p Ft(0)p
Fo(;)202 b Ft(1)p Fp(g)17769 33245 y Fj(k)18337 33685
y Ft(,)17124 36408 y(Pr)g([)o Fo(A)p Ft(\()p Fp(O)34
b Ft(\()p Fo(C)22678 36605 y Fj(\013;\014)24174 36408
y Ft(\))p Fo(;)202 b Fp(O)34 b Ft(\()p Fo(D)27658 36605
y Fj(\013;\014)29152 36408 y Ft(\)\))337 b(=)g(1])f(=)h(1)15575
b(\(1\))1882 39131 y(Observ)-34 b(e)409 b(that)h(an)-34
b(y)410 b(p)34 b(oly)17 b(\()p Fo(k)38 b Ft(\)-time)410
b(algorithm)f Fo(S)480 b Ft(whic)-34 b(h)410 b(has)g(oracle)f(access)f
(to)i Fo(C)40403 39328 y Fj(\013;\014)42307 39131 y Ft(and)h
Fo(D)45674 39328 y Fj(\013;\014)47577 39131 y Ft(has)f(only)0
40637 y(exp)34 b(onen)-34 b(tially)412 b(small)h(probabilit)-34
b(y)413 b(\(for)g(a)g(random)g Fo(\013)418 b Ft(and)413
b Fo(\014)64 b Ft(\))413 b(of)g(querying)f(either)g(oracle)g(at)h(a)f
(p)34 b(oin)-34 b(t)414 b(where)0 42142 y(its)346 b(v)-67
b(alue)345 b(is)g(nonzero.)519 b(Hence,)356 b(if)346
b(w)-34 b(e)346 b(let)f Fo(Z)20022 42339 y Fj(k)20937
42142 y Ft(b)34 b(e)346 b(a)f(T)-101 b(uring)346 b(mac)-34
b(hine)346 b(that)h(alw)-34 b(a)g(ys)346 b(outputs)i(0)43440
41702 y Fj(k)44009 42142 y Ft(,)357 b(then)346 b(for)g(ev)-34
b(ery)0 43648 y(PPT)404 b Fo(S)70 b Ft(,)10414 43994
y Ff(\014)10414 44721 y(\014)10414 45449 y(\014)10818
45388 y Ft(Pr)12320 44043 y Ff(h)12892 45388 y Fo(S)13705
44888 y Fj(C)14376 45044 y Fc(\013;\014)15715 44888 y
Fj(;D)16748 45044 y Fc(\013;\014)18144 45388 y Ft(\(1)19221
44888 y Fj(k)19790 45388 y Ft(\))337 b(=)g(1)22484 44043
y Ff(i)23325 45388 y Fp(\000)269 b Ft(Pr)26039 44043
y Ff(h)26612 45388 y Fo(S)27425 44888 y Fj(Z)28063 45044
y Fc(k)28570 44888 y Fj(;D)29603 45044 y Fc(\013;\014)30999
45388 y Ft(\(1)32076 44888 y Fj(k)32645 45388 y Ft(\))337
b(=)f(1)35338 44043 y Ff(i)35911 43994 y(\014)35911 44721
y(\014)35911 45449 y(\014)36652 45388 y Fp(\024)g Ft(2)38537
44888 y FG(\000)p Fl(\012\()p Fj(k)24 b Fl(\))41249 45388
y Fo(;)8865 b Ft(\(2\))0 48061 y(where)405 b(the)h(probabilities)f(are)
g(tak)-34 b(en)406 b(o)-34 b(v)g(er)405 b Fo(\013)410
b Ft(and)c Fo(\014)469 b Ft(selected)404 b(uniformly)i(in)f
Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)39587 47621 y Fj(k)40561
48061 y Ft(and)406 b(the)g(coin)f(tosses)h(of)0 49566
y Fo(S)70 b Ft(.)539 b(On)404 b(the)g(other)h(hand,)g(b)-34
b(y)404 b(the)h(de\014nition)g(of)f Fo(A)h Ft(w)-34 b(e)404
b(ha)-34 b(v)g(e:)17606 52290 y(Pr)202 b([)p Fo(A)p Ft(\()p
Fp(O)34 b Ft(\()p Fo(Z)23122 52487 y Fj(k)23692 52290
y Ft(\))p Fo(;)202 b Fp(O)34 b Ft(\()p Fo(D)27176 52487
y Fj(\013;\014)28670 52290 y Ft(\)\))337 b(=)g(1])f(=)h(0)16057
b(\(3\))0 55013 y(The)405 b(com)-34 b(bination)405 b(of)f(Equations)h
(\(1\),)g(\(2\),)f(and)h(\(3\))g(con)-34 b(tradict)405
b(the)g(fact)f(that)h Fp(O)438 b Ft(is)404 b(a)g(2-TM)h(obfuscator.)
1882 56518 y(In)307 b(the)g(ab)34 b(o)-34 b(v)g(e)307
b(pro)34 b(of,)326 b(w)-34 b(e)308 b(ignored)f(the)g(fact)g(that)h(w)
-34 b(e)308 b(had)f(to)h(truncate)f(the)h(running)g(times)e(of)i
Fo(A)f Ft(and)g Fo(D)50169 56715 y Fj(\013;\014)51663
56518 y Ft(.)0 58024 y(When)478 b(doing)h(so,)496 b(w)-34
b(e)479 b(m)-34 b(ust)479 b(mak)-34 b(e)478 b(sure)g(that)h(Equations)g
(\(1\))g(and)g(\(3\))g(still)e(hold.)761 b(Equation)479
b(\(1\))g(in)-34 b(v)g(olv)g(es)0 59529 y(executing)535
b(\(a\))g Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(D)11489
59726 y Fj(\013;\014)12984 59529 y Ft(\))p Fo(;)202 b
Fp(O)34 b Ft(\()p Fo(C)16330 59726 y Fj(\013;\014)17825
59529 y Ft(\)\),)568 b(whic)-34 b(h)536 b(in)f(turn)h(amoun)-34
b(ts)537 b(to)e(executing)g(\(b\))h Fp(O)34 b Ft(\()p
Fo(D)44453 59726 y Fj(\013;\014)45947 59529 y Ft(\)\()p
Fp(O)g Ft(\()p Fo(C)49225 59726 y Fj(\013;\014)50721
59529 y Ft(\)\).)0 61035 y(By)492 b(de\014nition)j(\(b\))f(has)f(the)h
(same)f(functionalit)-34 b(y)494 b(as)g Fo(D)26905 61232
y Fj(\013;\014)28399 61035 y Ft(\()p Fp(O)34 b Ft(\()p
Fo(C)31206 61232 y Fj(\013;\014)32701 61035 y Ft(\)\),)516
b(whic)-34 b(h)494 b(in)f(turn)h(in)-34 b(v)g(olv)g(es)493
b(executing)0 62540 y(\(c\))575 b Fp(O)34 b Ft(\()p Fo(C)4392
62737 y Fj(\013;\014)5887 62540 y Ft(\)\()p Fo(\013)t
Ft(\).)1053 b(Y)-101 b(et)576 b(the)f(functionalit)-34
b(y)576 b(requiremen)-34 b(t)575 b(of)g(the)h(obfuscator)g
(de\014nition)g(assures)f(us)h(that)0 64045 y(\(c\))465
b(has)h(the)f(same)g(functionalit)-34 b(y)466 b(as)g
Fo(C)18996 64242 y Fj(\013;\014)20490 64045 y Ft(\()p
Fo(\013)t Ft(\).)723 b(By)464 b(the)i(p)34 b(olynomial)464
b(slo)-34 b(wdo)g(wn)467 b(prop)34 b(ert)-34 b(y)466
b(of)f(obfuscators,)0 65551 y(execution)440 b(\(c\))g(only)f(tak)-34
b(es)441 b(p)34 b(oly)16 b(\(10)294 b Fp(\001)e Fo(k)38
b Ft(\))397 b(=)f(p)34 b(oly)17 b(\()p Fo(k)38 b Ft(\))440
b(steps,)449 b(whic)-34 b(h)441 b(means)f(that)h Fo(D)39701
65748 y Fj(\013;\014)41195 65551 y Ft(\()p Fp(O)34 b
Ft(\()p Fo(C)44002 65748 y Fj(\013;\014)45498 65551 y
Ft(\)\))440 b(need)g(only)0 67056 y(run)350 b(for)f(p)34
b(oly)17 b(\()p Fo(k)38 b Ft(\))350 b(steps.)521 b(Th)-34
b(us,)362 b(again)349 b(applying)h(the)g(p)34 b(olynomial)349
b(slo)-34 b(wdo)g(wn)352 b(prop)34 b(ert)-34 b(y)-101
b(,)360 b(execution)349 b(\(b\))i(tak)-34 b(es)0 68562
y(p)34 b(oly)16 b(\()p Fo(k)38 b Ft(\))474 b(steps,)489
b(whic)-34 b(h)474 b(\014nally)e(implies)g(that)h Fo(A)f
Ft(need)h(only)f(run)h(for)f(p)34 b(oly)17 b(\()p Fo(k)38
b Ft(\))473 b(steps.)744 b(The)472 b(same)h(reasoning)25394
74521 y(11)p eop
%%Page: 12 13
12 12 bop 0 1218 a Ft(holds)402 b(for)f(Equation)h(\(3\),)g(using)f
Fo(Z)16536 1415 y Fj(k)17507 1218 y Ft(instead)g(of)h
Fo(C)23929 1415 y Fj(\013;\014)25424 1218 y Ft(.)25761
778 y Fl(5)26824 1218 y Ft(Note)f(that)h(all)f(the)g(p)34
b(olynomials)401 b(in)-34 b(v)g(olv)g(ed)402 b(are)e
Fr(\014xe)-62 b(d)0 2723 y Ft(once)404 b(w)-34 b(e)404
b(\014x)h(the)f(p)34 b(olynomial)404 b Fo(p)p Ft(\()p
Fp(\001)p Ft(\))g(of)h(the)f(p)34 b(olynomial)404 b(slo)-34
b(wdo)g(wn)406 b(prop)34 b(ert)-34 b(y)-101 b(.)1882
4229 y(The)474 b(pro)34 b(of)473 b(for)g(the)h(2-circuit)f(case)f(is)h
(v)-34 b(ery)473 b(similar)f(to)h(the)h(2-TM)g(case,)489
b(with)475 b(a)e(related,)490 b(but)474 b(sligh)-34 b(tly)0
5734 y(di\013eren)g(t)446 b(subtlet)-34 b(y)-101 b(.)663
b(Supp)34 b(ose,)457 b(for)445 b(sak)-34 b(e)445 b(of)h(con)-34
b(tradiction,)457 b(that)446 b Fp(O)479 b Ft(is)445 b(a)h(2-circuit)e
(obfuscator.)664 b(F)-101 b(or)445 b Fo(k)f Fp(2)406
b Fg(N)0 7240 y Ft(and)415 b Fo(\013)t(;)202 b(\014)417
b Fp(2)354 b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)8913 6800
y Fj(k)9482 7240 y Ft(,)416 b(de\014ne)f Fo(Z)14574 7437
y Fj(k)15143 7240 y Ft(,)h Fo(C)16762 7437 y Fj(\013;\014)18671
7240 y Ft(and)f Fo(D)22042 7437 y Fj(\013;\014)23950
7240 y Ft(in)f(the)h(same)f(w)-34 b(a)g(y)415 b(as)f(ab)34
b(o)-34 b(v)g(e)415 b(but)g(as)f(circuits)g(rather)g(than)0
8745 y(TMs,)332 b(and)315 b(de\014ne)g(an)f(adv)-34 b(ersary)314
b Fo(A)g Ft(b)-34 b(y)314 b Fo(A)p Ft(\()p Fo(C)19 b(;)202
b(D)34 b Ft(\))337 b(=)f Fo(D)34 b Ft(\()p Fo(C)87 b
Ft(\).)509 b(\(Note)314 b(that)i(the)e(issues)g(of)g
Fo(A)g Ft(and)h Fo(D)45266 8942 y Fj(\013;\014)46760
8745 y Ft('s)f(running)0 10250 y(times)483 b(go)g(a)-34
b(w)g(a)g(y)484 b(in)f(this)g(setting,)503 b(since)482
b(circuits)h(can)g(alw)-34 b(a)g(ys)483 b(b)34 b(e)483
b(ev)-67 b(aluated)482 b(in)h(time)g(p)34 b(olynomial)482
b(in)h(their)0 11756 y(size.\))528 b(The)376 b(new)f(subtlet)-34
b(y)376 b(here)e(is)h(that)h(the)f(de\014nition)h(of)f
Fo(A)g Ft(as)g Fo(A)p Ft(\()p Fo(C)19 b(;)202 b(D)34
b Ft(\))337 b(=)g Fo(D)34 b Ft(\()p Fo(C)87 b Ft(\))375
b(only)f(mak)-34 b(es)375 b(sense)g(when)0 13261 y(the)363
b(input)h(length)f(of)f Fo(D)396 b Ft(is)362 b(larger)g(than)h(the)g
(size)f(of)h Fo(C)449 b Ft(\(note)363 b(that)h(one)f(can)f(alw)-34
b(a)g(ys)363 b(pad)h Fo(C)449 b Ft(to)363 b(a)f(larger)g(size\).)0
14767 y(Th)-34 b(us,)423 b(for)418 b(the)g(analogues)h(of)f(Equations)h
(\(1\))f(and)h(\(3\))g(to)f(hold,)k(the)c(input)h(length)g(of)f
Fo(D)42370 14964 y Fj(\013;\014)44282 14767 y Ft(m)-34
b(ust)419 b(b)34 b(e)418 b(larger)0 16272 y(than)406
b(the)f(sizes)g(of)g(the)g Fr(obfusc)-62 b(ations)401
b Ft(of)k Fo(C)20195 16469 y Fj(\013;\014)22095 16272
y Ft(and)h Fo(Z)25280 16469 y Fj(k)25849 16272 y Ft(.)541
b(Ho)-34 b(w)g(ev)g(er,)405 b(b)-34 b(y)405 b(the)g(p)34
b(olynomial)405 b(slo)-34 b(wdo)g(wn)407 b(prop)34 b(ert)-34
b(y)0 17778 y(of)404 b(obfuscators,)i(it)e(su\016ces)g(to)g(let)g
Fo(D)17667 17975 y Fj(\013;\014)19565 17778 y Ft(ha)-34
b(v)g(e)405 b(input)g(length)g(p)34 b(oly)16 b(\()p Fo(k)38
b Ft(\))405 b(and)g(the)g(pro)34 b(of)404 b(w)-34 b(orks)405
b(as)f(b)34 b(efore.)p 51114 19283 886 886 v 0 22529
a Fn(3.2)1495 b(Obfuscating)500 b(one)f(TM/circuit)0
24817 y Ft(Our)437 b(approac)-34 b(h)438 b(to)f(extending)g(the)h(t)-34
b(w)g(o-program)438 b(obfuscation)g(imp)34 b(ossibilit)-34
b(y)437 b(results)g(to)g(the)h(one-program)0 26323 y(de\014nitions)446
b(is)f(to)h(com)-34 b(bine)446 b(the)f(t)-34 b(w)g(o)447
b(programs)f(constructed)g(ab)34 b(o)-34 b(v)g(e)445
b(in)-34 b(to)446 b(one.)662 b(This)446 b(will)f(w)-34
b(ork)446 b(in)f(a)g(quite)0 27828 y(straigh)-34 b(tforw)g(ard)406
b(manner)f(for)f(TM)h(obfuscators,)g(but)g(will)f(require)e(new)j
(ideas)f(for)g(circuit)g(obfuscators.)0 31030 y Fz(Com)-39
b(bining)510 b(functions)f(and)g(programs.)1212 b Ft(F)-101
b(or)442 b(functions,)454 b(TMs,)e(or)442 b(circuits)g
Fo(f)40707 31212 y Fl(0)41233 31030 y Fo(;)202 b(f)42365
31212 y Fl(1)43291 31030 y Ft(:)400 b Fo(X)496 b Fp(!)401
b Fo(Y)269 b Ft(,)451 b(de\014ne)0 32800 y(their)462
b Fr(c)-62 b(ombination)460 b Fo(f)10424 32982 y Fl(0)10951
32800 y Ft(#)p Fo(f)12554 32982 y Fl(1)13514 32800 y
Ft(:)433 b Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)308 b(\002)g
Fo(X)529 b Fp(!)434 b Fo(Y)731 b Ft(b)-34 b(y)463 b(\()p
Fo(f)26226 32982 y Fl(0)26752 32800 y Ft(#)p Fo(f)28355
32982 y Fl(1)28882 32800 y Ft(\)\()p Fo(b;)202 b(x)p
Ft(\))32356 32113 y Fl(def)32533 32800 y Ft(=)486 b Fo(f)34555
32997 y Fj(b)35013 32800 y Ft(\()p Fo(x)p Ft(\).)714
b(Con)-34 b(v)g(ersely)-101 b(,)476 b(if)463 b(w)-34
b(e)462 b(are)g(giv)-34 b(en)0 34305 y(a)437 b(TM)g(\(resp.,)444
b(circuit\))436 b Fo(C)478 b Ft(:)391 b Fp(f)p Ft(0)p
Fo(;)202 b Ft(1)p Fp(g)291 b(\002)g Fo(X)486 b Fp(!)391
b Fo(Y)269 b Ft(,)444 b(w)-34 b(e)437 b(can)g(e\016cien)-34
b(tly)436 b(decomp)34 b(ose)436 b Fo(C)524 b Ft(in)-34
b(to)437 b Fo(C)43343 34487 y Fl(0)43869 34305 y Ft(#)p
Fo(C)45745 34487 y Fl(1)46708 34305 y Ft(b)-34 b(y)437
b(setting)0 36142 y Fo(C)866 36339 y Fj(b)1324 36142
y Ft(\()p Fo(x)p Ft(\))3259 35455 y Fl(def)3437 36142
y Ft(=)477 b Fo(C)87 b Ft(\()p Fo(b;)202 b(x)p Ft(\);)471
b(note)450 b(that)g Fo(C)15588 36324 y Fl(0)16563 36142
y Ft(and)g Fo(C)19831 36324 y Fl(1)20806 36142 y Ft(ha)-34
b(v)g(e)450 b(size)e(and)i(running)g(time)f(essen)-34
b(tially)449 b(the)g(same)g(as)g(that)i(of)0 37648 y
Fo(C)87 b Ft(.)559 b(Observ)-34 b(e)411 b(that)h(ha)-34
b(ving)412 b(oracle)e(access)g(to)i(a)f(com)-34 b(bined)411
b(function)i Fo(f)33441 37830 y Fl(0)33967 37648 y Ft(#)p
Fo(f)35570 37830 y Fl(1)36508 37648 y Ft(is)d(equiv)-67
b(alen)-34 b(t)411 b(to)g(ha)-34 b(ving)412 b(oracle)0
39153 y(access)403 b(to)i Fo(f)5657 39335 y Fl(0)6587
39153 y Ft(and)g Fo(f)9537 39335 y Fl(1)10467 39153 y
Ft(individually)-101 b(.)0 41987 y Fz(Theorem)464 b(3.5)606
b Fr(TM)434 b(obfusc)-62 b(ators)430 b(do)j(not)g(exist.)0
44821 y Fz(Pro)39 b(of)556 b(Sk)-39 b(etc)g(h:)1812 b
Ft(Supp)34 b(ose,)504 b(for)484 b(sak)-34 b(e)484 b(of)g(con)-34
b(tradiction,)505 b(that)485 b(there)f(exists)f(a)h(TM)g(obfuscator)i
Fp(O)34 b Ft(.)777 b(F)-101 b(or)0 46326 y Fo(\013)t(;)202
b(\014)401 b Fp(2)337 b(f)p Ft(0)p Fo(;)202 b Ft(1)p
Fp(g)6513 45886 y Fj(k)7081 46326 y Ft(,)317 b(let)294
b Fo(C)10242 46523 y Fj(\013;\014)11737 46326 y Ft(,)317
b Fo(D)13395 46523 y Fj(\013;\014)14889 46326 y Ft(,)f(and)296
b Fo(Z)18617 46523 y Fj(k)19481 46326 y Ft(b)34 b(e)294
b(the)i(TMs)f(de\014ned)h(in)f(the)g(pro)34 b(of)295
b(of)h(Prop)34 b(osition)295 b(3.4.)501 b(Com)-34 b(bining)0
47832 y(these,)404 b(w)-34 b(e)405 b(get)f(the)g(TMs)h
Fo(F)12981 48029 y Fj(\013;\014)14813 47832 y Ft(=)336
b Fo(C)16958 48029 y Fj(\013;\014)18453 47832 y Ft(#)p
Fo(D)20467 48029 y Fj(\013;\014)22365 47832 y Ft(and)405
b Fo(G)25675 48029 y Fj(\013;\014)27506 47832 y Ft(=)337
b Fo(Z)29613 48029 y Fj(k)30182 47832 y Ft(#)p Fo(C)32058
48029 y Fj(\013;\014)33553 47832 y Ft(.)1882 49337 y(W)-101
b(e)456 b(consider)g(an)h(adv)-34 b(ersary)456 b Fo(A)g
Ft(analogous)i(to)e(the)h(one)g(in)f(the)h(pro)34 b(of)457
b(of)g(Prop)34 b(osition)456 b(3.4,)469 b(augmen)-34
b(ted)0 50843 y(to)403 b(\014rst)h(decomp)34 b(ose)402
b(the)i(program)e(it)h(is)g(fed.)538 b(That)404 b(is,)f(on)g(input)h(a)
f(TM)g Fo(F)168 b Ft(,)404 b(algorithm)f Fo(A)g Ft(\014rst)g(decomp)34
b(oses)0 52348 y Fo(F)574 b Ft(in)-34 b(to)406 b Fo(F)4591
52530 y Fl(0)5117 52348 y Ft(#)p Fo(F)6906 52530 y Fl(1)7838
52348 y Ft(and)g(then)g(outputs)i Fo(F)18190 52530 y
Fl(1)18716 52348 y Ft(\()p Fo(F)19966 52530 y Fl(0)20493
52348 y Ft(\).)542 b(\(As)406 b(in)f(the)h(pro)34 b(of)405
b(of)h(Prop)34 b(osition)406 b(3.4,)f Fo(A)g Ft(actually)g(should)h(b)
34 b(e)0 53853 y(mo)g(di\014ed)339 b(to)f(run)h(in)f(time)g(p)34
b(oly)16 b(\()p Fp(j)p Fo(F)168 b Fp(j)p Ft(\).\))518
b(Let)338 b Fo(S)408 b Ft(b)34 b(e)338 b(the)g(PPT)h(sim)-34
b(ulator)338 b(for)h Fo(A)f Ft(guaran)-34 b(teed)339
b(b)-34 b(y)339 b(De\014nition)f(2.1.)0 55359 y(Just)405
b(as)f(in)g(the)h(pro)34 b(of)404 b(of)h(Prop)34 b(osition)404
b(3.4,)f(w)-34 b(e)405 b(ha)-34 b(v)g(e:)9450 58082 y(Pr)202
b([)p Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(F)14918 58279
y Fj(\013;\014)16413 58082 y Ft(\)\))338 b(=)e(1])h(=)f(1)404
b(and)607 b(Pr)202 b([)o Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p
Fo(G)30741 58279 y Fj(\013;\014)32236 58082 y Ft(\)\))338
b(=)e(1])1107 b(=)g(0)13731 58895 y Ff(\014)13731 59622
y(\014)13731 60349 y(\014)14135 60288 y Ft(Pr)15637 58943
y Ff(h)16209 60288 y Fo(S)17022 59788 y Fj(F)17617 59944
y Fc(\013;\014)19013 60288 y Ft(\(1)20090 59788 y Fj(k)20659
60288 y Ft(\))337 b(=)g(1)23353 58943 y Ff(i)24194 60288
y Fp(\000)270 b Ft(Pr)26908 58943 y Ff(h)27481 60288
y Fo(S)28294 59788 y Fj(G)29027 59944 y Fc(\013;\014)30422
60288 y Ft(\(1)31499 59788 y Fj(k)32068 60288 y Ft(\))337
b(=)g(1)34762 58943 y Ff(i)35334 58895 y(\014)35334 59622
y(\014)35334 60349 y(\014)36845 60288 y Fp(\024)1107
b Ft(2)39501 59788 y FG(\000)p Fl(\012\()p Fj(k)24 b
Fl(\))42213 60288 y Fo(;)0 63460 y Ft(where)438 b(the)g(probabilities)g
(are)f(tak)-34 b(en)438 b(o)-34 b(v)g(er)438 b(uniformly)g(selected)f
Fo(\013)t(;)202 b(\014)457 b Fp(2)393 b(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)37396 63020 y Fj(k)37965 63460 y Ft(,)446
b(and)438 b(the)g(coin)g(tosses)g(of)g Fo(A)p Ft(,)0
64965 y Fo(S)70 b Ft(,)404 b(and)h Fp(O)34 b Ft(.)538
b(This)405 b(con)-34 b(tradicts)405 b(De\014nition)f(2.1.)28777
b Fb(2)p 0 66020 20800 45 v 1382 66737 a Fw(5)1843 67160
y Fx(Another,)287 b(ev)-28 b(en)274 b(more)g(minor)f(subtlet)-28
b(y)275 b(that)e(w)-28 b(e)273 b(ignored)h(is)f(that,)288
b(strictly)274 b(sp)28 b(eaking,)289 b Fi(A)273 b Fx(only)h(has)g
(running)e(time)j(p)28 b(olynomial)0 68378 y(in)403 b(the)f
(description)h(of)f(the)h Fk(obfusc)-52 b(ations)402
b Fx(of)g Fi(C)19180 68496 y Fc(\013;\014)20519 68378
y Fx(,)418 b Fi(D)22065 68496 y Fc(\013;\014)23405 68378
y Fx(,)g(and)402 b Fi(Z)26856 68496 y Fc(k)27364 68378
y Fx(,)418 b(whic)-28 b(h)401 b(could)i(conceiv)-57 b(ably)404
b(b)28 b(e)403 b(shorter)f(than)g(the)g(original)0 69595
y(TM)384 b(descriptions.)583 b(But)384 b(a)g(coun)-28
b(ting)384 b(argumen)-28 b(t)383 b(sho)-28 b(ws)382 b(that)i(for)g(all)
g(but)g(an)g(exp)28 b(onen)-28 b(tially)386 b(small)f(fraction)f(of)f
(pairs)h(\()p Fi(\013)q(;)171 b(\014)52 b Fx(\))355 b
Fv(2)0 70813 y(f)p Fx(0)p Fi(;)171 b Fx(1)p Fv(g)2503
70390 y Fc(k)3239 70813 y Fv(\002)228 b(f)p Fx(0)p Fi(;)171
b Fx(1)p Fv(g)6766 70390 y Fc(k)7274 70813 y Fx(,)341
b Fv(O)28 b Fx(\()p Fi(C)9869 70931 y Fc(\013;\014)11210
70813 y Fx(\))341 b(and)g Fv(O)28 b Fx(\()p Fi(D)16023
70931 y Fc(\013;\014)17364 70813 y Fx(\))342 b(m)-28
b(ust)341 b(ha)-28 b(v)g(e)341 b(description)g(size)i(\012\()p
Fi(k)29 b Fx(\).)25394 74521 y Ft(12)p eop
%%Page: 13 14
13 13 bop 1882 1218 a Ft(There)517 b(is)f(a)h(di\016cult)-34
b(y)518 b(in)f(trying)f(to)i(carry)e(out)h(the)h(ab)34
b(o)-34 b(v)g(e)517 b(argumen)-34 b(t)518 b(in)f(the)g(circuit)f
(setting.)878 b(\(This)0 2723 y(di\016cult)-34 b(y)589
b(is)f(related)g(to)g(\(but)i(more)e(serious)g(than\))i(the)f(same)f
(subtlet)-34 b(y)590 b(regarding)e(the)h(circuit)e(setting)0
4229 y(discussed)493 b(earlier.\))800 b(In)492 b(the)h(ab)34
b(o)-34 b(v)g(e)492 b(pro)34 b(of,)514 b(the)492 b(adv)-34
b(ersary)492 b Fo(A)p Ft(,)513 b(on)493 b(input)g Fp(O)34
b Ft(\()p Fo(F)38058 4426 y Fj(\013;\014)39553 4229 y
Ft(\),)514 b(attempts)493 b(to)g(ev)-67 b(aluate)0 5734
y Fo(F)779 5916 y Fl(1)1305 5734 y Ft(\()p Fo(F)2555
5916 y Fl(0)3082 5734 y Ft(\),)459 b(where)447 b Fo(F)8676
5916 y Fl(0)9202 5734 y Ft(#)p Fo(F)10991 5916 y Fl(1)11927
5734 y Ft(=)409 b Fp(O)34 b Ft(\()p Fo(F)15528 5931 y
Fj(\013;\014)17023 5734 y Ft(\))410 b(=)f Fp(O)34 b Ft(\()p
Fo(C)21592 5931 y Fj(\013;\014)23087 5734 y Ft(#)p Fo(D)25101
5931 y Fj(\013;\014)26595 5734 y Ft(\).)669 b(In)448
b(order)g(for)f(this)i(to)f(mak)-34 b(e)447 b(sense)h(in)g(the)g
(circuit)0 7240 y(setting,)392 b(the)d(size)f(of)h(the)g(circuit)e
Fo(F)16631 7422 y Fl(0)17546 7240 y Ft(m)-34 b(ust)390
b(b)34 b(e)388 b(at)h(most)g(the)g(input)h(length)f(of)g
Fo(F)37709 7422 y Fl(1)38623 7240 y Ft(\(whic)-34 b(h)390
b(is)e(the)h(same)g(as)g(the)0 8745 y(input)436 b(length)e(of)h
Fo(D)9411 8942 y Fj(\013;\014)10905 8745 y Ft(\).)630
b(But,)441 b(since)434 b(the)h(output)h Fo(F)25024 8927
y Fl(0)25551 8745 y Ft(#)p Fo(F)27340 8927 y Fl(1)28300
8745 y Ft(of)f(the)g(obfuscator)g(can)g(b)34 b(e)434
b(p)34 b(olynomially)433 b(larger)0 10250 y(than)f(its)f(input)i
Fo(C)8698 10447 y Fj(\013;\014)10193 10250 y Ft(#)p Fo(D)12207
10447 y Fj(\013;\014)13701 10250 y Ft(,)k(w)-34 b(e)432
b(ha)-34 b(v)g(e)431 b(no)h(suc)-34 b(h)432 b(guaran)-34
b(tee.)620 b(F)-101 b(urthermore,)438 b(note)431 b(that)i(if)e(w)-34
b(e)431 b(compute)h Fo(F)51137 10432 y Fl(0)51663 10250
y Ft(,)0 12154 y Fo(F)779 12336 y Fl(1)1748 12154 y Ft(in)442
b(the)h(w)-34 b(a)g(y)443 b(w)-34 b(e)443 b(describ)34
b(ed)442 b(ab)34 b(o)-34 b(v)g(e)442 b(\(i.e.,)451 b
Fo(F)22107 12351 y Fj(b)22565 12154 y Ft(\()p Fo(x)p
Ft(\))24496 11467 y Fl(def)24673 12154 y Ft(=)473 b Fp(O)34
b Ft(\()p Fo(F)28338 12351 y Fj(\013;\014)29833 12154
y Ft(\)\()p Fo(b;)202 b(x)p Ft(\)\))443 b(then)h(w)-34
b(e'll)441 b(ha)-34 b(v)g(e)443 b Fp(j)p Fo(F)43493 12336
y Fl(0)44019 12154 y Fp(j)400 b Ft(=)h Fp(j)p Fo(F)47216
12336 y Fl(1)47742 12154 y Fp(j)442 b Ft(and)h(so)0 13660
y Fo(F)779 13842 y Fl(0)1709 13660 y Ft(will)404 b(necessarily)f(b)34
b(e)403 b(larger)h(than)h Fo(F)18741 13842 y Fl(1)19267
13660 y Ft('s)f(input)h(length.)1882 15165 y(T)-101 b(o)414
b(get)g(around)h(this,)i(w)-34 b(e)414 b(mo)34 b(dify)414
b Fo(D)19422 15362 y Fj(\013;\014)21330 15165 y Ft(in)g(a)g(w)-34
b(a)g(y)415 b(that)g(will)f(allo)-34 b(w)414 b Fo(A)p
Ft(,)i(when)f(giv)-34 b(en)414 b Fo(D)43335 15362 y Fj(\013;\014)45243
15165 y Ft(and)h(a)f(circuit)0 16671 y Fo(C)87 b Ft(,)354
b(to)342 b(test)h(whether)g Fo(C)87 b Ft(\()p Fo(\013)t
Ft(\))338 b(=)f Fo(\014)405 b Ft(ev)-34 b(en)342 b(when)h
Fo(C)429 b Ft(is)342 b(larger)e(than)k(the)e(input)h(length)g(of)f
Fo(D)40894 16868 y Fj(\013;\014)42388 16671 y Ft(.)518
b(Of)342 b(course,)354 b(oracle)0 18176 y(access)452
b(to)i Fo(D)6166 18373 y Fj(\013;\014)8113 18176 y Ft(should)g(not)g
(rev)-34 b(eal)452 b Fo(\013)458 b Ft(and)453 b Fo(\014)64
b Ft(,)465 b(b)34 b(ecause)453 b(w)-34 b(e)453 b(do)h(not)f(w)-34
b(an)g(t)455 b(the)f(sim)-34 b(ulator)453 b Fo(S)524
b Ft(to)453 b(b)34 b(e)453 b(able)g(to)0 19681 y(test)k(whether)g
Fo(C)87 b Ft(\()p Fo(\013)t Ft(\))425 b(=)e Fo(\014)520
b Ft(giv)-34 b(en)456 b(just)i(oracle)d(access)g(to)i
Fo(C)543 b Ft(and)457 b Fo(D)32003 19878 y Fj(\013;\014)33497
19681 y Ft(.)695 b(W)-101 b(e)456 b(will)f(construct)i(suc)-34
b(h)457 b(functions)0 21187 y Fo(D)1004 21384 y Fj(\013;\014)2902
21187 y Ft(based)405 b(on)f(pseudorandom)i(functions)g([GGM86].)0
23807 y Fz(Lemma)464 b(3.6)606 b Fr(If)437 b(one-way)g(functions)e
(exist,)i(then)g(for)g(every)f Fo(k)381 b Fp(2)344 b
Fg(N)438 b Fr(and)f Fo(\013)t(;)202 b(\014)408 b Fp(2)343
b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)42029 23367 y Fj(k)42598
23807 y Fr(,)438 b(ther)-62 b(e)436 b(is)h(a)g(distri-)0
25312 y(bution)432 b Fp(D)4652 25509 y Fj(\013;\014)6580
25312 y Fr(on)i(cir)-62 b(cuits)431 b(such)h(that:)1433
27635 y(1.)605 b(Every)433 b Fo(D)370 b Fp(2)337 b Ft(Supp)q(\()p
Fp(D)13119 27832 y Fj(\013;\014)14614 27635 y Ft(\))434
b Fr(is)f(a)g(cir)-62 b(cuit)432 b(of)h(size)f Ft(p)34
b(oly)17 b(\()p Fo(k)38 b Ft(\))p Fr(.)1433 30066 y(2.)605
b(Ther)-62 b(e)347 b(is)g(a)g(p)-62 b(olynomial-time)344
b(algorithm)h Fo(A)j Fr(such)e(that)g(for)g(any)i(cir)-62
b(cuit)345 b Fo(C)87 b Fr(,)365 b(and)347 b(any)g Fo(D)370
b Fp(2)336 b Ft(Supp)r(\()p Fp(D)49662 30263 y Fj(\013;\014)51157
30066 y Ft(\))p Fr(,)3030 31571 y Fo(A)3939 31131 y Fj(D)4793
31571 y Ft(\()p Fo(C)19 b(;)202 b Ft(1)7294 31131 y Fj(k)7864
31571 y Ft(\))337 b(=)f(1)434 b Fr(i\013)g Fo(C)87 b
Ft(\()p Fo(\013)t Ft(\))338 b(=)e Fo(\014)64 b Fr(.)1433
34168 y(3.)605 b(F)-93 b(or)369 b(any)f(PPT)i Fo(S)70
b Fr(,)381 b Ft(Pr)13435 33187 y Ff(\002)13940 34168
y Fo(S)14753 33728 y Fj(D)15606 34168 y Ft(\(1)16683
33728 y Fj(k)17253 34168 y Ft(\))337 b(=)f Fo(\013)20119
33187 y Ff(\003)20962 34168 y Ft(=)g(neg)17 b(\()p Fo(k)38
b Ft(\))p Fr(,)383 b(wher)-62 b(e)368 b(the)f(pr)-62
b(ob)g(ability)365 b(is)j(taken)g(over)f Fo(\013)t(;)202
b(\014)47079 33481 y Fd(R)46756 34168 y Fp( )128 b(f)p
Ft(0)p Fo(;)202 b Ft(1)p Fp(g)51059 33728 y Fj(k)51628
34168 y Fr(,)3030 35962 y Fo(D)4660 35275 y Fd(R)4337
35962 y Fp( )269 b(D)6753 36159 y Fj(\013;\014)8248 35962
y Fr(,)433 b(and)g(the)g(c)-62 b(oin)433 b(tosses)e(of)i
Fo(S)70 b Fr(.)0 38582 y Fz(Pro)39 b(of:)1388 b Ft(Basically)-101
b(,)379 b(the)e(construction)f(implemen)-34 b(ts)377
b(a)e(priv)-67 b(ate-k)-34 b(ey)375 b(\\homomorphic)h(encryption")g(sc)
-34 b(heme.)0 40087 y(More)366 b(precisely)-101 b(,)373
b(the)367 b(functions)h(in)f Fp(D)17813 40284 y Fj(\013;\014)19675
40087 y Ft(will)f(consist)h(of)g(three)g(parts.)527 b(The)367
b(\014rst)h(part)f(giv)-34 b(es)366 b(out)i(an)f(encryp-)0
41593 y(tion)479 b(of)h(the)f(bits)g(of)g Fo(\013)484
b Ft(\(under)c(some)f(priv)-67 b(ate-k)-34 b(ey)477 b(encryption)i(sc)
-34 b(heme\).)764 b(The)479 b(second)g(part)g(pro)-34
b(vides)479 b(the)0 43098 y(abilit)-34 b(y)375 b(to)h(p)34
b(erform)375 b(binary)g(Bo)34 b(olean)374 b(op)34 b(erations)376
b(on)f(encrypted)h(bits,)381 b(and)376 b(the)g(third)f(part)h(tests)g
(whether)g(a)0 44603 y(sequence)439 b(of)h(encryptions)h(consists)f(of)
g(encryptions)g(of)h(the)f(bits)g(of)g Fo(\014)64 b Ft(.)646
b(These)440 b(op)34 b(erations)440 b(will)f(enable)h(one)0
46109 y(to)375 b(e\016cien)-34 b(tly)373 b(test)i(whether)g(a)f(giv)-34
b(en)374 b(circuit)f Fo(C)461 b Ft(satis\014es)375 b
Fo(C)87 b Ft(\()p Fo(\013)t Ft(\))338 b(=)f Fo(\014)64
b Ft(,)379 b(while)374 b(k)-34 b(eeping)374 b Fo(\013)379
b Ft(and)c Fo(\014)438 b Ft(hidden)375 b(when)0 47614
y(only)404 b(oracle)f(access)g(to)i Fo(C)491 b Ft(and)405
b Fo(D)15947 47811 y Fj(\013;\014)17845 47614 y Ft(is)f(pro)-34
b(vided.)1882 49120 y(W)-101 b(e)282 b(b)34 b(egin)282
b(with)h(an)-34 b(y)283 b(one-bit)g(\(probabilistic\))g(priv)-67
b(ate-k)-34 b(ey)281 b(encryption)h(sc)-34 b(heme)283
b(\(Enc)p Fo(;)202 b Ft(Dec)o(\))283 b(that)g(satis\014es)0
50625 y(indistinguishabilit)-34 b(y)389 b(under)f Fr(chosen)417
b(plaintext)500 b Ft(and)388 b Fr(nonadaptive)416 b(chosen)g
(ciphertext)500 b Ft(attac)-34 b(ks.)534 b(Informally)-101
b(,)0 52131 y(this)480 b(means)h(that)g(an)f(encryption)g(of)g(0)g
(should)h(b)34 b(e)479 b(indistinguishable)j(from)e(an)g(encryption)g
(of)g(1)g(ev)-34 b(en)480 b(for)0 53636 y(adv)-34 b(ersaries)439
b(that)h(ha)-34 b(v)g(e)440 b(access)f(to)g(encryption)h(and)g
(decryption)g(oracles)e(prior)h(to)h(receiving)d(the)j(c)-34
b(hallenge)0 55142 y(ciphertext,)498 b(and)480 b(access)f(to)h(just)h
(an)f(encryption)g(oracle)f(after)g(receiving)g(the)h(c)-34
b(hallenge)479 b(ciphertext.)765 b(\(See)0 56647 y([KY00])445
b(for)i(formal)f(de\014nitions.\))666 b(W)-101 b(e)445
b(note)i(that)g(suc)-34 b(h)447 b(encryptions)f(sc)-34
b(hemes)447 b(exist)e(if)h(one-w)-34 b(a)g(y)447 b(functions)0
58323 y(exist;)391 b(indeed,)e(the)d(\\standard")h(encryption)f(sc)-34
b(heme)386 b(Enc)27855 58511 y Fj(K)28766 58323 y Ft(\()p
Fo(b)p Ft(\))338 b(=)e(\()p Fo(r)-34 b(;)202 b(f)33961
58511 y Fj(K)34873 58323 y Ft(\()p Fo(r)34 b Ft(\))233
b Fp(\010)f Fo(b)p Ft(\),)389 b(where)d Fo(r)44133 57636
y Fw(R)43820 58323 y Fp( )233 b(f)p Ft(0)p Fo(;)202 b
Ft(1)p Fp(g)48228 57883 y FG(j)p Fj(K)63 b FG(j)50047
58323 y Ft(and)0 59829 y Fo(f)593 60017 y Fj(K)1909 59829
y Ft(is)404 b(a)g(pseudorandom)i(function,)f(has)g(this)f(prop)34
b(ert)-34 b(y)-101 b(.)1882 61334 y(No)-34 b(w)390 b(w)-34
b(e)390 b(consider)f(a)g(\\homomorphic)i(encryption")e(algorithm)h
(Hom,)i(whic)-34 b(h)390 b(tak)-34 b(es)390 b(as)g(input)g(a)g(priv)-67
b(ate-)0 62840 y(k)-34 b(ey)313 b Fo(K)400 b Ft(and)314
b(t)-34 b(w)g(o)314 b(ciphertexts)f Fo(c)g Ft(and)h Fo(d)f
Ft(\(w.r.t.)1016 b(this)314 b(k)-34 b(ey)312 b Fo(K)87
b Ft(\),)332 b(and)314 b(a)f(binary)g(b)34 b(o)g(olean)313
b(op)34 b(eration)313 b Fp(\014)g Ft(\(sp)34 b(eci\014ed)0
64345 y(b)-34 b(y)405 b(its)f(2)269 b Fp(\002)g Ft(2)404
b(truth)i(table\).)539 b(W)-101 b(e)403 b(de\014ne)14014
67186 y(Hom)16539 67374 y Fj(K)17451 67186 y Ft(\()p
Fo(c;)202 b(d;)g Fp(\014)p Ft(\))21838 66498 y Fl(def)22016
67186 y Ft(=)447 b(Enc)25443 67374 y Fj(K)26355 67186
y Ft(\(Dec)28829 67374 y Fj(K)29741 67186 y Ft(\()p Fo(c)p
Ft(\))269 b Fp(\014)h Ft(Dec)34693 67374 y Fj(K)35604
67186 y Ft(\()p Fo(d)p Ft(\)\))p Fo(:)0 69694 y Ft(It)421
b(can)f(b)34 b(e)420 b(sho)-34 b(wn)423 b(that)e(suc)-34
b(h)422 b(an)f(encryption)g(sc)-34 b(heme)420 b(retains)h(its)f
(securit)-34 b(y)421 b(ev)-34 b(en)420 b(if)h(the)g(adv)-34
b(ersary)420 b(is)g(giv)-34 b(en)0 71200 y(access)403
b(to)i(a)f(Hom)g(oracle.)538 b(This)404 b(is)g(formalized)g(in)g(the)h
(follo)-34 b(wing)405 b(claim:)25394 74521 y(13)p eop
%%Page: 14 15
14 14 bop 0 1218 a Fz(Claim)464 b(3.7)606 b Fr(F)-93
b(or)434 b(every)e(PPT)i Fo(A)p Fr(,)5650 2910 y Ff(\014)5650
3638 y(\014)6054 3941 y Ft(Pr)7556 2959 y Ff(\002)8061
3941 y Fo(A)8970 3440 y Fl(Hom)10929 3589 y Fc(K)11712
3440 y Fj(;)p Fl(Enc)13554 3589 y Fc(K)14392 3941 y Ft(\(Enc)16900
4129 y Fj(K)17812 3941 y Ft(\(0\)\))338 b(=)e(1)22054
2959 y Ff(\003)22829 3941 y Fp(\000)269 b Ft(Pr)25543
2959 y Ff(\002)26048 3941 y Fo(A)26957 3440 y Fl(Hom)28916
3589 y Fc(K)29699 3440 y Fj(;)p Fl(Enc)31541 3589 y Fc(K)32379
3941 y Ft(\(Enc)34887 4129 y Fj(K)35798 3941 y Ft(\(1\)\))338
b(=)f(1)40041 2959 y Ff(\003)40546 2910 y(\014)40546
3638 y(\014)41287 3941 y Fp(\024)f Ft(neg)17 b(\()p Fo(k)38
b Ft(\))p Fo(:)3030 6775 y Fz(Pro)h(of)330 b(of)f(claim:)1139
b Ft(Supp)34 b(ose)288 b(there)f(w)-34 b(ere)287 b(a)g(PPT)g
Fo(A)g Ft(violating)f(the)i(claim.)498 b(First,)310 b(w)-34
b(e)288 b(argue)e(that)3030 8280 y(w)-34 b(e)274 b(can)g(replace)e(the)
i(resp)34 b(onses)274 b(to)g(all)f(of)h Fo(A)p Ft('S)f(Hom)26832
8468 y Fj(K)27743 8280 y Ft(-oracle)f(queries)h(with)h(encryptions)g
(of)g(0)g(with)3030 9786 y(only)368 b(a)g(negligible)f(e\013ect)h(on)g
Fo(A)p Ft('s)g(distinguishing)h(gap.)527 b(This)368 b(follo)-34
b(ws)369 b(from)f(indistinguishabilit)-34 b(y)3030 11291
y(under)322 b(c)-34 b(hosen)321 b(plain)-34 b(text)322
b(and)f(ciphertext)g(attac)-34 b(ks)321 b(and)h(a)e(h)-34
b(ybrid)322 b(argumen)-34 b(t:)498 b(Consider)321 b(h)-34
b(ybrids)3030 12796 y(where)564 b(the)g(\014rst)g Fo(i)f
Ft(oracle)g(queries)g(are)g(answ)-34 b(ered)564 b(according)g(to)g(Hom)
37848 12984 y Fj(K)39323 12796 y Ft(and)g(the)g(rest)g(with)3030
14302 y(encryptions)373 b(of)f(0.)528 b(An)-34 b(y)372
b(adv)-67 b(an)-34 b(tage)373 b(in)f(distinguishing)h(t)-34
b(w)g(o)373 b(adjacen)-34 b(t)373 b(h)-34 b(ybrids)373
b(m)-34 b(ust)373 b(b)34 b(e)372 b(due)g(to)3030 15807
y(distinguishing)435 b(an)e(encryption)g(of)g(1)f(from)h(an)g
(encryption)g(of)g(0.)625 b(The)433 b(resulting)g(distinguisher)3030
17313 y(can)463 b(b)34 b(e)463 b(implemen)-34 b(ted)464
b(using)f(oracle)f(access)h(to)g(encryption)g(and)h(decryption)g
(oracles)e(prior)g(to)3030 18818 y(receiving)403 b(the)h(c)-34
b(hallenge)404 b(ciphertext)g(\(and)h(an)g(encryption)f(oracle)f
(afterw)-34 b(ard\).)4848 20324 y(Once)326 b(w)-34 b(e)326
b(ha)-34 b(v)g(e)326 b(replaced)e(the)i(Hom)21530 20512
y Fj(K)22441 20324 y Ft(-oracle)f(resp)34 b(onses)325
b(with)i(encryptions)e(of)h(0,)341 b(w)-34 b(e)326 b(ha)-34
b(v)g(e)326 b(an)3030 21829 y(adv)-34 b(ersary)436 b(that)h(can)f
(distinguish)i(an)e(encryption)h(of)f(0)g(from)g(an)h(encryption)f(of)h
(1)f(when)h(giv)-34 b(en)3030 23335 y(access)433 b(to)h(just)g(an)g
(encryption)f(oracle.)625 b(This)434 b(con)-34 b(tradicts)434
b(indistinguishabilit)-34 b(y)435 b(under)f(c)-34 b(hosen)3030
24840 y(plain)g(text)405 b(attac)-34 b(k.)36280 b Fb(2)1882
27674 y Ft(No)-34 b(w)508 b(w)-34 b(e)508 b(return)f(to)h(the)g
(construction)g(of)g(our)g(circuit)e(family)h Fp(D)33637
27871 y Fj(\013;\014)35132 27674 y Ft(.)848 b(F)-101
b(or)507 b(a)h(k)-34 b(ey)506 b Fo(K)87 b Ft(,)533 b(let)508
b Fo(E)46736 27862 y Fj(K)10 b(;\013)48967 27674 y Ft(b)34
b(e)508 b(an)0 29179 y(algorithm)550 b(whic)-34 b(h,)587
b(on)550 b(input)h Fo(i)f Ft(outputs)i(Enc)22467 29367
y Fj(K)23378 29179 y Ft(\()p Fo(\013)24624 29361 y Fj(i)25001
29179 y Ft(\),)586 b(where)550 b Fo(\013)30821 29361
y Fj(i)31747 29179 y Ft(is)g(the)g Fo(i)p Ft('th)g(bit)g(of)g
Fo(\013)t Ft(.)977 b(Let)550 b Fo(B)46681 29376 y Fj(K)10
b(;\014)48925 29179 y Ft(b)34 b(e)550 b(an)0 30685 y(algorithm)393
b(whic)-34 b(h)393 b(when)h(fed)f(a)f Fo(k)38 b Ft(-tuple)393
b(of)g(ciphertexts)g(\()p Fo(c)27849 30867 y Fl(1)28375
30685 y Fo(;)202 b(:)g(:)g(:)f(;)h(c)31594 30882 y Fj(k)32162
30685 y Ft(\))393 b(outputs)h(1)f(if)f(for)h(all)f Fo(i)p
Ft(,)i(Dec)46231 30873 y Fj(K)47143 30685 y Ft(\()p Fo(c)48139
30867 y Fj(i)48514 30685 y Ft(\))338 b(=)e Fo(\014)51288
30867 y Fj(i)51663 30685 y Ft(,)0 32190 y(where)404 b
Fo(\014)4191 32372 y Fl(1)4716 32190 y Fo(;)202 b(:)g(:)g(:)f(;)h(\014)
8096 32387 y Fj(k)9069 32190 y Ft(are)403 b(the)i(bits)f(of)h
Fo(\014)64 b Ft(.)538 b(A)404 b(random)h(circuit)e(from)h
Fp(D)31884 32387 y Fj(\013;\014)33783 32190 y Ft(will)g(essen)-34
b(tially)403 b(b)34 b(e)404 b(the)h(algorithm)17955 35312
y Fo(D)18959 35509 y Fj(K)10 b(;\013;\014)21787 34625
y Fl(def)21965 35312 y Ft(=)447 b Fo(E)24250 35500 y
Fj(K)10 b(;\013)25975 35312 y Ft(#Hom)29510 35500 y Fj(K)30422
35312 y Ft(#)p Fo(B)32351 35509 y Fj(K)g(;\014)0 38035
y Ft(\(for)396 b(a)f(uniformly)g(selected)f(k)-34 b(ey)395
b Fo(K)87 b Ft(\).)536 b(One)395 b(minor)g(complication)g(is)f(that)j
Fo(D)35990 38232 y Fj(K)10 b(;\013;\014)38944 38035 y
Ft(is)395 b(actually)f(a)h Fr(pr)-62 b(ob)g(abilistic)0
39540 y Ft(algorithm,)591 b(since)553 b Fo(E)10063 39728
y Fj(K)10 b(;\013)12342 39540 y Ft(and)554 b(Hom)17374
39728 y Fj(K)18839 39540 y Ft(emplo)-34 b(y)554 b(probabilistic)f
(encryption,)592 b(whereas)553 b(the)i(lemma)e(requires)0
41046 y(deterministic)487 b(functions.)789 b(This)488
b(can)f(b)34 b(e)487 b(solv)-34 b(ed)487 b(in)g(the)g(usual)h(w)-34
b(a)g(y)-101 b(,)509 b(b)-34 b(y)487 b(using)h(pseudorandom)h
(functions.)0 42551 y(Let)447 b Fo(q)453 b Ft(=)409 b
Fo(q)43 b Ft(\()p Fo(k)38 b Ft(\))449 b(b)34 b(e)447
b(the)h(input)h(length)f(of)g Fo(D)20480 42748 y Fj(K)10
b(;\013;\014)23487 42551 y Ft(and)448 b Fo(m)410 b Ft(=)f
Fo(m)p Ft(\()p Fo(k)38 b Ft(\))448 b(the)g(maxim)-34
b(um)449 b(n)-34 b(um)g(b)34 b(er)448 b(of)g(random)g(bits)0
44057 y(used)506 b(b)-34 b(y)505 b Fo(D)5657 44254 y
Fj(K)10 b(;\013;\014)8721 44057 y Ft(on)505 b(an)-34
b(y)506 b(input.)842 b(W)-101 b(e)504 b(can)h(select)g(a)g
(pseudorandom)h(function)h Fo(f)39565 44260 y Fj(K)40421
44008 y Fy(0)41279 44057 y Ft(:)e Fp(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)45084 43617 y Fj(q)46096 44057 y Fp(!)505
b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)50776 43617 y Fj(m)51663
44057 y Ft(,)0 45562 y(and)390 b(let)g Fo(D)5117 45122
y FG(0)5083 45937 y Fj(K)10 b(;\013;\014)46 b(;K)8703
45685 y Fy(0)9447 45562 y Ft(b)34 b(e)389 b(the)h(\(deterministic\))g
(algorithm,)i(whic)-34 b(h)391 b(on)f(input)g Fo(x)337
b Fp(2)g(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)40694 45122
y Fj(q)41590 45562 y Ft(ev)-67 b(aluates)389 b Fo(D)47805
45759 y Fj(K)10 b(;\013;\014)50364 45562 y Ft(\()p Fo(x)p
Ft(\))0 47068 y(using)405 b(randomness)g Fo(f)10381 47271
y Fj(K)11237 47019 y Fy(0)11591 47068 y Ft(\()p Fo(x)p
Ft(\).)1882 48573 y(De\014ne)369 b(the)h(distribution)h
Fp(D)15164 48770 y Fj(\013;\014)17028 48573 y Ft(to)f(b)34
b(e)369 b Fo(D)21128 48133 y FG(0)21094 48948 y Fj(K)10
b(;\013;\014)46 b(;K)24714 48696 y Fy(0)25069 48573 y
Ft(,)376 b(o)-34 b(v)g(er)369 b(uniformly)h(selected)f(k)-34
b(eys)369 b Fo(K)456 b Ft(and)371 b Fo(K)45874 48133
y FG(0)46184 48573 y Ft(.)527 b(W)-101 b(e)369 b(argue)0
50079 y(that)455 b(this)g(distribution)h(has)f(the)f(prop)34
b(erties)454 b(stated)i(in)e(the)h(lemma.)688 b(By)454
b(construction,)468 b(eac)-34 b(h)454 b Fo(D)46790 49639
y FG(0)46756 50454 y Fj(K)10 b(;\013;\014)46 b(;K)50376
50202 y Fy(0)51185 50079 y Ft(is)0 51584 y(computable)405
b(b)-34 b(y)404 b(circuit)g(of)g(size)g(p)34 b(oly)16
b(\()p Fo(k)38 b Ft(\),)405 b(so)f(Prop)34 b(ert)-34
b(y)404 b(1)g(is)g(satis\014ed.)1882 53090 y(F)-101 b(or)351
b(Prop)34 b(ert)-34 b(y)351 b(2,)361 b(consider)351 b(an)g(algorithm)h
Fo(A)f Ft(that)h(on)g(input)g Fo(C)438 b Ft(and)352 b(oracle)e(access)h
(to)g Fo(D)43836 52650 y FG(0)43802 53465 y Fj(K)10 b(;\013;\014)46
b(;K)47422 53213 y Fy(0)48128 53090 y Ft(\(whic)-34 b(h,)0
54595 y(as)564 b(usual,)604 b(w)-34 b(e)564 b(can)g(view)f(as)h(access)
f(to)h(\(deterministic)g(v)-34 b(ersions)563 b(of)94
b(\))565 b(the)f(three)g(separate)g(oracles)f Fo(E)49939
54783 y Fj(K)10 b(;\013)51663 54595 y Ft(,)0 56100 y(Hom)2525
56288 y Fj(K)3437 56100 y Ft(,)431 b(and)c Fo(B)7503
56288 y Fj(K)10 b(;\013)9229 56100 y Ft(\),)432 b(pro)34
b(ceeds)426 b(as)g(follo)-34 b(ws:)584 b(First,)432 b(with)427
b Fo(k)464 b Ft(oracle)426 b(queries)f(to)i(the)g Fo(E)40799
56288 y Fj(K)10 b(;\013)42950 56100 y Ft(oracle,)430
b Fo(A)d Ft(obtains)0 57606 y(encryptions)367 b(of)h(eac)-34
b(h)367 b(of)h(the)f(bits)h(of)f Fo(\013)t Ft(.)527 b(Then,)375
b Fo(A)367 b Ft(uses)h(the)f(Hom)31072 57794 y Fj(K)32350
57606 y Ft(oracle)f(to)i(do)f(a)g(gate-b)-34 b(y-gate)368
b(em)-34 b(ulation)0 59111 y(of)478 b(the)f(computation)i(of)e
Fo(C)87 b Ft(\()p Fo(\013)t Ft(\),)497 b(in)477 b(whic)-34
b(h)478 b Fo(A)f Ft(obtains)i(encryptions)e(of)h(the)f(v)-67
b(alues)477 b(at)g(eac)-34 b(h)478 b(gate)f(of)g Fo(C)87
b Ft(.)758 b(In)0 60617 y(particular,)366 b Fo(A)356
b Ft(obtains)i(encryptions)f(of)g(the)g(v)-67 b(alues)356
b(at)h(eac)-34 b(h)357 b(output)h(gate)f(of)g Fo(C)444
b Ft(\(on)357 b(input)h Fo(\013)t Ft(\).)524 b(It)356
b(then)i(feeds)0 62122 y(these)391 b(output)i(encryptions)f(to)f
Fo(D)16019 62319 y Fj(K)10 b(;\014)17712 62122 y Ft(,)394
b(and)d(outputs)i(the)f(resp)34 b(onse)391 b(to)g(this)h(oracle)e
(query)-101 b(.)533 b(By)390 b(construction,)0 63628
y Fo(A)404 b Ft(outputs)i(1)e(i\013)g Fo(C)87 b Ft(\()p
Fo(\013)t Ft(\))339 b(=)d Fo(\014)64 b Ft(.)1882 65133
y(Finally)-101 b(,)486 b(w)-34 b(e)470 b(v)-34 b(erify)470
b(Prop)34 b(ert)-34 b(y)470 b(3.)736 b(Let)470 b Fo(S)540
b Ft(b)34 b(e)470 b(an)-34 b(y)471 b(PPT)f(algorithm.)737
b(W)-101 b(e)470 b(m)-34 b(ust)471 b(sho)-34 b(w)472
b(that)f Fo(S)540 b Ft(has)471 b(only)0 66639 y(a)455
b(negligible)f(probabilit)-34 b(y)456 b(of)f(outputting)i
Fo(\013)j Ft(when)c(giv)-34 b(en)455 b(oracle)f(access)g(to)i
Fo(D)37718 66199 y FG(0)37684 67014 y Fj(K)10 b(;\013;\014)46
b(;K)41304 66762 y Fy(0)42113 66639 y Ft(\(o)-34 b(v)g(er)456
b(the)f(c)-34 b(hoice)455 b(of)0 68144 y Fo(K)87 b Ft(,)496
b Fo(\013)t Ft(,)i Fo(\014)64 b Ft(,)496 b Fo(K)6262
67704 y FG(0)6572 68144 y Ft(,)h(and)479 b(the)f(coin)g(tosses)h(of)f
Fo(S)70 b Ft(\).)762 b(By)477 b(the)i(pseudorandomness)h(of)e
Fo(f)38515 68348 y Fj(K)39371 68096 y Fy(0)39726 68144
y Ft(,)496 b(w)-34 b(e)479 b(can)f(replace)f(oracle)0
69650 y(access)326 b(to)h(the)g(function)g Fo(D)12626
69210 y FG(0)12592 70025 y Fj(K)10 b(;\013;\014)46 b(;K)16212
69773 y Fy(0)16894 69650 y Ft(with)327 b(oracle)e(access)h(to)h(the)g
(probabilistic)f(algorithm)h Fo(D)43243 69847 y Fj(K)10
b(;\013;\014)46128 69650 y Ft(with)328 b(only)e(a)0 71155
y(negligible)369 b(e\013ect)g(on)h Fo(S)70 b Ft('s)369
b(success)h(probabilit)-34 b(y)-101 b(.)527 b(Oracle)369
b(access)g(to)h Fo(D)32621 71352 y Fj(K)10 b(;\013;\014)35549
71155 y Ft(is)370 b(equiv)-67 b(alen)-34 b(t)369 b(to)h(oracle)f
(access)g(to)25394 74521 y(14)p eop
%%Page: 15 16
15 15 bop 0 1218 a Fo(E)895 1406 y Fj(K)10 b(;\013)2620
1218 y Ft(,)361 b(Hom)5844 1406 y Fj(K)6755 1218 y Ft(,)h(and)352
b Fo(B)10677 1415 y Fj(K)10 b(;\014)12372 1218 y Ft(.)521
b(Since)351 b Fo(\014)416 b Ft(is)351 b(indep)34 b(enden)-34
b(t)353 b(of)g Fo(\013)j Ft(and)d Fo(K)87 b Ft(,)362
b(the)352 b(probabilit)-34 b(y)352 b(that)h Fo(S)422
b Ft(queries)351 b Fo(B)48877 1415 y Fj(K)10 b(;\014)50923
1218 y Ft(at)0 2723 y(a)323 b(p)34 b(oin)-34 b(t)324
b(where)e(its)h(v)-67 b(alue)322 b(is)h(nonzero)g(\(i.e.,)338
b(at)323 b(a)g(sequence)f(of)i(encryptions)f(of)g(the)g(bits)h(of)f
Fo(\014)64 b Ft(\))323 b(is)f(exp)34 b(onen)-34 b(tially)0
4229 y(small,)370 b(so)362 b(w)-34 b(e)362 b(can)g(remo)-34
b(v)g(e)361 b Fo(S)70 b Ft('s)362 b(queries)f(to)h Fo(B)21335
4426 y Fj(K)10 b(;\014)23392 4229 y Ft(with)362 b(only)g(a)g
(negligible)f(e\013ect)g(on)i(the)f(success)g(probabilit)-34
b(y)-101 b(.)0 5734 y(Oracle)416 b(access)h(to)g Fo(E)9840
5922 y Fj(K)10 b(;\013)11982 5734 y Ft(is)417 b(equiv)-67
b(alen)-34 b(t)417 b(to)g(giving)g Fo(S)487 b Ft(p)34
b(olynomially)417 b(man)-34 b(y)417 b(encryptions)h(of)g(eac)-34
b(h)417 b(of)h(the)f(bits)0 7240 y(of)502 b Fo(\013)t
Ft(.)832 b(Th)-34 b(us,)528 b(w)-34 b(e)502 b(m)-34 b(ust)503
b(argue)f(that)h Fo(S)571 b Ft(cannot)503 b(compute)g
Fo(\013)j Ft(with)d(nonnegligible)e(probabilit)-34 b(y)503
b(from)f(these)0 8745 y(encryptions)509 b(and)h(oracle)d(access)h(to)i
(Hom)20483 8933 y Fj(K)21394 8745 y Ft(.)853 b(This)509
b(follo)-34 b(ws)509 b(from)g(the)h(fact)f(that)h(the)f(encryption)g
(sc)-34 b(heme)0 10250 y(remains)404 b(secure)f(in)h(the)h(presence)f
(of)g(a)g(Hom)21444 10438 y Fj(K)22760 10250 y Ft(oracle)f(\(Claim)h
(3.7\))h(and)g(a)f(h)-34 b(ybrid)404 b(argumen)-34 b(t.)p
51114 10250 886 886 v 1882 13084 a(No)g(w)405 b(w)-34
b(e)404 b(can)h(pro)-34 b(v)g(e)404 b(the)g(imp)34 b(ossibilit)-34
b(y)404 b(of)h(circuit)e(obfuscators.)0 15918 y Fz(Theorem)464
b(3.8)606 b Fr(If)434 b(one-way)e(functions)g(exist,)h(then)g(cir)-62
b(cuit)431 b(obfusc)-62 b(ators)430 b(do)j(not)g(exist.)0
18752 y Fz(Pro)39 b(of:)1700 b Ft(Supp)34 b(ose,)472
b(for)459 b(sak)-34 b(e)458 b(of)g(con)-34 b(tradiction,)473
b(that)459 b(there)f(exists)g(a)g(circuit)f(obfuscator)j
Fp(O)34 b Ft(.)700 b(F)-101 b(or)458 b Fo(k)465 b Fp(2)427
b Fg(N)0 20257 y Ft(and)e Fo(\013)t(;)202 b(\014)434
b Fp(2)370 b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)8956 19818
y Fj(k)9524 20257 y Ft(,)429 b(let)424 b Fo(Z)12888 20454
y Fj(k)13881 20257 y Ft(and)h Fo(C)17124 20454 y Fj(\013;\014)19042
20257 y Ft(b)34 b(e)424 b(the)h(circuits)e(de\014ned)i(in)f(the)g(pro)
34 b(of)425 b(of)f(Prop)34 b(osition)424 b(3.4,)k(and)d(let)0
21763 y Fp(D)935 21960 y Fj(\013;\014)2887 21763 y Ft(b)34
b(e)456 b(the)i(distribution)g(on)g(circuits)e(giv)-34
b(en)457 b(b)-34 b(y)457 b(Lemma)g(3.6.)697 b(F)-101
b(or)457 b(eac)-34 b(h)457 b Fo(k)463 b Fp(2)426 b Fg(N)p
Ft(,)470 b(consider)457 b(the)g(follo)-34 b(wing)0 23268
y(t)g(w)g(o)406 b(distributions)f(on)g(circuits)e(of)i(size)e(p)34
b(oly)16 b(\()p Fo(k)38 b Ft(\):)0 25940 y Fp(F)871 26137
y Fj(k)1440 25940 y Fz(:)606 b Ft(Cho)34 b(ose)405 b
Fo(\013)k Ft(and)c Fo(\014)467 b Ft(uniformly)405 b(in)f
Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)21246 25501 y Fj(k)21814
25940 y Ft(,)404 b Fo(D)24175 25253 y Fw(R)23862 25940
y Fp( )269 b(D)26278 26137 y Fj(\013;\014)27773 25940
y Ft(.)538 b(Output)406 b Fo(C)33824 26137 y Fj(\013;\014)35319
25940 y Ft(#)p Fo(D)34 b Ft(.)0 28679 y Fp(G)721 28876
y Fj(k)1290 28679 y Fz(:)606 b Ft(Cho)34 b(ose)405 b
Fo(\013)k Ft(and)404 b Fo(\014)468 b Ft(uniformly)404
b(in)h Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)21096 28239
y Fj(k)21664 28679 y Ft(,)404 b Fo(D)24024 27992 y Fw(R)23712
28679 y Fp( )269 b(D)26128 28876 y Fj(\013;\014)27623
28679 y Ft(.)538 b(Output)406 b Fo(Z)33635 28876 y Fj(k)34204
28679 y Ft(#)p Fo(D)34 b Ft(.)1882 31181 y(Let)412 b
Fo(A)g Ft(b)34 b(e)411 b(the)i(PPT)f(algorithm)g(guaran)-34
b(teed)414 b(b)-34 b(y)412 b(Prop)34 b(ert)-34 b(y)412
b(2)g(in)g(Lemma)g(3.6,)h(and)g(consider)f(a)g(PPT)g
Fo(A)51689 30741 y FG(0)0 32686 y Ft(whic)-34 b(h,)533
b(on)507 b(input)g(a)g(circuit)f Fo(F)168 b Ft(,)532
b(decomp)34 b(oses)507 b Fo(F)676 b Ft(=)507 b Fo(F)26225
32868 y Fl(0)26751 32686 y Ft(#)p Fo(F)28540 32868 y
Fl(1)29573 32686 y Ft(and)h(ev)-67 b(aluates)506 b Fo(A)38270
32247 y Fj(F)38865 32370 y Fw(1)39382 32686 y Ft(\()p
Fo(F)40632 32868 y Fl(0)41158 32686 y Fo(;)202 b Ft(1)42303
32247 y Fj(k)42872 32686 y Ft(\),)532 b(where)507 b Fo(k)544
b Ft(is)506 b(the)0 34192 y(input)363 b(length)g(of)f
Fo(F)8969 34374 y Fl(0)9495 34192 y Ft(.)525 b(Th)-34
b(us,)371 b(when)363 b(fed)g(a)f(circuit)f(from)h Fp(O)34
b Ft(\()p Fp(F)28662 34389 y Fj(k)29231 34192 y Ft(\))363
b(\(resp.,)370 b Fp(O)34 b Ft(\()p Fp(G)35936 34389 y
Fj(k)36505 34192 y Ft(\)\),)370 b Fo(A)39063 33752 y
FG(0)39736 34192 y Ft(is)362 b(ev)-67 b(aluating)361
b Fo(A)47604 33752 y Fj(D)48458 34192 y Ft(\()p Fo(C)19
b(;)202 b Ft(1)50959 33752 y Fj(k)51529 34192 y Ft(\))0
35697 y(where)355 b Fo(D)388 b Ft(computes)356 b(the)f(same)g(function)
h(as)f(some)g(circuit)f(from)i Fp(D)31821 35894 y Fj(\013;\014)33670
35697 y Ft(and)g Fo(C)442 b Ft(computes)355 b(the)h(same)f(function)0
37203 y(as)404 b Fo(C)2354 37400 y Fj(\013;\014)4253
37203 y Ft(\(resp.,)g Fo(Z)8794 37400 y Fj(k)9363 37203
y Ft(\).)539 b(Therefore,)404 b(b)-34 b(y)404 b(Prop)34
b(ert)-34 b(y)404 b(2)g(in)g(Lemma)g(3.6,)f(w)-34 b(e)405
b(ha)-34 b(v)g(e:)0 42649 y(W)-101 b(e)404 b(no)-34 b(w)405
b(argue)f(that)h(for)f(an)-34 b(y)405 b(PPT)f(algorithm)g
Fo(S)13242 44213 y Ff(\014)13242 44940 y(\014)13242 45668
y(\014)13646 45607 y Ft(Pr)15147 44262 y Ff(h)15720 45607
y Fo(S)16533 45107 y FG(F)17205 45263 y Fc(k)17768 45607
y Ft(\(1)18845 45107 y Fj(k)19414 45607 y Ft(\))337 b(=)g(1)22108
44262 y Ff(i)22950 45607 y Fp(\000)269 b Ft(Pr)25663
44262 y Ff(h)26236 45607 y Fo(S)27049 45107 y FG(G)27608
45263 y Fc(k)28171 45607 y Ft(\(1)29248 45107 y Fj(k)29817
45607 y Ft(\))337 b(=)g(1)32511 44262 y Ff(i)33083 44213
y(\014)33083 44940 y(\014)33083 45668 y(\014)33824 45607
y Fp(\024)g Ft(2)35710 45107 y FG(\000)p Fl(\012\()p
Fj(k)24 b Fl(\))38422 45607 y Fo(;)0 48565 y Ft(whic)-34
b(h)470 b(will)f(con)-34 b(tradict)470 b(the)g(de\014nition)g(of)g
(circuit)f(obfuscators.)735 b(Ha)-34 b(ving)470 b(oracle)e(access)h(to)
h(a)f(circuit)g(from)0 50071 y Fp(F)871 50268 y Fj(k)1821
50071 y Ft(\(resp)34 b(ectiv)-34 b(ely)-101 b(,)384 b
Fp(G)9840 50268 y Fj(k)10409 50071 y Ft(\))d(is)g(equiv)-67
b(alen)-34 b(t)380 b(to)i(ha)-34 b(ving)381 b(oracle)f(access)h(to)g
Fo(C)32898 50268 y Fj(\013;\014)34774 50071 y Ft(\(resp.,)k
Fo(Z)39296 50268 y Fj(k)39866 50071 y Ft(\))c(and)h Fo(D)44625
49383 y Fw(R)44312 50071 y Fp( )223 b(D)46682 50268 y
Fj(\013;\014)48177 50071 y Ft(,)385 b(where)0 51576 y
Fo(\013)t(;)202 b(\014)504 b Ft(are)440 b(selected)f(uniformly)i(in)f
Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)19097 51136 y Fj(k)19666
51576 y Ft(.)646 b(Prop)34 b(ert)-34 b(y)440 b(3)g(of)g(Lemma)g(3.6)g
(implies)f(that)j(the)e(probabilit)-34 b(y)441 b(that)0
53082 y Fo(S)485 b Ft(queries)414 b(the)h(\014rst)g(oracle)f(at)h
Fo(\013)k Ft(is)c(negligible,)h(and)f(hence)g Fo(S)484
b Ft(cannot)416 b(distinguish)g(that)g(oracle)e(b)34
b(eing)414 b Fo(C)50505 53279 y Fj(\013;\014)0 54587
y Ft(from)404 b(it)g(b)34 b(eing)404 b Fo(Z)8170 54784
y Fj(k)8740 54587 y Ft(.)p 51114 54587 V 1882 57421 a(W)-101
b(e)370 b(can)i(remo)-34 b(v)g(e)370 b(the)i(assumption)h(that)f(one-w)
-34 b(a)g(y)372 b(functions)h(exist)d(for)h Fr(e\016cient)f
Ft(circuit)g(obfuscators)i(via)0 58926 y(the)405 b(follo)-34
b(wing)404 b(\(easy\))h(lemma.)0 61760 y Fz(Lemma)464
b(3.9)606 b Fr(If)434 b(e\016cient)e(obfusc)-62 b(ators)429
b(exist,)k(then)f(one-way)h(functions)f(exist.)0 64594
y Fz(Pro)39 b(of)493 b(Sk)-39 b(etc)g(h:)1578 b Ft(Supp)34
b(ose)430 b(that)h Fp(O)463 b Ft(is)429 b(an)h(e\016cien)-34
b(t)429 b(obfuscator)i(as)e(p)34 b(er)429 b(De\014nition)h(2.2.)614
b(F)-101 b(or)430 b Fo(\013)383 b Fp(2)c(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)51431 64154 y Fj(k)0 66099 y Ft(and)405
b Fo(b)337 b Fp(2)g(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)p
Ft(,)403 b(let)h Fo(C)10679 66296 y Fj(\013;b)12340 66099
y Ft(:)336 b Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)15976
65659 y Fj(k)16882 66099 y Fp(!)336 b(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)404 b Ft(b)34 b(e)404 b(the)h(circuit)e(de\014ned)i(b)-34
b(y)19023 69274 y Fo(C)19889 69471 y Fj(\013;b)21213
69274 y Ft(\()p Fo(x)p Ft(\))23118 68587 y Fl(def)23296
69274 y Ft(=)24686 67929 y Ff(n)25696 68639 y Fo(b)1298
b(x)336 b Ft(=)h Fo(\013)25696 70145 y Ft(0)1212 b(otherwise.)25394
74521 y(15)p eop
%%Page: 16 17
16 16 bop 0 1423 a Ft(No)-34 b(w)422 b(de\014ne)g Fo(f)6890
1620 y Fj(k)7460 1423 y Ft(\()p Fo(\013)t(;)202 b(b;)g(r)34
b Ft(\))11642 736 y Fl(def)11819 1423 y Ft(=)459 b Fp(O)34
b Ft(\()p Fo(C)15557 1620 y Fj(\013;b)16881 1423 y Ft(;)202
b Fo(r)34 b Ft(\),)425 b(i.e.)590 b(the)422 b(obfuscation)h(of)f
Fo(C)32198 1620 y Fj(\013;b)33943 1423 y Ft(using)g(coin)g(tosses)g
Fo(r)34 b Ft(.)590 b(W)-101 b(e)421 b(will)g(argue)0
2929 y(that)468 b Fo(f)572 b Ft(=)5239 2020 y Ff(S)6249
3287 y Fj(k)24 b FG(2)p Fe(N)8286 2929 y Fo(f)8879 3126
y Fj(k)9915 2929 y Ft(is)467 b(a)g(one-w)-34 b(a)g(y)468
b(function.)728 b(Clearly)466 b Fo(f)27288 3126 y Fj(k)28324
2929 y Ft(can)h(b)34 b(e)467 b(ev)-67 b(aluated)467 b(in)g(time)f(p)34
b(oly)17 b(\()p Fo(k)38 b Ft(\).)728 b(Since)466 b(the)0
4434 y(bit)408 b Fo(b)h Ft(is)e(information-theoretically)h(determined)
h(b)-34 b(y)408 b Fo(f)26204 4631 y Fj(k)26773 4434 y
Ft(\()p Fo(\013)t(;)202 b(b;)g(r)34 b Ft(\),)410 b(to)e(sho)-34
b(w)410 b(that)f Fo(f)539 b Ft(is)407 b(one-w)-34 b(a)g(y)409
b(it)f(su\016ces)h(to)0 5940 y(sho)-34 b(w)405 b(that)h
Fo(b)e Ft(is)g(a)g Fr(har)-62 b(d-c)g(or)g(e)431 b(bit)403
b Ft(of)h Fo(f)130 b Ft(.)540 b(T)-101 b(o)404 b(pro)-34
b(v)g(e)404 b(this,)h(w)-34 b(e)404 b(\014rst)h(observ)-34
b(e)404 b(that)h(for)f(an)-34 b(y)405 b(PPT)f Fo(S)70
b Ft(,)17301 8895 y(Pr)17317 9712 y Fj(\013;b)18803 7549
y Ff(h)19375 8895 y Fo(S)20188 8394 y Fj(C)20859 8550
y Fc(\013;b)22120 8895 y Ft(\(1)23197 8394 y Fj(k)23766
8895 y Ft(\))337 b(=)f Fo(b)26373 7549 y Ff(i)27283 8895
y Fp(\024)28695 8075 y Ft(1)p 28695 8616 607 49 v 28695
9726 a(2)29703 8895 y(+)269 b(neg)17 b(\()p Fo(k)38 b
Ft(\))p Fo(:)0 11965 y Ft(By)403 b(the)i(virtual)f(blac)-34
b(k)404 b(b)34 b(o)-34 b(x)404 b(prop)34 b(ert)-34 b(y)404
b(of)h Fp(O)34 b Ft(,)403 b(it)h(follo)-34 b(ws)405 b(that)g(for)f(an)
-34 b(y)405 b(PPT)f Fo(A)p Ft(,)9874 14920 y(Pr)9534
15737 y Fj(\013;b;r)11716 14920 y Ft([)p Fo(A)p Ft(\()p
Fo(f)130 b Ft(\()p Fo(\013)t(;)202 b(b;)g(r)34 b Ft(\)\))339
b(=)e Fo(b)p Ft(])f(=)677 b(Pr)22619 15737 y Fj(\013;b;r)24801
14920 y Ft([)p Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(C)28854
15117 y Fj(\013;b)30178 14920 y Ft(;)202 b Fo(r)34 b
Ft(\)\))337 b(=)f Fo(b)p Ft(])h Fp(\024)36462 14100 y
Ft(1)p 36462 14641 V 36462 15751 a(2)37470 14920 y(+)269
b(neg)18 b(\()p Fo(k)38 b Ft(\))p Fo(:)0 17990 y Ft(This)405
b(demonstrates)g(that)g Fo(b)f Ft(is)g(indeed)h(a)f(hard-core)g(bit)g
(of)h Fo(f)130 b Ft(,)404 b(and)h(hence)f(that)i Fo(f)534
b Ft(is)404 b(one-w)-34 b(a)g(y)-101 b(.)5952 b Fb(2)0
20614 y Fz(Corollary)464 b(3.10)606 b Fr(E\016cient)433
b(cir)-62 b(cuit)431 b(obfusc)-62 b(ators)430 b(do)j(not)g(exist)g
(\(unc)-62 b(onditional)62 b(ly\).)1882 23237 y Ft(As)404
b(stated)h(ab)34 b(o)-34 b(v)g(e,)404 b(our)g(imp)34
b(ossibilit)-34 b(y)404 b(pro)34 b(of)405 b(can)f(b)34
b(e)404 b(cast)g(in)g(terms)g(of)h(\\unobfuscatable)h(functions":)0
25861 y Fz(Theorem)464 b(3.11)h(\(unobfuscatable)g(functions\))607
b Fr(If)350 b(one-way)e(functions)f(exist,)365 b(then)349
b(ther)-62 b(e)348 b(exists)g(an)h(un-)0 27366 y(obfusc)-62
b(atable)429 b(function)j(ensemble.)0 29990 y Fz(Pro)39
b(of:)1510 b Ft(Let)414 b Fp(F)8336 30187 y Fj(k)9319
29990 y Ft(and)h Fp(G)12407 30187 y Fj(k)13390 29990
y Ft(b)34 b(e)413 b(the)i(distributions)g(on)g(functions)g(in)f(the)h
(pro)34 b(of)414 b(of)g(Theorem)g(3.8,and)h(let)e Fp(H)51431
30187 y Fj(k)0 31495 y Ft(b)34 b(e)511 b(the)i(distribution)g(that,)539
b(with)513 b(probabilit)-34 b(y)512 b(1)p Fo(=)p Ft(2)g(outputs)h(a)f
(sample)g(of)g Fp(F)37462 31692 y Fj(k)38543 31495 y
Ft(and)g(with)h(probabilit)-34 b(y)512 b(1)p Fo(=)p Ft(2)0
33001 y(outputs)406 b(a)e(sample)g(of)h Fp(G)11610 33198
y Fj(k)12179 33001 y Ft(.)538 b(W)-101 b(e)404 b(claim)f(that)i
Fp(fH)22630 33198 y Fj(k)23199 33001 y Fp(g)23805 33198
y Fj(k)24 b FG(2)p Fe(N)26045 33001 y Ft(is)404 b(an)g(unobfuscatable)i
(function)g(ensem)-34 b(ble.)1882 34506 y(The)545 b(fact)h(that)g
Fp(fH)11443 34703 y Fj(k)12011 34506 y Fp(g)12617 34703
y Fj(k)24 b FG(2)p Fe(N)14998 34506 y Ft(is)545 b(e\016cien)-34
b(tly)544 b(computable)i(is)f(ob)-34 b(vious.)962 b(W)-101
b(e)544 b(de\014ne)i Fo(\031)43 b Ft(\()p Fo(f)130 b
Ft(\))547 b(to)e(b)34 b(e)545 b(1)g(if)g Fo(f)702 b Fp(2)0
35102 y Ff(S)1010 36370 y Fj(k)1781 36012 y Ft(Supp)r(\()p
Fp(F)5817 36209 y Fj(k)6386 36012 y Ft(\))429 b(and)g(0)f(otherwise)h
(\(note)g(that)g(\()22433 35102 y Ff(S)23444 36370 y
Fj(k)24215 36012 y Ft(Supp)q(\()p Fp(F)28250 36209 y
Fj(k)28820 36012 y Ft(\)\))286 b Fp(\\)g Ft(\()31613
35102 y Ff(S)32623 36370 y Fj(k)33394 36012 y Ft(Supp)r(\()p
Fp(G)37280 36209 y Fj(k)37849 36012 y Ft(\)\))378 b(=)f
Fp(;)428 b Ft(and)h(so)f Fo(\031)43 b Ft(\()p Fo(f)130
b Ft(\))380 b(=)c(0)429 b(for)0 37517 y(an)-34 b(y)443
b Fo(f)533 b Fp(2)4663 36608 y Ff(S)5673 37875 y Fj(k)6444
37517 y Ft(Supp)r(\()p Fp(G)10330 37714 y Fj(k)10899
37517 y Ft(\)\).)656 b(The)443 b(algorithm)g Fo(A)21837
37077 y FG(0)22591 37517 y Ft(giv)-34 b(en)443 b(in)g(the)g(pro)34
b(of)443 b(of)h(Theorem)f(3.8)f(sho)-34 b(ws)445 b(that)f
Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))445 b(can)0 39023
y(b)34 b(e)384 b(computed)h(in)f(p)34 b(olynomial)383
b(time)h(from)g(an)-34 b(y)384 b(circuit)f(computing)j
Fo(f)467 b Fp(2)337 b Ft(Supp)r(\()p Fp(H)38812 39220
y Fj(k)39380 39023 y Ft(\).)533 b(Because)383 b(oracle)g(access)0
40528 y(to)487 b Fp(F)2435 40725 y Fj(k)3491 40528 y
Ft(cannot)h(b)34 b(e)486 b(distinguished)i(from)f(oracle)f(access)g(to)
h Fp(G)29201 40725 y Fj(k)30257 40528 y Ft(\(as)g(sho)-34
b(wn)488 b(in)f(the)g(pro)34 b(of)487 b(of)g(Theorem)g(3.8\),)0
42204 y(it)496 b(follo)-34 b(ws)496 b(that)h Fo(\031)43
b Ft(\()p Fo(f)130 b Ft(\))497 b(cannot)g(b)34 b(e)495
b(computed)i(from)f(an)g(oracle)f(for)h Fo(f)34119 41517
y Fw(R)33806 42204 y Fp( )330 b(H)36372 42401 y Fj(k)37436
42204 y Ft(with)497 b(probabilit)-34 b(y)496 b(noticeably)0
43709 y(greater)403 b(than)j(1)p Fo(=)p Ft(2.)p 51114
43709 886 886 v 0 47495 a Fs(4)1793 b(Extensions)0 50246
y Fn(4.1)1495 b(T)-125 b(otally)500 b(unobfuscatable)g(functions)0
52534 y Ft(Some)486 b(of)h(the)f(extensions)h(of)f(our)g(imp)34
b(ossibilit)-34 b(y)486 b(result)g(require)f(a)h(somewhat)i(stronger)e
(form)g(of)h(unobfus-)0 54039 y(catable)536 b(functions,)570
b(in)536 b(whic)-34 b(h)537 b(it)e(is)h(not)h(only)e(p)34
b(ossible)536 b(to)g(compute)h Fo(\031)43 b Ft(\()p Fo(f)130
b Ft(\))538 b(from)e(an)-34 b(y)536 b(circuit)f(for)h
Fo(f)130 b Ft(,)570 b(but)0 55545 y(ev)-34 b(en)462 b(to)h(reco)-34
b(v)g(er)461 b(the)i(\\original")f(circuit)g(for)g Fo(f)130
b Ft(.)715 b(This)463 b(can)f(b)34 b(e)462 b(ac)-34 b(hiev)g(ed)463
b(b)-34 b(y)463 b(a)f(sligh)-34 b(t)463 b(mo)34 b(di\014cation)463
b(of)g(our)0 57050 y(construction.)573 b(It)416 b(will)e(also)i(b)34
b(e)415 b(useful)h(to)g(extend)f(the)h(construction)g(so)g(that)g(not)h
(only)e(the)h(one)f(bit)h Fo(\031)43 b Ft(\()p Fo(f)130
b Ft(\))417 b(is)0 58556 y(unpredictable)464 b(giv)-34
b(en)462 b(oracle)g(access)g(to)h Fo(f)130 b Ft(,)479
b(but)463 b(rather)g(that)h(there)f(are)f(man)-34 b(y)464
b(bits)f(of)g(information)h(ab)34 b(out)0 60061 y Fo(f)550
b Ft(whic)-34 b(h)420 b(are)f(completely)g(pseudorandom.)586
b(These)420 b(prop)34 b(erties)418 b(are)h(captured)h(b)-34
b(y)420 b(the)g(de\014nition)g(b)34 b(elo)-34 b(w.)585
b(In)0 61567 y(this)490 b(de\014nition,)513 b(it)489
b(will)h(b)34 b(e)489 b(con)-34 b(v)g(enien)g(t)491 b(to)g(iden)-34
b(tify)490 b(the)h(functions)g Fo(f)621 b Ft(in)490 b(our)g(family)f
(with)i(the)f(canonical)0 63072 y(circuits)404 b(that)h(compute)g
(them.)0 65696 y Fz(De\014nition)466 b(4.1)606 b Fr(A)864
b Ft(totally)418 b(unobfuscatable)i(function)f(ensem)-34
b(ble)447 b Fr(is)f(an)g(ensemble)f Fp(fH)42692 65893
y Fj(k)43260 65696 y Fp(g)43866 65893 y Fj(k)24 b FG(2)p
Fe(N)46148 65696 y Fr(of)446 b(distribu-)0 67201 y(tions)433
b Fp(H)4029 67398 y Fj(k)5031 67201 y Fr(on)h Ft(circuits)f
Fr(\(fr)-62 b(om,)432 b(say,)g Fp(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)20252 66761 y Fj(l)20533 66904 y Fw(in)21271
66761 y Fl(\()p Fj(k)24 b Fl(\))23006 67201 y Fr(to)432
b Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)27424 66761 y
Fj(l)27705 66872 y Fw(out)28935 66761 y Fl(\()p Fj(k)24
b Fl(\))30236 67201 y Fr(\))433 b(satisfying:)1818 69694
y Fp(\017)606 b Fr(\(e\016cient)j(c)-62 b(omputability\))607
b(Every)j(cir)-62 b(cuit)609 b Fo(f)789 b Fp(2)658 b
Ft(Supp)q(\()p Fp(H)31098 69891 y Fj(k)31667 69694 y
Ft(\))612 b Fr(is)e(of)h(size)e Ft(p)34 b(oly)17 b(\()p
Fo(k)38 b Ft(\))p Fr(.)1090 b(Mor)-62 b(e)g(over,)653
b Fo(f)51111 69007 y Fd(R)50788 69694 y Fp( )3030 71200
y Ft(Supp)r(\()p Fp(H)7219 71397 y Fj(k)7788 71200 y
Ft(\))434 b Fr(c)-62 b(an)433 b(b)-62 b(e)433 b(sample)-62
b(d)431 b(uniformly)i(in)h(time)e Ft(p)34 b(oly)17 b(\()p
Fo(k)38 b Ft(\))p Fr(.)25394 74521 y Ft(16)p eop
%%Page: 17 18
17 17 bop 1818 1218 a Fp(\017)606 b Fr(\(unobfusc)-62
b(atability\))304 b(Ther)-62 b(e)308 b(exists)h(a)g(p)-62
b(oly-time)307 b(c)-62 b(omputable)306 b(function)j Fo(\031)380
b Ft(:)38249 309 y Ff(S)39259 1576 y Fj(k)24 b FG(2)p
Fe(N)41297 1218 y Ft(Supp)r(\()p Fp(H)45486 1415 y Fj(k)46055
1218 y Ft(\))337 b Fp(!)g(f)p Ft(0)p Fo(;)202 b Ft(1)p
Fp(g)51375 778 y FG(\003)51900 1218 y Fr(,)3030 2723
y(such)432 b(that)4100 5225 y(1.)605 b Fo(\031)43 b Ft(\()p
Fo(f)130 b Ft(\))436 b Fr(is)d(pseudor)-62 b(andom)430
b(given)j(black-b)-62 b(ox)430 b(ac)-62 b(c)g(ess)432
b(to)h Fo(f)130 b Fr(:)558 b(F)-93 b(or)433 b(any)h(PPT)g
Fo(S)13443 6856 y Ff(\014)13443 7584 y(\014)13443 8311
y(\014)13443 9038 y(\014)13443 9765 y(\014)14597 8977
y Ft(Pr)13847 10252 y Fj(f)14584 9740 y Fd(R)14397 10252
y FG( )o(H)16138 10408 y Fc(k)16646 8977 y Ft([)p Fo(S)17796
8477 y Fj(f)18401 8977 y Ft(\()p Fo(\031)43 b Ft(\()p
Fo(f)130 b Ft(\)\))340 b(=)c(1])269 b Fp(\000)3184 b
Ft(Pr)25785 10252 y Fj(f)26523 9740 y Fd(R)26335 10252
y FG( H)28077 10408 y Fc(k)28585 10252 y Fj(;z)29509
9740 y Fd(R)29321 10252 y FG( f)p Fl(0)p Fj(;)p Fl(1)p
FG(g)32403 10000 y Fc(k)32914 8977 y Ft([)p Fo(S)34064
8477 y Fj(f)34668 8977 y Ft(\()p Fo(z)53 b Ft(\))338
b(=)e(1])38787 6856 y Ff(\014)38787 7584 y(\014)38787
8311 y(\014)38787 9038 y(\014)38787 9765 y(\014)39528
8977 y Fp(\024)g Ft(neg)17 b(\()p Fo(k)38 b Ft(\))4100
12986 y Fr(2.)605 b Fo(f)598 b Fr(is)467 b(e)-62 b(asy)465
b(to)h(r)-62 b(e)g(c)g(onstruct)465 b(given)h(any)h(other)f(cir)-62
b(cuit)465 b(for)h Fo(f)130 b Fr(:)625 b(Ther)-62 b(e)467
b(exists)e(a)i(PPT)h Fo(A)f Fr(such)f(that)5697 14492
y(for)433 b(any)g Fo(f)468 b Fp(2)12162 13583 y Ff(S)13172
14850 y Fj(k)13943 14492 y Ft(Supp)q(\()p Fp(H)18131
14689 y Fj(k)18700 14492 y Ft(\))434 b Fr(and)f(for)g(any)h(cir)-62
b(cuit)431 b Fo(C)521 b Fr(that)432 b(c)-62 b(omputes)431
b(the)h(same)h(function)f(as)h Fo(f)26276 17215 y(A)p
Ft(\()p Fo(C)87 b Ft(\))338 b(=)e Fo(f)5697 19938 y Fr(,)0
22772 y Ft(Note)k(that)h(totally)f(unobfuscatable)i(functions)g(imply)d
(unobfuscatable)k(functions:)507 b(giv)-34 b(en)340 b(oracle)f(access)h
(to)g(a)0 24277 y(totally)394 b(unobfuscatable)i Fo(f)130
b Ft(,)397 b(pseudorandomness)f(implies)e(that)h(the)f(\014rst)h(bit)f
(of)h Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))396 b(cannot)f(b)34
b(e)394 b(computed)0 25783 y(with)j(probabilit)-34 b(y)397
b(noticeably)g(more)f(than)i(1)p Fo(=)p Ft(2,)g(and)f(giv)-34
b(en)396 b(an)-34 b(y)397 b(circuit)f(for)h Fo(f)130
b Ft(,)398 b(one)f(can)g(e\016cien)-34 b(tly)396 b(\014nd)h(the)0
27288 y(canonical)404 b(circuit)f(for)i Fo(f)130 b Ft(,)404
b(from)g(whic)-34 b(h)405 b(one)g(can)f(compute)h Fo(\031)43
b Ft(\()p Fo(f)130 b Ft(\))406 b(\(and)f(in)f(particular,)g(its)g
(\014rst)h(bit\).)0 30122 y Fz(Theorem)464 b(4.2)h(\(totally)f
(unobfuscatable)i(functions\))607 b Fr(If)354 b(one-way)f(functions)g
(exist,)369 b(then)354 b(ther)-62 b(e)352 b(exists)0
31628 y(a)433 b(total)62 b(ly)431 b(unobfusc)-62 b(atable)429
b(function)j(ensemble.)0 34461 y Fz(Pro)39 b(of)399 b(Sk)-39
b(etc)g(h:)1310 b Ft(The)348 b(\014rst)g(step)g(is)f(to)g(observ)-34
b(e)348 b(that)g(the)g(ensem)-34 b(ble)347 b Fp(D)34364
34658 y Fj(\013;\014)36206 34461 y Ft(of)h(Lemma)f(3.6)g(can)g(b)34
b(e)348 b(mo)34 b(di\014ed)0 35967 y(so)603 b(that)g(Prop)34
b(ert)-34 b(y)602 b(2)h(instead)g(sa)-34 b(ys)603 b Fo(A)19086
35527 y Fj(D)19939 35967 y Ft(\()p Fo(C)19 b(;)202 b
Ft(1)22440 35527 y Fj(k)23010 35967 y Ft(\))668 b(=)f
Fo(\013)607 b Ft(if)602 b Fo(C)87 b Ft(\()p Fo(\013)t
Ft(\))669 b(=)e Fo(\014)f Ft(and)604 b Fo(A)38220 35527
y Fj(D)39073 35967 y Ft(\()p Fo(C)19 b(;)202 b Ft(1)41574
35527 y Fj(k)42144 35967 y Ft(\))668 b(=)f(0)45499 35527
y Fj(k)46670 35967 y Ft(otherwise.)0 37472 y(\(T)-101
b(o)507 b(ac)-34 b(hiev)g(e)506 b(this,)532 b(replace)505
b Fo(B)14629 37669 y Fj(K)10 b(;\014)16830 37472 y Ft(with)507
b Fo(B)20673 37033 y FG(0)20612 37841 y Fj(K)10 b(;\013;\014)23679
37472 y Ft(whic)-34 b(h)507 b(outputs)h Fo(\013)j Ft(when)c(fed)g(a)g
(sequence)e(of)i(ciphertexts)0 39179 y(\()p Fo(c)996
39361 y Fl(1)1522 39179 y Fo(;)202 b(:)g(:)g(:)f(;)h(c)4741
39376 y Fj(k)5309 39179 y Ft(\))404 b(whose)h(decryptions)f(are)g(the)g
(bits)h(of)g Fo(\014)467 b Ft(and)405 b(outputs)h(0)32691
38739 y Fj(k)33664 39179 y Ft(otherwise.\))1882 40685
y(No)-34 b(w)405 b(our)f(totally)g(unobfuscatable)i(function)g(ensem)
-34 b(ble)404 b Fp(H)29926 40882 y Fj(k)30899 40685 y
Ft(is)f(de\014ned)j(as)e(follo)-34 b(ws.)0 43186 y Fp(H)1024
43383 y Fj(k)1593 43186 y Fz(:)606 b Ft(Cho)34 b(ose)404
b Fo(\013)t(;)202 b(\014)64 b(;)202 b(\015)471 b Ft(uniformly)404
b(in)h Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)20410 42746
y Fj(k)20978 43186 y Ft(,)404 b Fo(D)23338 42499 y Fw(R)23026
43186 y Fp( )269 b(D)25442 43383 y Fj(\013;\014)26937
43186 y Ft(.)538 b(Output)406 b Fo(C)32988 43383 y Fj(\013;\014)34483
43186 y Ft(#)p Fo(D)34 b Ft(#)p Fo(C)38407 43433 y Fj(\013;)p
Fl(\()p Fj(D)26 b(;\015)52 b Fl(\))41656 43186 y Ft(.)0
45688 y(\(Ab)34 b(o)-34 b(v)g(e,)498 b Fo(C)5505 45935
y Fj(\013;)p Fl(\()p Fj(D)26 b(;\015)52 b Fl(\))9233
45688 y Ft(is)478 b(the)i(circuit)e(whic)-34 b(h)479
b(on)g(input)h Fo(\013)k Ft(outputs)d(\()p Fo(D)34 b(;)202
b(\015)67 b Ft(\),)496 b(and)480 b(on)f(all)f(other)h(inputs)h(outputs)
0 47371 y(0)606 46931 y FG(j)p Fl(\()p Fj(D)26 b(;\015)52
b Fl(\))p FG(j)3512 47371 y Ft(.\))1882 48876 y(E\016ciency)413
b(is)i(clearly)e(satis\014ed.)571 b(F)-101 b(or)415 b(unobfuscatabilit)
-34 b(y)-101 b(,)419 b(w)-34 b(e)415 b(de\014ne)g Fo(\031)43
b Ft(\()p Fo(C)37024 49073 y Fj(\013;\014)38520 48876
y Ft(#)p Fo(D)34 b Ft(#)p Fo(C)42444 49123 y Fj(\013;)p
Fl(\()p Fj(D)26 b(;\015)52 b Fl(\))45693 48876 y Ft(\))354
b(=)g Fo(\015)67 b Ft(.)571 b(Let's)0 50382 y(v)-34 b(erify)364
b(that)h Fo(\015)431 b Ft(is)364 b(pseudorandom)j(giv)-34
b(en)364 b(oracle)f(access.)525 b(As)364 b(in)h(the)g(pro)34
b(of)364 b(of)h(Theorem)f(3.11,)372 b(it)365 b(follo)-34
b(ws)365 b(from)0 51887 y(Prop)34 b(ert)-34 b(y)365 b(3)g(of)h(Lemma)e
(3.6)h(that)i(a)e(PPT)g(algorithm)h(giv)-34 b(en)365
b(oracle)f(access)h(to)g Fo(C)37963 52084 y Fj(\013;\014)39458
51887 y Ft(#)p Fo(D)34 b Ft(#)p Fo(C)43382 52134 y Fj(\013;)p
Fl(\()p Fj(D)26 b(;\015)52 b Fl(\))46631 51887 y Ft(.)525
b(will)365 b(only)0 53393 y(query)403 b Fo(C)4236 53639
y Fj(\013;)p Fl(\()p Fj(D)26 b(;\015)52 b Fl(\))7890
53393 y Ft(with)405 b(negligible)e(probabilit)-34 b(y)405
b(and)g(hence)f Fo(\015)471 b Ft(is)403 b(indistinguishable)j(from)e
(uniform.)1882 54898 y(Finally)-101 b(,)368 b(let's)360
b(sho)-34 b(w)362 b(that)g(giv)-34 b(en)360 b(an)-34
b(y)361 b(circuit)f Fo(C)24323 54458 y FG(0)24994 54898
y Ft(computing)i(the)f(same)f(function)i(as)f Fo(C)42996
55095 y Fj(\013;\014)44490 54898 y Ft(#)p Fo(D)34 b Ft(#)p
Fo(C)48414 55145 y Fj(\013;)p Fl(\()p Fj(D)26 b(;\015)52
b Fl(\))51663 54898 y Ft(,)0 56404 y(w)-34 b(e)285 b(can)f(reconstruct)
h(the)f(latter)g(circuit.)498 b(First,)308 b(w)-34 b(e)285
b(can)f(decomp)34 b(ose)284 b Fo(C)33245 55964 y FG(0)33893
56404 y Ft(=)336 b Fo(C)36125 55964 y Fl(1)36651 56404
y Ft(#)p Fo(D)38699 55964 y FG(0)39009 56404 y Ft(#)p
Fo(C)40972 55964 y Fl(2)41498 56404 y Ft(.)498 b(Since)284
b Fo(D)46416 55964 y FG(0)47010 56404 y Ft(computes)0
57909 y(the)383 b(same)g(function)h(as)e Fo(D)416 b Ft(and)384
b Fo(C)15983 57469 y Fl(1)16509 57909 y Ft(\()p Fo(\013)t
Ft(\))338 b(=)e Fo(\014)64 b Ft(,)386 b(w)-34 b(e)384
b(ha)-34 b(v)g(e)383 b Fo(A)26766 57469 y Fj(D)27564
57157 y Fy(0)27917 57909 y Ft(\()p Fo(C)29341 57469 y
Fl(1)29868 57909 y Ft(\))337 b(=)f Fo(\013)t Ft(,)388
b(where)382 b Fo(A)h Ft(is)f(the)h(algorithm)g(from)g(\(the)0
59415 y(mo)34 b(di\014ed\))355 b(Prop)34 b(ert)-34 b(y)354
b(2)h(of)f(Lemma)g(3.6.)522 b(Giv)-34 b(en)354 b Fo(\013)t
Ft(,)365 b(w)-34 b(e)355 b(can)g(obtain)g Fo(\014)400
b Ft(=)337 b Fo(C)35294 58975 y Fl(1)35820 59415 y Ft(\()p
Fo(\013)t Ft(\))356 b(and)f(\()p Fo(D)34 b(;)202 b(\015)67
b Ft(\))336 b(=)h Fo(C)45987 58975 y Fl(2)46512 59415
y Ft(\()p Fo(\013)t Ft(\),)366 b(whic)-34 b(h)0 60920
y(allo)g(ws)404 b(us)h(to)f(reconstruct)h Fo(C)13856
61117 y Fj(\013;\014)15351 60920 y Ft(#)p Fo(D)34 b Ft(#)p
Fo(C)19275 61167 y Fj(\013;)p Fl(\()p Fj(D)26 b(;\015)52
b Fl(\))22524 60920 y Ft(.)28233 b Fb(2)0 64166 y Fn(4.2)1495
b(Appro)-42 b(ximate)501 b(obfuscators)0 66454 y Ft(One)414
b(of)h(the)f(most)h(reasonable)f(w)-34 b(a)g(ys)416 b(to)e(w)-34
b(eak)g(en)415 b(the)g(de\014nition)g(of)g(obfuscators,)j(is)c(to)g
(relax)f(the)i(condition)0 67960 y(that)359 b(the)f(obfuscated)i
(circuit)d(m)-34 b(ust)359 b(compute)f Fr(exactly)463
b Ft(the)358 b(same)g(function)h(as)f(the)g(original)f(circuit.)523
b(Rather,)0 69465 y(w)-34 b(e)405 b(can)f(allo)-34 b(w)404
b(the)h(obfuscated)g(circuit)f(to)g(only)g Fr(appr)-62
b(oximate)493 b Ft(the)404 b(original)g(circuit.)25394
74521 y(17)p eop
%%Page: 18 19
18 18 bop 1882 1218 a Ft(W)-101 b(e)538 b(m)-34 b(ust)540
b(b)34 b(e)538 b(careful)g(in)h(de\014ning)h(\\appro)-34
b(ximation".)943 b(W)-101 b(e)538 b(do)h(not)g(w)-34
b(an)g(t)541 b(to)e(lose)f(the)h(notion)h(of)f(an)0 2723
y(obfuscator)354 b(as)e(a)h Fr(gener)-62 b(al)384 b(purp)-62
b(ose)441 b Ft(scram)-34 b(bling)353 b(algorithm)f(and)h(therefore)g(w)
-34 b(e)352 b(w)-34 b(an)g(t)355 b(a)d(de\014nition)h(of)g(appro)-34
b(x-)0 4229 y(imation)500 b(that)g(will)f(b)34 b(e)499
b(strong)i(enough)f(to)g(guaran)-34 b(tee)500 b(that)h(the)e
(obfuscated)i(circuit)e(can)h(still)f(b)34 b(e)499 b(used)h(in)0
5734 y(the)g(place)g(of)g(the)g(original)f(circuit)g(in)h
Fr(any)522 b(applic)-62 b(ation)p Ft(.)822 b(Consider)500
b(the)g(case)g(of)g(a)g(signature)g(v)-34 b(eri\014cation)0
7240 y(algorithm)478 b Fo(V)6306 7428 y Fj(K)7217 7240
y Ft(.)759 b(A)477 b(p)34 b(olynomial-time)477 b(algorithm)h(cannot)g
(\014nd)h(an)f(input)g(on)g(whic)-34 b(h)478 b Fo(V)42006
7428 y Fj(K)43395 7240 y Ft(do)34 b(es)478 b(not)g(output)0
8745 y(0)496 b(\(without)i(kno)-34 b(wing)496 b(the)h(signature)f(k)-34
b(ey\).)813 b(Ho)-34 b(w)g(ev)g(er,)519 b(w)-34 b(e)496
b(clearly)f(do)h(not)g(w)-34 b(an)g(t)498 b(this)e(to)h(mean)f(that)g
(the)0 10250 y(constan)-34 b(t)406 b(zero)d(function)j(is)e(an)g(appro)
-34 b(ximation)405 b(of)g Fo(V)25273 10438 y Fj(K)26184
10250 y Ft(.)0 13363 y Fz(4.2.1)1394 b(De\014nition)466
b(and)g(Imp)39 b(ossibilit)-39 b(y)464 b(Result)0 15651
y Ft(In)409 b(order)g(to)g(a)-34 b(v)g(oid)409 b(the)h(ab)34
b(o)-34 b(v)g(e)409 b(pitfalls)g(w)-34 b(e)410 b(c)-34
b(ho)34 b(ose)409 b(a)f(de\014nition)j(of)e(appro)-34
b(ximation)410 b(that)g(allo)-34 b(ws)409 b(the)h(obfus-)0
17157 y(cated)276 b(circuit)e(to)i(deviate)f(on)h(a)g(particular)f
(input)i(from)f(the)g(original)e(circuit)h(only)g(with)i(negligible)d
(probabilit)-34 b(y)0 18662 y(and)428 b(allo)-34 b(ws)427
b(this)h(ev)-34 b(en)g(t)427 b(to)h(dep)34 b(end)427
b(on)h(only)f(the)g(coin)g(tosses)h(of)f(the)g(obfuscating)i(algorithm)
e(\(rather)h(than)0 20168 y(o)-34 b(v)g(er)404 b(the)h(c)-34
b(hoice)403 b(of)i(a)f(randomly)g(c)-34 b(hosen)405 b(input\).)0
22435 y Fz(De\014nition)466 b(4.3)606 b Fr(F)-93 b(or)451
b(any)f(function)g Fo(f)498 b Ft(:)368 b Fp(f)p Ft(0)p
Fo(;)202 b Ft(1)p Fp(g)23063 21995 y Fj(n)24057 22435
y Fp(!)367 b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)28599
21995 y Fj(k)29168 22435 y Fr(,)455 b Fo(\017)368 b(>)f
Ft(0)p Fr(,)455 b(the)449 b(r)-62 b(andom)451 b(variable)d
Fo(C)537 b Fr(is)450 b(c)-62 b(al)62 b(le)-62 b(d)449
b(an)0 23941 y Fo(\017)p Fr(-)p Ft(appro)-34 b(ximate)404
b(implemen)-34 b(tation)435 b Fr(of)e Fo(f)565 b Fr(if)433
b(the)f(fol)62 b(lowing)432 b(holds:)1433 25971 y(1.)605
b Fo(C)521 b Fr(r)-62 b(anges)433 b(over)f(cir)-62 b(cuits)431
b(fr)-62 b(om)434 b Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)21010
25531 y Fj(n)22069 25971 y Fr(to)433 b Fp(f)p Ft(0)p
Fo(;)202 b Ft(1)p Fp(g)26488 25531 y Fj(k)1433 28284
y Fr(2.)605 b(F)-93 b(or)434 b(any)f Fo(x)337 b Fp(2)f(f)p
Ft(0)p Fo(;)202 b Ft(1)p Fp(g)12753 27844 y Fj(n)13813
28284 y Fr(,)434 b Ft(Pr)15918 28472 y Fj(C)16707 28284
y Ft([)p Fo(C)87 b Ft(\()p Fo(x)p Ft(\))337 b(=)f Fo(f)130
b Ft(\()p Fo(x)p Ft(\)])338 b Fp(\025)f Ft(1)269 b Fp(\000)g
Fo(\017)1882 30551 y Ft(W)-101 b(e)433 b(then)i(de\014ne)f(a)g
(strongly)g(unobfuscatable)i(function)f(ensem)-34 b(ble)434
b(to)h(b)34 b(e)433 b(an)i(unobfuscatable)h(function)0
32057 y(ensem)-34 b(ble)393 b(where)g(the)h(hard)f(prop)34
b(ert)-34 b(y)394 b Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))395
b(can)e(b)34 b(e)393 b(computed)h(not)g(only)f(from)g(an)-34
b(y)393 b(circuit)g(that)h(computes)0 33562 y Fo(f)535
b Ft(but)405 b(also)f(from)g(an)-34 b(y)405 b(appro)-34
b(ximate)405 b(implemen)-34 b(tation)405 b(of)f Fo(f)130
b Ft(.)0 35830 y Fz(De\014nition)466 b(4.4)606 b Fr(A)411
b Ft(strongly)380 b(unobfuscatable)i(function)e(ensem)-34
b(ble)412 b Fp(fH)34691 36027 y Fj(k)35259 35830 y Fp(g)35865
36027 y Fj(k)24 b FG(2)p Fe(N)38112 35830 y Fr(is)410
b(de\014ne)-62 b(d)411 b(in)g(the)f(same)g(way)0 37335
y(as)464 b(an)g(unobfusc)-62 b(atable)460 b(function)k(ensemble,)470
b(exc)-62 b(ept)463 b(that)g(Part)h(2)g(of)g(the)g(\\unobfusc)-62
b(atability")458 b(c)-62 b(ondition)463 b(is)0 38841
y(r)-62 b(eplac)g(e)g(d)431 b(with)i(the)f(fol)62 b(lowing:)1433
42376 y(2.)605 b Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))449
b Fr(is)e(e)-62 b(asy)445 b(to)h(c)-62 b(ompute)445 b(with)h(ac)-62
b(c)g(ess)445 b(to)h(a)h(cir)-62 b(cuit)445 b(that)g(appr)-62
b(oximates)444 b Fo(f)130 b Fr(:)585 b(Ther)-62 b(e)446
b(exists)g(a)g(PPT)i Fo(A)3030 43882 y Fr(and)439 b(a)f(p)-62
b(olynomial)437 b Fo(p)p Ft(\()p Fp(\001)p Ft(\))i Fr(such)e(that)h
(for)g(any)h Fo(f)477 b Fp(2)26624 42972 y Ff(S)27634
44240 y Fj(n)p FG(2)p Fe(N)29730 43882 y Ft(Supp)q(\()p
Fp(H)33918 44064 y Fj(n)34544 43882 y Ft(\))440 b Fr(and)e(for)h(any)f
(r)-62 b(andom)439 b(variable)d Fo(C)3030 45387 y Fr(that)c(is)h(an)h
Fo(\017)p Fr(-appr)-62 b(oximate)431 b(implementation)g(of)i
Fo(f)19442 47544 y Ft(Pr[)p Fo(A)p Ft(\()p Fo(C)87 b
Ft(\))337 b(=)f Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\)])339
b Fp(\025)d Ft(1)270 b Fp(\000)f Fo(\017)g Fp(\001)g
Fo(p)p Ft(\()p Fo(n)p Ft(\))1882 50105 y(Our)404 b(main)g(theorem)g(in)
g(this)h(section)f(is)g(the)h(follo)-34 b(wing:)0 52372
y Fz(Theorem)464 b(4.5)606 b Fr(If)446 b(one-way)d(functions)h(exist,)i
(then)f(ther)-62 b(e)443 b(exists)h(a)h(str)-62 b(ongly)443
b(unobfusc)-62 b(atable)440 b(function)k(en-)0 53878
y(semble.)1882 56145 y Ft(Similarly)351 b(to)j(the)f(w)-34
b(a)g(y)354 b(that)g(Theorem)f(3.11)f(implies)h(Theorem)g(3.8,)362
b(Theorem)353 b(4.5)g(implies)f(that,)364 b(assum-)0
57651 y(ing)462 b(the)g(existence)e(of)i(one-w)-34 b(a)g(y)463
b(functions,)477 b(an)462 b(ev)-34 b(en)461 b(w)-34 b(eak)g(er)462
b(de\014nition)g(of)g(circuit)f(obfuscators)i(\(one)f(that)0
59156 y(allo)-34 b(ws)461 b(the)h(obfuscated)g(circuit)e(to)i(only)e
(appro)-34 b(ximate)462 b(the)f(original)g(circuit\))f(is)h(imp)34
b(ossible)460 b(to)i(meet.)708 b(W)-101 b(e)0 60662 y(note)355
b(that)h(it)f(some)g(\(but)h(not)f(all\))g(applications)g(of)g
(obfuscators,)366 b(a)355 b(w)-34 b(eak)g(er)354 b(notion)i(of)f(appro)
-34 b(ximation)356 b(migh)-34 b(t)0 62167 y(su\016ce.)971
b(Sp)34 b(eci\014cally)-101 b(,)583 b(in)548 b(some)g(cases)g(it)h
(su\016ces)f(for)g(the)h(obfuscator)h(to)e(only)g(appro)-34
b(ximately)549 b(preserv)-34 b(e)0 63673 y(functionalit)g(y)417
b(with)g(resp)34 b(ect)415 b(to)h(a)g(particular)g(distribution)h(on)f
(inputs,)k(suc)-34 b(h)416 b(as)g(the)g(uniform)h(distribution.)0
65178 y(W)-101 b(e)404 b(do)g(not)h(kno)-34 b(w)405 b(whether)g(suc)-34
b(h)405 b(obfuscators)g(exist,)f(and)h(lea)-34 b(v)g(e)403
b(it)h(as)h(an)f(op)34 b(en)404 b(problem.)1882 66684
y(W)-101 b(e)450 b(shall)h(pro)-34 b(v)g(e)451 b(this)h(theorem)f(in)g
(the)g(follo)-34 b(wing)452 b(stages.)679 b(First)451
b(w)-34 b(e)451 b(will)f(see)h(wh)-34 b(y)452 b(the)f(pro)34
b(of)451 b(of)g(Theo-)0 68189 y(rem)360 b(3.11)g(do)34
b(es)360 b(not)h(apply)f(directly)f(to)i(the)f(case)g(of)h(appro)-34
b(ximate)361 b(implemen)-34 b(tations.)525 b(Then)361
b(w)-34 b(e)360 b(shall)g(de\014ne)0 69694 y(a)527 b(construct)h
(called)e Fr(invoker-r)-62 b(andomizable)542 b(pseudor)-62
b(andom)544 b(functions)p Ft(,)555 b(whic)-34 b(h)528
b(will)f(help)g(us)h(mo)34 b(dify)526 b(the)0 71200 y(original)403
b(pro)34 b(of)405 b(to)f(hold)h(in)f(this)h(case.)25394
74521 y(18)p eop
%%Page: 19 20
19 19 bop 0 1218 a Fz(4.2.2)1394 b(Generalizing)465 b(the)g(Pro)39
b(of)464 b(of)h(Theorem)f(3.11)h(to)f(the)h(Appro)-39
b(ximate)465 b(Case)0 3506 y Ft(The)386 b(\014rst)h(question)f(is)g
(whether)g(the)h(pro)34 b(of)386 b(of)g(Theorem)g(3.11)g(already)f(sho)
-34 b(ws)387 b(that)g(the)g(ensem)-34 b(ble)386 b Fp(fH)48990
3703 y Fj(k)49558 3506 y Fp(g)50164 3703 y Fj(k)24 b
FG(2)p Fe(N)0 5011 y Ft(de\014ned)492 b(there)f(is)g(actually)g(a)g
Fr(str)-62 b(ongly)597 b Ft(unobfuscatable)493 b(function)f(ensem)-34
b(ble.)800 b(As)491 b(w)-34 b(e)492 b(explain)f(b)34
b(elo)-34 b(w,)512 b(the)0 6517 y(answ)-34 b(er)405 b(is)f(no.)1882
8022 y(T)-101 b(o)419 b(see)e(wh)-34 b(y)-101 b(,)423
b(let)418 b(us)h(recall)e(the)h(de\014nition)i(of)e(the)h(ensem)-34
b(ble)419 b Fp(fH)32854 8219 y Fj(k)33422 8022 y Fp(g)34028
8219 y Fj(k)24 b FG(2)p Fe(N)36282 8022 y Ft(that)420
b(is)e(de\014ned)h(there)g(and)g(uses)0 9528 y(the)505
b(distributions)i Fp(F)10248 9725 y Fj(k)11322 9528 y
Ft(and)f Fp(G)14501 9725 y Fj(k)15574 9528 y Ft(that)g(are)f(de\014ned)
h(in)f(the)g(pro)34 b(of)506 b(of)f(Theorem)g(3.8.)841
b(The)505 b(distribution)h Fp(H)51431 9725 y Fj(k)0 11033
y Ft(is)438 b(de\014ned)h(b)-34 b(y)439 b(taking)g(an)f(elemen)-34
b(t)439 b(from)f Fp(F)20952 11230 y Fj(k)21959 11033
y Ft(or)g Fp(G)24199 11230 y Fj(k)24768 11033 y Ft(,)447
b(with)439 b(probabilit)-34 b(y)439 b(1)p Fo(=)p Ft(2)f(eac)-34
b(h.)641 b(The)439 b(distribution)h Fp(F)50178 11230
y Fj(k)51185 11033 y Ft(is)0 12709 y(de\014ned)392 b(b)-34
b(y)392 b(c)-34 b(ho)34 b(osing)392 b Fo(\013)t(;)202
b(\014)13367 12022 y Fw(R)13054 12709 y Fp( )244 b(f)p
Ft(0)p Fo(;)202 b Ft(1)p Fp(g)17473 12269 y Fj(k)18433
12709 y Ft(,)394 b(a)d(function)h Fo(D)26489 12022 y
Fw(R)26176 12709 y Fp( )244 b(D)28567 12906 y Fj(\013;\014)30453
12709 y Ft(and)392 b(outputting)i Fo(C)39711 12906 y
Fj(\013;\014)41205 12709 y Ft(#)p Fo(D)34 b Ft(.)534
b(Similarly)-101 b(,)392 b Fp(G)50225 12906 y Fj(k)51185
12709 y Ft(is)0 14452 y(de\014ned)449 b(b)-34 b(y)449
b(c)-34 b(ho)34 b(osing)448 b Fo(\013)t(;)202 b(\014)13593
13765 y Fw(R)13280 14452 y Fp( )299 b(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)17754 14012 y Fj(k)18771 14452 y Ft(,)459
b Fo(D)21215 13765 y Fw(R)20903 14452 y Fp( )298 b(D)23348
14649 y Fj(\013;\014)25291 14452 y Ft(and)449 b(outputting)i
Fo(Z)34624 14649 y Fj(k)35193 14452 y Ft(#)p Fo(D)34
b Ft(.)670 b(The)448 b(prop)34 b(ert)-34 b(y)449 b Fo(\031)491
b Ft(is)448 b(de\014ned)0 15957 y(simply)404 b(to)g(distinguish)i
(functions)f(in)f Fp(F)19012 16154 y Fj(k)19986 15957
y Ft(from)g(those)h(in)f Fp(G)28158 16154 y Fj(k)28727
15957 y Ft(.)1882 17463 y(That)413 b(pro)34 b(of)413
b(ga)-34 b(v)g(e)413 b(an)g(algorithm)f Fo(A)18967 17023
y FG(0)19690 17463 y Ft(whic)-34 b(h)414 b(computes)f
Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))414 b(giv)-34 b(en)413
b(a)f(circuit)g(computing)i(an)-34 b(y)413 b(function)0
18968 y Fo(f)521 b Ft(from)391 b Fp(H)12 b Ft(.)534 b(Let)390
b(us)h(see)f(wh)-34 b(y)391 b Fo(A)14973 18528 y FG(0)15675
18968 y Ft(migh)-34 b(t)391 b(fail)g(when)g(giv)-34 b(en)391
b(only)f(an)h(appro)-34 b(ximate)391 b(implemen)-34 b(tation)392
b(of)f Fo(f)130 b Ft(.)535 b(On)0 20474 y(input)287 b(a)f(circuit)f
Fo(F)168 b Ft(,)310 b Fo(A)10165 20034 y FG(0)10761 20474
y Ft(w)-34 b(orks)286 b(as)g(follo)-34 b(ws:)480 b(It)286
b(decomp)34 b(oses)285 b Fo(F)455 b Ft(in)-34 b(to)286
b(t)-34 b(w)g(o)288 b(circuits)d Fo(F)505 b Ft(=)337
b Fo(F)40726 20656 y Fl(1)41252 20474 y Ft(#)p Fo(F)43041
20656 y Fl(2)43567 20474 y Ft(.)499 b Fo(F)45182 20656
y Fl(2)45994 20474 y Ft(is)286 b(used)g(only)0 21979
y(in)425 b(a)f(blac)-34 b(k-b)34 b(o)-34 b(x)425 b(manner,)k(but)d(the)
f(queries)f Fo(A)22081 21539 y FG(0)22816 21979 y Ft(mak)-34
b(es)424 b(to)h(it)g(dep)34 b(end)425 b(on)g(the)g(gate)g(structure)g
(of)g(the)g(circuit)0 23485 y Fo(F)779 23667 y Fl(1)1305
23485 y Ft(.)524 b(The)361 b(problem)f(is)g(that)i(a)e(vicious)f(appro)
-34 b(ximate)361 b(implemen)-34 b(tation)362 b(for)e(a)g(function)i
Fo(C)41900 23682 y Fj(\013;\014)43395 23485 y Ft(#)p
Fo(D)370 b Fp(2)337 b Ft(Supp)q(\()p Fp(F)50959 23682
y Fj(k)51529 23485 y Ft(\))0 24990 y(ma)-34 b(y)332 b(w)-34
b(ork)333 b(in)f(the)h(follo)-34 b(wing)333 b(w)-34 b(a)g(y:)503
b(c)-34 b(ho)34 b(ose)333 b(a)f(random)h(circuit)f Fo(F)30316
25172 y Fl(1)31174 24990 y Ft(out)h(of)g(some)f(set)h
Fp(C)403 b Ft(of)332 b(exp)34 b(onen)-34 b(tially)332
b(man)-34 b(y)0 26495 y(circuits)388 b(that)i(compute)f
Fo(C)12614 26692 y Fj(\013;\014)14109 26495 y Ft(,)i(and)e(tak)-34
b(e)389 b Fo(F)20568 26677 y Fl(2)21483 26495 y Ft(that)g(computes)h
Fo(D)34 b Ft(.)532 b(Then)389 b(see)f(at)h(whic)-34 b(h)390
b(p)34 b(oin)-34 b(ts)389 b Fo(A)45927 26056 y FG(0)46626
26495 y Ft(queries)f Fo(F)51474 26677 y Fl(2)0 28001
y Ft(when)471 b(giv)-34 b(en)470 b Fo(F)7241 28183 y
Fl(1)7767 28001 y Ft(#)p Fo(F)9556 28183 y Fl(2)10552
28001 y Ft(as)g(input.)15270 27561 y Fl(6)16533 28001
y Ft(As)g(these)h(places)e(dep)34 b(end)471 b(on)f Fo(F)32005
28183 y Fl(1)32531 28001 y Ft(,)486 b(it)470 b(is)g(p)34
b(ossible)470 b(that)h(for)f(eac)-34 b(h)470 b Fo(F)48727
28183 y Fl(1)49700 28001 y Fp(2)446 b(C)71 b Ft(,)0 29506
y(there)475 b(is)f(a)h(p)34 b(oin)-34 b(t)476 b Fo(x)p
Ft(\()p Fo(F)10721 29688 y Fl(1)11247 29506 y Ft(\))f(suc)-34
b(h)476 b(that)g Fo(A)18604 29067 y FG(0)19390 29506
y Ft(will)e(query)g Fo(F)25970 29688 y Fl(2)26971 29506
y Ft(at)h(the)h(p)34 b(oin)-34 b(t)475 b Fo(x)p Ft(\()p
Fo(F)35860 29688 y Fl(1)36387 29506 y Ft(\),)492 b(but)476
b Fo(x)p Ft(\()p Fo(F)41923 29688 y Fl(1)42450 29506
y Ft(\))455 b Fp(6)p Ft(=)f Fo(x)p Ft(\()p Fo(F)46884
29067 y FG(0)46716 29831 y Fl(1)47242 29506 y Ft(\))476
b(for)e(an)-34 b(y)0 31012 y Fo(F)947 30572 y FG(0)779
31337 y Fl(1)1831 31012 y Fp(2)525 b(C)416 b(n)345 b(f)p
Fo(F)6554 31194 y Fl(1)7080 31012 y Fp(g)p Ft(.)878 b(If)518
b(the)f(appro)-34 b(ximate)518 b(implemen)-34 b(tation)519
b(c)-34 b(hanges)518 b(the)g(v)-67 b(alue)516 b(of)i
Fo(F)40630 31194 y Fl(2)41674 31012 y Ft(at)f Fo(x)p
Ft(\()p Fo(F)45211 31194 y Fl(1)45738 31012 y Ft(\),)545
b(then)519 b Fo(A)50875 30572 y FG(0)51185 31012 y Ft('s)0
32517 y(computation)406 b(on)e Fo(F)9533 32699 y Fl(1)10059
32517 y Ft(#)p Fo(F)11848 32699 y Fl(2)12779 32517 y
Ft(is)g(corrupted.)1882 34023 y(One)415 b(w)-34 b(a)g(y)416
b(to)g(solv)-34 b(e)415 b(this)i(problem)e(w)-34 b(ould)417
b(b)34 b(e)415 b(to)h(mak)-34 b(e)415 b(the)h(queries)f(that)i
Fo(A)38118 33583 y FG(0)38844 34023 y Ft(mak)-34 b(es)415
b(to)h Fo(F)44770 34205 y Fl(2)45712 34023 y Fr(indep)-62
b(endent)0 35528 y Ft(of)514 b(the)f(structure)h(of)g
Fo(F)11263 35710 y Fl(1)11789 35528 y Ft(.)866 b(If)513
b Fo(A)15222 35088 y FG(0)16046 35528 y Ft(had)h(this)f(prop)34
b(ert)-34 b(y)-101 b(,)541 b(then)514 b(giv)-34 b(en)513
b(an)g Fo(\017)p Ft(-appro)-34 b(ximate)514 b(implemen)-34
b(tation)515 b(of)0 37034 y Fo(C)866 37231 y Fj(\013;\014)2361
37034 y Ft(#)p Fo(D)34 b Ft(,)439 b(eac)-34 b(h)433 b(query)g(of)g
Fo(A)13659 36594 y FG(0)14403 37034 y Ft(w)-34 b(ould)434
b(ha)-34 b(v)g(e)434 b(only)f(an)g Fo(\017)g Ft(c)-34
b(hance)434 b(to)f(get)g(an)h(incorrect)e(answ)-34 b(er)434
b(and)g(o)-34 b(v)g(erall)432 b Fo(A)51689 36594 y FG(0)0
38539 y Ft(w)-34 b(ould)478 b(succeed)f(with)g(probabilit)-34
b(y)478 b(1)317 b Fp(\000)h Fo(\017)g Fp(\001)f Fo(p)p
Ft(\()p Fo(k)38 b Ft(\))478 b(for)f(some)g(p)34 b(olynomial)476
b Fo(p)p Ft(\()p Fp(\001)p Ft(\).)757 b(\(Note)477 b(that)h(the)f
(probabilit)-34 b(y)0 40045 y(that)405 b Fo(F)3405 40227
y Fl(1)3932 40045 y Ft(\()p Fo(\013)t Ft(\))g(c)-34 b(hanges)405
b(is)f(at)g(most)h Fo(\017)p Ft(.\))1882 41550 y(W)-101
b(e)398 b(will)g(not)h(b)34 b(e)398 b(able)h(to)g(ac)-34
b(hiev)g(e)398 b(this,)i(but)g(something)f(sligh)-34
b(tly)399 b(w)-34 b(eak)g(er)398 b(that)i(still)e(su\016ces.)537
b(Let's)398 b(lo)34 b(ok)0 43056 y(more)374 b(closely)f(at)h(the)h
(structure)g(of)g Fp(D)17817 43253 y Fj(\013;\014)19685
43056 y Ft(whic)-34 b(h)376 b(is)e(de\014ned)h(in)f(the)h(pro)34
b(of)374 b(of)h(Lemma)f(3.6.)528 b(W)-101 b(e)374 b(de\014ned)h(there)0
44561 y(the)405 b(algorithm)17955 46331 y Fo(D)18959
46528 y Fj(K)10 b(;\013;\014)21787 45644 y Fl(def)21965
46331 y Ft(=)447 b Fo(E)24250 46519 y Fj(K)10 b(;\013)25975
46331 y Ft(#Hom)29510 46519 y Fj(K)30422 46331 y Ft(#)p
Fo(B)32351 46528 y Fj(K)g(;\014)0 48556 y Ft(and)539
b(turned)g(it)g(in)-34 b(to)539 b(a)f(deterministic)g(function)i(b)-34
b(y)539 b(using)g(a)f(pseudorandom)i(function)g Fo(f)44014
48116 y FG(0)43884 48915 y Fj(K)45334 48556 y Ft(and)f(de\014ning)0
50061 y Fo(D)1038 49621 y FG(0)1004 50436 y Fj(K)10 b(;\013;\014)46
b(;K)4624 50184 y Fy(0)5474 50061 y Ft(to)496 b(b)34
b(e)496 b(the)g(deterministic)f(algorithm)h(that)h(on)f(input)h
Fo(x)489 b Fp(2)g(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)37218
49621 y Fj(q)38221 50061 y Ft(ev)-67 b(aluates)495 b
Fo(D)44542 50258 y Fj(K)10 b(;\013;\014)47101 50061 y
Ft(\()p Fo(x)p Ft(\))496 b(using)0 51689 y(randomness)514
b Fo(f)7318 51893 y Fj(K)8174 51641 y Fy(0)8528 51689
y Ft(\()p Fo(x)p Ft(\).)866 b(W)-101 b(e)512 b(then)i(de\014ned)g
Fp(D)21651 51886 y Fj(\013;\014)23659 51689 y Ft(to)f(b)34
b(e)513 b Fo(D)28046 51249 y FG(0)28012 52064 y Fj(K)10
b(;\013;\014)46 b(;K)31632 51812 y Fy(0)32505 51689 y
Ft(=)518 b Fo(E)34931 51249 y FG(0)34861 52064 y Fj(K)10
b(;\013;K)37646 51812 y Fy(0)38002 51689 y Ft(#Hom)41537
51211 y FG(0)41537 52026 y Fj(K)g(;K)43457 51774 y Fy(0)43812
51689 y Ft(#)p Fo(B)45741 51886 y Fj(K)g(;\014)47948
51689 y Ft(for)513 b(uni-)0 53194 y(formly)404 b(selected)f(priv)-67
b(ate)404 b(k)-34 b(ey)403 b Fo(K)491 b Ft(and)405 b(seed)f
Fo(K)22252 52755 y FG(0)22563 53194 y Ft(.)1882 54700
y(No)-34 b(w)516 b(our)f(algorithm)g Fo(A)13568 54260
y FG(0)14394 54700 y Ft(\(that)h(uses)g(the)f(algorithm)g
Fo(A)g Ft(de\014ned)h(in)g(Lemma)e(3.6\))h(treats)h Fo(F)46663
54882 y Fl(2)47704 54700 y Ft(as)f(three)0 56205 y(oracles:)610
b Fo(E)70 b Ft(,)449 b Fo(H)98 b Ft(,)449 b(and)441 b
Fo(B)502 b Ft(,)449 b(where)440 b(if)g Fo(F)18237 56387
y Fl(2)19204 56205 y Ft(computes)h Fo(D)431 b Ft(=)397
b Fo(E)28374 55766 y FG(0)28304 56581 y Fj(K)10 b(;\013;K)31089
56328 y Fy(0)31444 56205 y Ft(#Hom)34979 55728 y FG(0)34979
56543 y Fj(K)g(;K)36899 56291 y Fy(0)37254 56205 y Ft(#)p
Fo(B)39183 56402 y Fj(K)g(;\014)41318 56205 y Ft(then)441
b Fo(E)510 b Ft(is)440 b(the)h(oracle)0 57833 y(to)430
b Fo(E)2472 57393 y FG(0)2402 58208 y Fj(K)10 b(;\013;K)5187
57956 y Fy(0)5543 57833 y Ft(,)435 b Fo(H)528 b Ft(is)430
b(the)g(oracle)f(to)h(Hom)18772 57355 y FG(0)18772 58171
y Fj(K)10 b(;K)20692 57918 y Fy(0)21477 57833 y Ft(and)430
b Fo(B)491 b Ft(is)430 b(the)g(oracle)f(to)h Fo(B)34584
58030 y Fj(K)10 b(;\014)36279 57833 y Ft(.)615 b(The)431
b(queries)d(to)j Fo(E)499 b Ft(are)429 b(at)i(the)0 59339
y(places)416 b(1)p Fo(;)202 b(:)g(:)g(:)f(;)h(k)455 b
Ft(and)417 b(so)g(are)g(indep)34 b(enden)-34 b(t)418
b(of)f(the)g(structure)g(of)h Fo(F)31639 59521 y Fl(1)32165
59339 y Ft(.)576 b(The)417 b(queries)f(that)i Fo(A)f
Ft(mak)-34 b(es)417 b(to)g(the)g Fo(H)0 60844 y Ft(oracle,)403
b(ho)-34 b(w)g(ev)g(er,)404 b(do)h(dep)34 b(end)405 b(on)f(the)h
(structure)f(of)h Fo(F)25850 61026 y Fl(1)26376 60844
y Ft(.)1882 62350 y(Recall)508 b(that)j(an)-34 b(y)510
b(query)f Fo(A)15152 61910 y FG(0)15972 62350 y Ft(mak)-34
b(es)510 b(to)g(the)g Fo(H)607 b Ft(oracle)509 b(are)g(of)h(the)g(form)
f(\()p Fo(c;)202 b(d;)g Fp(\014)p Ft(\))509 b(where)h
Fo(c)f Ft(and)h Fo(d)f Ft(are)0 63855 y(ciphertexts)600
b(of)h(some)f(bits,)649 b(and)601 b Fp(\014)f Ft(is)g(a)h(4-bit)f
(description)h(of)f(a)h(binary)f(b)34 b(o)g(olean)600
b(function.)1128 b(Just)601 b(for)0 65361 y(motiv)-67
b(ation,)533 b(supp)34 b(ose)508 b(that)h Fo(A)14861
64921 y FG(0)15678 65361 y Ft(has)f(the)g(follo)-34 b(wing)508
b(abilit)-34 b(y:)745 b(giv)-34 b(en)507 b(an)h(encryption)f
Fo(c)p Ft(,)533 b Fo(A)43296 64921 y FG(0)44113 65361
y Ft(can)508 b(generate)f(a)0 66866 y(random)442 b(encryption)f(of)g
(the)h(same)f(bit)g(\(i.e.,)449 b(distributed)443 b(according)e(to)g
(Enc)37190 67054 y Fj(K)38102 66866 y Ft(\(Dec)40576
67054 y Fj(K)41488 66866 y Ft(\()p Fo(c)p Ft(\))p Fo(;)202
b(r)34 b Ft(\))441 b(for)g(uniformly)p 0 67988 20800
45 v 1382 68705 a Fw(6)1843 69128 y Fx(Recall)305 b(that)g
Fi(A)7842 68705 y Fy(0)8444 69128 y Fx(is)g(not)f(some)h(giv)-28
b(en)305 b(algorithm)g(that)f(w)-28 b(e)304 b(m)-28 b(ust)304
b(treat)h(as)f(a)g(blac)-28 b(k-b)28 b(o)-28 b(x)305
b(but)g(rather)e(a)i(sp)28 b(eci\014c)305 b(algorithm)g(that)0
70346 y(w)-28 b(e)341 b(de\014ned)g(ourselv)-28 b(es.)25394
74521 y Ft(19)p eop
%%Page: 20 21
20 20 bop 0 1218 a Ft(selected)337 b Fo(r)34 b Ft(\).)516
b(F)-101 b(or)338 b(instance,)351 b(this)338 b(w)-34
b(ould)339 b(b)34 b(e)337 b(true)h(if)f(the)h(encryption)g(sc)-34
b(heme)338 b(w)-34 b(ere)337 b(\\random)i(self-reducible.")0
2723 y(Supp)34 b(ose)408 b(no)-34 b(w)408 b(that,)g(b)34
b(efore)407 b(querying)f(the)h Fo(H)505 b Ft(oracle)406
b(with)h(\()p Fo(c;)202 b(d;)g Fp(\014)p Ft(\),)407 b
Fo(A)34529 2283 y FG(0)35246 2723 y Ft(generates)g Fo(c)41104
2283 y FG(0)41414 2723 y Fo(;)202 b(d)42584 2283 y FG(0)43301
2723 y Ft(that)408 b(are)e(random)0 4229 y(encryptions)494
b(of)h(the)f(same)g(bits)g(as)g Fo(c;)202 b(d)493 b Ft(and)i(query)e
(the)i(oracle)d(with)j(\()p Fo(c)35118 3789 y FG(0)35429
4229 y Fo(;)202 b(d)36599 3789 y FG(0)36909 4229 y Fo(;)g
Fp(\014)p Ft(\))494 b(instead.)808 b(W)-101 b(e)494 b(claim)f(that)0
5734 y(if)503 b Fo(F)1989 5916 y Fl(2)3018 5734 y Ft(is)g(an)h
Fo(\017)p Ft(-appro)-34 b(ximate)504 b(implemen)-34 b(tation)504
b(of)f Fo(D)34 b Ft(,)527 b(then)504 b(for)f(an)-34 b(y)504
b(suc)-34 b(h)504 b(query)-101 b(,)527 b(there)503 b(is)g(at)g(most)h
(a)f(64)p Fo(\017)0 7240 y Ft(probabilit)-34 b(y)370
b(for)f(the)g(answ)-34 b(er)370 b(to)f(b)34 b(e)369 b(wrong)h
Fr(even)401 b(if)f Ft(\()p Fo(c;)202 b(d;)g Fp(\014)p
Ft(\))401 b Fr(dep)-62 b(end)400 b(on)h(the)f(cir)-62
b(cuit)400 b Fo(F)168 b Fr(.)527 b Ft(The)370 b(reason)f(is)g(that)0
8745 y(the)380 b(distribution)h(of)g(the)f(mo)34 b(di\014ed)380
b(query)f(\()p Fo(c)21335 8305 y FG(0)21646 8745 y Fo(;)202
b(d)22816 8305 y FG(0)23126 8745 y Fo(;)g Fp(\014)p Ft(\))379
b(dep)34 b(ends)381 b(only)e(on)i(\(Dec)36890 8933 y
Fj(K)37802 8745 y Ft(\()p Fo(c)p Ft(\))p Fo(;)202 b Ft(Dec)41811
8933 y Fj(K)42723 8745 y Ft(\()p Fo(d)p Ft(\))p Fo(;)g
Fp(\014)p Ft(\),)384 b(and)d(there)0 10250 y(are)450
b(only)g(2)300 b Fp(\001)g Ft(2)h Fp(\001)e Ft(2)8468
9811 y Fl(4)9408 10250 y Ft(=)414 b(64)450 b(p)34 b(ossibilities)450
b(for)g(the)h(latter.)678 b(F)-101 b(or)450 b(eac)-34
b(h)450 b(of)h(the)g(64)f(p)34 b(ossibilities,)462 b(the)450
b(probabilit)-34 b(y)0 11756 y(of)373 b(an)g(incorrect)f(answ)-34
b(er)374 b(\(o)-34 b(v)g(er)373 b(the)g(c)-34 b(hoice)372
b(of)h Fo(F)168 b Ft(\))374 b(is)f(at)g(most)g Fo(\017)p
Ft(.)528 b(Cho)34 b(osing)374 b(\(Dec)38487 11944 y Fj(K)39398
11756 y Ft(\()p Fo(c)p Ft(\))p Fo(;)202 b Ft(Dec)43408
11944 y Fj(K)44319 11756 y Ft(\()p Fo(d)p Ft(\))p Fo(;)g
Fp(\014)p Ft(\))373 b(after)g Fo(F)0 13261 y Ft(to)404
b(maximize)f(the)i(probabilit)-34 b(y)405 b(of)f(an)h(incorrect)e(answ)
-34 b(er)405 b(m)-34 b(ultiplies)404 b(this)h(probabilit)-34
b(y)404 b(b)-34 b(y)405 b(at)f(most)h(64.)1882 14767
y(W)-101 b(e)524 b(shall)h(no)-34 b(w)527 b(use)e(this)g(motiv)-67
b(ation)526 b(to)f(\014x)g(the)h(function)g Fo(D)558
b Ft(so)526 b(that)g Fo(A)37953 14327 y FG(0)38789 14767
y Ft(will)e(essen)-34 b(tially)525 b(ha)-34 b(v)g(e)525
b(this)0 16272 y(desired)450 b(abilit)-34 b(y)450 b(of)h(randomly)f
(self-reducing)h(an)-34 b(y)450 b(encryption)h(to)f(a)h(random)g
(encryption)f(of)h(the)g(same)f(bit.)0 17778 y(Recall)406
b(that)h(Hom)8810 17300 y FG(0)8810 18115 y Fj(K)10 b(;K)10730
17863 y Fy(0)11085 17778 y Ft(\()p Fo(c;)202 b(d;)g Fp(\014)p
Ft(\))340 b(=)g(Enc)18864 17966 y Fj(K)19776 17778 y
Ft(\(Dec)22251 17966 y Fj(K)23162 17778 y Ft(\()p Fo(c)p
Ft(\))271 b Fp(\014)g Ft(Dec)28117 17966 y Fj(K)29029
17778 y Ft(\()p Fo(d)p Ft(\);)202 b Fo(f)31734 17981
y Fj(K)32590 17729 y Fy(0)32944 17778 y Ft(\()p Fo(c;)g(d;)g
Fp(\014)p Ft(\)\).)546 b(No)-34 b(w,)408 b(a)e(naiv)-34
b(e)407 b(approac)-34 b(h)408 b(to)0 19283 y(ensure)391
b(that)h(an)-34 b(y)391 b(query)g(returns)g(a)g(random)g(encryption)h
(of)f(Dec)31009 19471 y Fj(K)31921 19283 y Ft(\()p Fo(c)p
Ft(\))243 b Fp(\014)g Ft(Dec)36820 19471 y Fj(K)37731
19283 y Ft(\()p Fo(d)p Ft(\))392 b(w)-34 b(ould)392 b(b)34
b(e)391 b(to)g(c)-34 b(hange)391 b(the)0 20789 y(de\014nition)431
b(of)e(Hom)9344 20311 y FG(0)10084 20789 y Ft(to)g(the)h(follo)-34
b(wing:)590 b(Hom)21868 20311 y FG(0)21868 21126 y Fj(K)10
b(;K)23788 20874 y Fy(0)24143 20789 y Ft(\()p Fo(c;)202
b(d;)g Fp(\014)p Fo(;)g(r)34 b Ft(\))378 b(=)h(Enc)33119
20977 y Fj(K)34030 20789 y Ft(\(Dec)36505 20977 y Fj(K)37416
20789 y Ft(\()p Fo(c)p Ft(\))287 b Fp(\014)f Ft(Dec)42402
20977 y Fj(K)43313 20789 y Ft(\()p Fo(d)p Ft(\);)202
b Fo(r)34 b Ft(\).)615 b(Then)431 b(w)-34 b(e)0 22294
y(c)g(hange)510 b Fo(A)5021 21854 y FG(0)5841 22294 y
Ft(to)g(an)g(algorithm)f Fo(A)15756 21854 y FG(00)16831
22294 y Ft(that)i(c)-34 b(ho)34 b(oses)509 b(a)h(uniform)g
Fo(r)546 b Fp(2)512 b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)35137
21854 y Fj(n)36272 22294 y Ft(and)511 b(thereb)-34 b(y)509
b(ensures)h(that)g(the)0 23800 y(result)393 b(is)f(a)h(random)g
(encryption)g(of)g(Dec)19399 23988 y Fj(K)20311 23800
y Ft(\()p Fo(c)p Ft(\))246 b Fp(\014)h Ft(Dec)25217 23988
y Fj(K)26128 23800 y Ft(\()p Fo(d)p Ft(\).)536 b(The)393
b(problem)g(is)f(that)i(this)f(construction)h(w)-34 b(ould)0
25305 y(no)429 b(longer)f(satisfy)g(Prop)34 b(ert)-34
b(y)428 b(3)g(of)h(Lemma)f(3.6)g(\(securit)-34 b(y)428
b(against)h(a)g(sim)-34 b(ulator)428 b(with)h(oracle)f(access\).)610
b(This)0 26811 y(is)365 b(b)34 b(ecause)365 b(the)h(sim)-34
b(ulator)366 b(could)g(no)-34 b(w)366 b(con)-34 b(trol)366
b(the)g(random)g(coins)f(of)h(the)g(encryption)f(sc)-34
b(heme)366 b(and)g(use)f(this)0 28316 y(to)404 b(break)g(it.)539
b(Our)404 b(solution)h(will)e(b)34 b(e)404 b(to)g(rede\014ne)h(Hom)
26044 27838 y FG(0)26758 28316 y Ft(in)g(the)f(follo)-34
b(wing)405 b(w)-34 b(a)g(y:)9262 31039 y(Hom)11787 30539
y FG(0)11787 31354 y Fj(K)10 b(;K)13707 31102 y Fy(0)14062
31039 y Ft(\()p Fo(c;)202 b(d;)g Fp(\014)p Fo(;)g(r)34
b Ft(\))335 b(=)i(Enc)22953 31227 y Fj(K)23864 31039
y Ft(\(Dec)26339 31227 y Fj(K)27250 31039 y Ft(\()p Fo(c)p
Ft(\))270 b Fp(\014)f Ft(Dec)32202 31227 y Fj(K)33114
31039 y Ft(\()p Fo(d)p Ft(\);)202 b Fo(f)35819 31243
y Fj(K)36675 30991 y Fy(0)37029 31039 y Ft(\()p Fo(c;)g(d;)g
Fp(\014)p Fo(;)g(r)34 b Ft(\)\))0 33762 y(but)405 b(require)e(an)i
(additional)f(sp)34 b(ecial)404 b(prop)34 b(ert)-34 b(y)404
b(from)g(the)h(pseudorandom)h(function)f Fo(f)40997 33966
y Fj(K)41853 33714 y Fy(0)42207 33762 y Ft(.)0 36964
y Fz(4.2.3)1394 b(In)-39 b(v)g(ok)g(er-Randomizable)465
b(Pseudorandom)g(F)-116 b(unctions)0 39252 y Ft(The)405
b(prop)34 b(ert)-34 b(y)404 b(w)-34 b(e)404 b(w)-34 b(ould)406
b(lik)-34 b(e)403 b(the)h(pseudorandom)i(function)g Fo(f)30509
39456 y Fj(K)31365 39204 y Fy(0)32123 39252 y Ft(to)e(p)34
b(ossess)405 b(is)f(the)g(follo)-34 b(wing:)0 42086 y
Fz(De\014nition)466 b(4.6)606 b Fr(A)431 b(function)d(ensemble)h
Fp(f)p Fo(f)21332 42290 y Fj(K)22188 42037 y Fy(0)22542
42086 y Fp(g)23148 42333 y Fj(K)24004 42081 y Fy(0)24302
42333 y FG(2f)p Fl(0)p Fj(;)p Fl(1)p FG(g)27070 42081
y Fy(\003)28038 42086 y Fr(\()p Fo(f)29127 42290 y Fj(K)29983
42037 y Fy(0)30673 42086 y Ft(:)337 b Fp(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)34310 41646 y Fj(q)32 b Fl(+)p Fj(n)36456
42086 y Fp(!)337 b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)40968
41646 y Fj(n)42025 42086 y Fr(,)430 b Fo(n)g Fr(,)p Fo(q)473
b Fr(p)-62 b(olynomial)62 b(ly)0 43591 y(r)-62 b(elate)g(d)394
b(to)h Fp(j)p Fo(K)6721 43152 y FG(0)7031 43591 y Fp(j)p
Fr(\))h(is)f(c)-62 b(al)62 b(le)-62 b(d)394 b(an)489
b Ft(in)-34 b(v)g(ok)g(er-randomizable)363 b(pseudorandom)i(function)f
(ensem)-34 b(ble)396 b Fr(if)g(the)f(fol)62 b(lowing)0
45097 y(holds:)1433 47599 y(1.)605 b Fp(f)p Fo(f)4229
47802 y Fj(K)5085 47550 y Fy(0)5439 47599 y Fp(g)6045
47845 y Fj(K)6901 47593 y Fy(0)7200 47845 y FG(2f)p Fl(0)p
Fj(;)p Fl(1)p FG(g)9968 47593 y Fy(\003)10939 47599 y
Fr(is)433 b(a)g(pseudor)-62 b(andom)431 b(function)h(ensemble)1433
50100 y(2.)605 b(F)-93 b(or)366 b(any)f Fo(x)336 b Fp(2)h(f)p
Ft(0)p Fo(;)202 b Ft(1)p Fp(g)12617 49661 y Fj(q)13489
50100 y Fr(,)379 b(if)365 b Fo(r)399 b Fr(is)364 b(chosen)g(uniformly)g
(in)i Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)31267 49661
y Fj(n)32258 50100 y Fr(then)365 b Fo(f)35478 50304 y
Fj(K)36334 50052 y Fy(0)36688 50100 y Ft(\()p Fo(x;)202
b(r)34 b Ft(\))365 b Fr(is)g(distribute)-62 b(d)361 b(uniformly)3030
51606 y(\(and)433 b(so)g(indep)-62 b(endently)431 b(of)i
Fo(x)p Fr(\))g(in)h Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)22546
51166 y Fj(n)23172 51606 y Fr(.)1882 54440 y Ft(F)-101
b(ortunately)g(,)404 b(w)-34 b(e)405 b(can)f(pro)-34
b(v)g(e)404 b(the)h(follo)-34 b(wing)405 b(lemma:)0 57273
y Fz(Lemma)464 b(4.7)606 b Fr(If)388 b(pseudor)-62 b(andom)385
b(functions)h(exist)h(then)g(ther)-62 b(e)386 b(exist)h(invoker-r)-62
b(andomizable)383 b(pseudor)-62 b(andom)0 58779 y(functions.)0
61613 y Fz(Pro)39 b(of)642 b(Sk)-39 b(etc)g(h:)2135 b
Ft(Supp)34 b(ose)560 b(that)g Fp(f)p Fo(g)19431 61816
y Fj(K)20287 61564 y Fy(0)20641 61613 y Fp(g)21247 61859
y Fj(K)22103 61607 y Fy(0)22402 61859 y FG(2f)p Fl(0)p
Fj(;)p Fl(1)p FG(g)25170 61607 y Fy(\003)26266 61613
y Ft(is)f(a)g(pseudorandom)h(function)h(ensem)-34 b(ble)559
b(and)h(that)0 63118 y Fp(f)p Fo(p)1216 63306 y Fj(S)1893
63118 y Fp(g)2499 63365 y Fj(S)50 b FG(2f)p Fl(0)p Fj(;)p
Fl(1)p FG(g)5889 63113 y Fy(\003)6731 63118 y Ft(is)305
b(a)g(pseudorandom)i(function)f(ensem)-34 b(ble)306 b(in)f(whic)-34
b(h)306 b(for)f(an)-34 b(y)305 b Fo(S)407 b Fp(2)337
b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)40350 62678 y FG(\003)41180
63118 y Ft(,)325 b Fo(p)42452 63306 y Fj(S)43434 63118
y Ft(is)305 b(a)g(p)34 b(erm)-34 b(utation)0 64624 y(\(the)510
b(existence)f(of)h(suc)-34 b(h)510 b(ensem)-34 b(bles)510
b(is)f(implied)h(b)-34 b(y)509 b(the)h(existence)f(of)h(ordinary)f
(pseudorandom)j(function)0 66129 y(ensem)-34 b(bles)404
b([LR88]\).)1882 67635 y(W)-101 b(e)403 b(de\014ne)i(the)g(function)g
(ensem)-34 b(ble)404 b Fp(f)p Fo(f)20663 67838 y Fj(K)21519
67586 y Fy(0)21874 67635 y Fp(g)22480 67881 y Fj(K)23336
67629 y Fy(0)23634 67881 y FG(2f)p Fl(0)p Fj(;)p Fl(1)p
FG(g)26402 67629 y Fy(\003)27343 67635 y Ft(in)h(the)f(follo)-34
b(wing)405 b(w)-34 b(a)g(y:)20313 70854 y Fo(f)20906
71058 y Fj(K)21762 70806 y Fy(0)22116 70854 y Ft(\()p
Fo(x;)202 b(r)34 b Ft(\))25140 70167 y Fl(def)25318 70854
y Ft(=)447 b Fo(p)27318 71101 y Fj(g)27766 71356 y Fc(K)28493
71165 y Fy(0)28847 71101 y Fl(\()p Fj(x)p Fl(\))30163
70854 y Ft(\()p Fo(r)34 b Ft(\))25394 74521 y(20)p eop
%%Page: 21 22
21 21 bop 0 1218 a Ft(It)599 b(is)g(clear)g(that)h(this)g(ensem)-34
b(ble)600 b(satis\014es)f(Prop)34 b(ert)-34 b(y)599 b(2)h(of)f
(De\014nition)h(4.6)g(as)f(for)g(an)-34 b(y)600 b Fo(x)p
Ft(,)648 b(the)600 b(function)0 2723 y Fo(r)370 b Fp(7!)337
b Fo(f)3059 2927 y Fj(K)3915 2675 y Fy(0)4269 2723 y
Ft(\()p Fo(x;)202 b(r)34 b Ft(\))404 b(is)g(a)g(p)34
b(erm)-34 b(utation.)1882 4229 y(What)332 b(needs)g(to)g(b)34
b(e)331 b(sho)-34 b(wn)333 b(is)f(that)g(it)g(is)f(a)h(pseudorandom)h
(function)g(ensem)-34 b(ble.)515 b(W)-101 b(e)331 b(do)g(this)i(b)-34
b(y)332 b(sho)-34 b(wing)0 5734 y(that)405 b(for)f(an)-34
b(y)405 b(PPT)f Fo(D)34 b Ft(,)403 b(the)i(follo)-34
b(wing)405 b(probabilities)f(are)f(iden)-34 b(tical)405
b(up)f(to)h(a)f(negligible)f(factor.)1481 8129 y(1.)606
b(Pr)4330 8333 y Fj(K)5186 8081 y Fy(0)5540 8129 y Ft([)p
Fo(D)6915 7689 y Fj(f)7367 7944 y Fc(K)8094 7753 y Fy(0)8502
8129 y Ft(\(1)9579 7689 y Fj(k)10148 8129 y Ft(\))337
b(=)g(1])403 b(\(where)i Fo(k)375 b Ft(=)336 b Fp(j)p
Fo(K)21297 7689 y FG(0)21608 8129 y Fp(j)p Ft(\).)1481
10484 y(2.)606 b(Pr)4330 10672 y Fj(G)5119 10484 y Ft([)p
Fo(D)6494 9995 y Fl(\()p Fj(x;R)5 b Fl(\))p FG(7!)p Fj(p)10146
10188 y Fc(G)p Fw(\()p Fc(x)p Fw(\))11935 9995 y Fl(\()p
Fj(R)g Fl(\))13437 10484 y Ft(\(1)14514 10045 y Fj(k)15083
10484 y Ft(\))337 b(=)g(1],)403 b(where)h Fo(G)g Ft(is)g(a)g(true)g
(random)h(function.)1481 12840 y(3.)606 b(Pr)4330 13028
y Fj(P)4927 13151 y Fw(1)5388 13028 y Fj(;:::)q(;P)7291
13139 y Fc(t)7719 12840 y Ft([)p Fo(D)9094 12400 y Fj(P)9691
12523 y Fw(1)10150 12400 y Fj(;:::)r(;P)12054 12511 y
Fc(t)12481 12840 y Ft(\(1)13558 12400 y Fj(k)14127 12840
y Ft(\))337 b(=)g(1],)351 b(where)339 b Fo(t)e Ft(=)f
Fo(t)p Ft(\()p Fo(k)38 b Ft(\))340 b(is)e(a)h(b)34 b(ound)341
b(on)e(the)g(n)-34 b(um)g(b)34 b(er)340 b(of)g(queries)e(that)i
Fo(D)372 b Ft(mak)-34 b(es)3030 14345 y(and)496 b(eac)-34
b(h)495 b(time)f Fo(D)528 b Ft(mak)-34 b(es)495 b(a)g(query)f(with)h(a)
g(new)g(v)-67 b(alue)494 b(of)h Fo(x)g Ft(w)-34 b(e)495
b(use)g(a)g(new)g(random)g(function)h Fo(P)51287 14527
y Fj(i)51663 14345 y Ft(.)3030 15851 y(\(This)405 b(requires)e(a)h(h)
-34 b(ybrid)405 b(argumen)-34 b(t\).)1481 18206 y(4.)606
b(Pr)4330 18394 y Fj(F)5111 18206 y Ft([)p Fo(D)6486
17766 y Fj(F)7267 18206 y Ft(\(1)8344 17766 y Fj(k)8913
18206 y Ft(\))337 b(=)g(1],)403 b(where)h Fo(F)573 b
Ft(is)404 b(a)g(truly)g(random)h(function.)51094 20916
y Fb(2)0 24049 y Fz(4.2.4)1394 b(Finishing)467 b(the)e(Pro)39
b(of)464 b(of)g(Theorem)g(4.5)0 26337 y Ft(No)-34 b(w,)444
b(supp)34 b(ose)436 b(w)-34 b(e)436 b(use)f(a)h(pseudorandom)h
(function)g Fo(f)26204 26541 y Fj(K)27060 26289 y Fy(0)27849
26337 y Ft(that)g(is)e(in)-34 b(v)g(ok)g(er-randomizable,)443
b(and)436 b(mo)34 b(dify)436 b(the)0 27843 y(algorithm)567
b Fo(A)6597 27403 y FG(0)7474 27843 y Ft(so)g(that)h(all)e(its)h
(queries)e(\()p Fo(c;)202 b(d;)g Fp(\014)p Ft(\))566
b(to)h(the)h Fo(H)664 b Ft(oracle)566 b(are)g(augmen)-34
b(ted)568 b(to)f(b)34 b(e)567 b(of)g(the)g(form)0 29348
y(\()p Fo(c;)202 b(d;)g Fp(\014)p Fo(;)g(r)34 b Ft(\),)406
b(where)h Fo(r)441 b Ft(is)406 b(c)-34 b(hosen)408 b(uniformly)f(and)h
(indep)34 b(enden)-34 b(tly)408 b(for)f(eac)-34 b(h)407
b(query)-101 b(.)547 b(Then)408 b(the)f(result)g(of)h(eac)-34
b(h)0 30854 y(suc)g(h)324 b(query)e(is)g(a)h Fr(r)-62
b(andom)322 b Ft(encryption)h(of)g(Dec)21563 31042 y
Fj(K)22474 30854 y Ft(\()p Fo(c)p Ft(\))107 b Fp(\014)g
Ft(Dec)27101 31042 y Fj(K)28012 30854 y Ft(\()p Fo(d)p
Ft(\).)512 b(Therefore,)339 b(as)323 b(argued)g(ab)34
b(o)-34 b(v)g(e,)339 b Fo(A)46118 30414 y FG(0)46751
30854 y Ft(nev)-34 b(er)323 b(gets)0 32359 y(a)420 b(wrong)h(answ)-34
b(er)421 b(from)g(the)g Fo(H)518 b Ft(oracle)419 b(with)i(probabilit)
-34 b(y)421 b(at)g(least)f(1)281 b Fp(\000)f Fo(p)p Ft(\()p
Fo(k)38 b Ft(\))280 b Fp(\001)g Fo(\017)p Ft(,)424 b(for)d(some)f(p)34
b(olynomial)420 b Fo(p)p Ft(\()p Fp(\001)p Ft(\).)0 33865
y(Indeed,)474 b(this)460 b(holds)h(b)34 b(ecause)459
b(aside)h(from)g(the)g(\014rst)h(queries)e(whic)-34 b(h)461
b(are)e(\014xed)h(and)h(therefore)e(indep)34 b(enden)-34
b(t)0 35370 y(of)371 b(the)h(gate)f(structure)g(of)g
Fo(F)13286 35552 y Fl(1)13813 35370 y Ft(,)377 b(all)370
b(other)h(queries)f(are)h(of)g(the)g(form)g(\()p Fo(c;)202
b(d;)g Fp(\014)p Fo(;)g(r)34 b Ft(\))370 b(where)h Fo(c)f
Ft(and)i Fo(d)e Ft(are)h(uniformly)0 36875 y(distributed)353
b(and)g(indep)34 b(enden)-34 b(t)353 b(encryptions)g(of)f(some)g(bits)g
Fo(a)g Ft(and)g Fo(b)p Ft(,)362 b(and)353 b Fo(r)385
b Ft(is)352 b(uniformly)g(distributed.)522 b(Only)0 38381
y(\()p Fo(a;)202 b(b;)g Fp(\014)p Ft(\))363 b(dep)34
b(end)363 b(on)g(the)h(gate)f(structure)g(of)g Fo(F)22203
38563 y Fl(1)22730 38381 y Ft(,)370 b(and)364 b(there)f(are)f(only)h
(64)g(p)34 b(ossibilities)362 b(for)h(them.)525 b(Assuming)0
39886 y Fo(A)909 39447 y FG(0)1791 39886 y Ft(nev)-34
b(er)571 b(gets)g(an)h(incorrect)f(answ)-34 b(er)572
b(from)f(the)h Fo(H)669 b Ft(oracle,)612 b(its)572 b(last)f(query)g(to)
g(the)h Fo(B)633 b Ft(oracle)570 b(will)h(b)34 b(e)571
b(a)0 41392 y(uniformly)395 b(distributed)g(encryption)g(of)g
Fo(\014)19789 41574 y Fl(1)20314 41392 y Fo(;)202 b(:)g(:)g(:)f(;)h
(\014)23694 41589 y Fj(k)24262 41392 y Ft(,)396 b(whic)-34
b(h)396 b(is)e(indep)34 b(enden)-34 b(t)396 b(of)f(the)g(structure)f
(of)h Fo(F)47309 41574 y Fl(1)47836 41392 y Ft(,)h(and)f(so)0
42897 y(has)405 b(only)f(an)g Fo(\017)g Ft(probabilit)-34
b(y)405 b(to)f(b)34 b(e)404 b(incorrect.)538 b(This)404
b(completes)g(the)h(pro)34 b(of.)1882 44403 y(One)474
b(p)34 b(oin)-34 b(t)476 b(to)f(note)g(is)f(that)i(w)-34
b(e)475 b(ha)-34 b(v)g(e)475 b(con)-34 b(v)g(erted)475
b(our)f(deterministic)h(algorithm)f Fo(A)42417 43963
y FG(0)43203 44403 y Ft(of)g(Theorem)h(3.11)0 45908 y(in)-34
b(to)405 b(a)f Fr(pr)-62 b(ob)g(abilistic)468 b Ft(algorithm.)0
49085 y Fn(4.3)1495 b(Imp)42 b(ossibilit)-42 b(y)500
b(of)e(the)g(applications)0 51373 y Ft(So)375 b(far,)380
b(w)-34 b(e)375 b(ha)-34 b(v)g(e)375 b(only)g(pro)-34
b(v)g(ed)375 b(imp)34 b(ossibilit)-34 b(y)374 b(of)h(some)g(natural)g
(and)g(arguably)g(minimalistic)f(de\014nitions)i(for)0
52879 y(obfuscation.)605 b(Y)-101 b(et)426 b(it)g(migh)-34
b(t)426 b(seem)g(that)h(there's)e(still)h(hop)34 b(e)425
b(for)h(a)g(di\013eren)-34 b(t)426 b(de\014nition)h(of)f(obfuscation,)
433 b(one)0 54384 y(that)401 b(will)e(not)i(b)34 b(e)400
b(imp)34 b(ossible)399 b(to)i(meet)e(but)j(w)-34 b(ould)401
b(still)e(b)34 b(e)400 b(useful)g(for)g(some)g(in)-34
b(tended)402 b(applications.)537 b(W)-101 b(e'll)0 55890
y(sho)-34 b(w)439 b(no)-34 b(w)439 b(that)f(this)g(is)g(not)g(the)g
(case)f(for)h(man)-34 b(y)438 b(of)g(the)g(applications)g(w)-34
b(e)438 b(describ)34 b(ed)437 b(in)h(the)g(in)-34 b(tro)34
b(duction.)0 57395 y(Rather,)310 b(an)-34 b(y)286 b(de\014nition)h(of)f
(obfuscator)h(that)g(w)-34 b(ould)287 b(b)34 b(e)286
b(strong)g(enough)h(to)f(pro)-34 b(vide)286 b(them,)310
b(will)285 b(b)34 b(e)285 b(imp)34 b(ossible)0 58901
y(to)404 b(meet.)1882 60406 y(Note)e(that)h(w)-34 b(e)403
b(do)f(not)h(pro)-34 b(v)g(e)402 b(that)i(the)e(applications)h
(themselv)-34 b(es)402 b(are)g(imp)34 b(ossible)401 b(to)i(meet,)f(but)
h(rather)0 61912 y(that)498 b(there)e(do)34 b(es)497
b(not)g(exist)g(an)g(obfuscator)21221 61472 y Fl(7)22244
61912 y Ft(that)h(can)f(b)34 b(e)496 b(used)i(to)f(ac)-34
b(hiev)g(e)496 b(them)h(in)g(the)g(w)-34 b(a)g(ys)498
b(that)f(are)0 63417 y(describ)34 b(ed)404 b(in)g(Section)g(1.1.)538
b(Our)404 b(results)h(in)f(the)g(section)g(also)g(extend)h(to)f(appro)
-34 b(ximate)405 b(obfuscators.)1882 64923 y(Consider,)591
b(for)553 b(example,)590 b(the)553 b(application)h(to)g(transforming)g
(priv)-67 b(ate-k)-34 b(ey)552 b(encryption)h(to)h(public-k)-34
b(ey)0 66428 y(ones.)691 b(The)455 b(circuit)9817 66122
y Ff(f)9691 66428 y Fo(E)10586 66625 y Fj(k)11610 66428
y Ft(in)g(the)g(follo)-34 b(wing)456 b(de\014nition)g(can)f(b)34
b(e)454 b(view)-34 b(ed)455 b(as)g(an)g(encryption-k)-34
b(ey)455 b(in)g(the)g(corre-)0 67934 y(sp)34 b(onding)405
b(public-k)-34 b(ey)404 b(encryption)g(sc)-34 b(heme.)p
0 68842 20800 45 v 1382 69559 a Fw(7)1843 69982 y Fx(By)448
b(this,)475 b(w)-28 b(e)447 b(mean)h(an)-28 b(y)447 b(algorithm)h(that)
g(satis\014es)g(the)g(syn)-28 b(tactic)448 b(requiremen)-28
b(ts)448 b(of)f(De\014nition)i(2.2)f(\(functionalit)-28
b(y)448 b(and)0 71200 y(p)28 b(olynomial)343 b(slo)-28
b(wdo)g(wn\).)25394 74521 y Ft(21)p eop
%%Page: 22 23
22 22 bop 0 1218 a Fz(De\014nition)466 b(4.8)606 b Fr(A)405
b(private-key)400 b(encryption)i(scheme)g Ft(\()p Fo(G;)202
b(E)70 b(;)202 b(D)34 b Ft(\))403 b Fr(is)h(c)-62 b(al)62
b(le)-62 b(d)527 b Ft(unobfuscatable)406 b Fr(if)e(ther)-62
b(e)402 b(exists)0 2723 y(a)433 b(PPT)i Fo(A)e Fr(such)f(that)18243
4229 y Ft(Pr)16772 5503 y Fj(K)17816 4992 y Fd(R)17628
5503 y FG( )p Fj(G)p Fl(\(1)20138 5251 y Fc(k)20647 5503
y Fl(\))21013 4229 y Ft([)p Fo(A)p Ft(\()22758 3922 y
Ff(g)22730 4229 y Fo(E)23625 4417 y Fj(K)24536 4229 y
Ft(\))338 b(=)e Fo(K)87 b Ft(])337 b Fp(\025)f Ft(1)269
b Fp(\000)h Ft(neg)17 b(\()p Fo(k)38 b Ft(\))0 7892 y
Fr(wher)-62 b(e)3451 7585 y Ff(g)3423 7892 y Fo(E)4318
8080 y Fj(K)5663 7892 y Fr(is)433 b(any)g(cir)-62 b(cuit)432
b(that)g(c)-62 b(omputes)431 b(the)h(encryption)g(function)g(with)h
(private)f(key)g Fo(K)87 b Fr(.)1882 10726 y Ft(Note)462
b(that)h(an)g(unobfuscatable)h(encryption)e(sc)-34 b(heme)462
b(is)g(unobfuscatable)i(in)e(a)g(v)-34 b(ery)461 b(strong)i(sense.)713
b(An)0 12231 y(adv)-34 b(ersary)448 b(is)h(able)g(to)g(completely)f
(break)g(the)i(system)e(giv)-34 b(en)449 b Fr(any)556
b Ft(circuit)448 b(that)i(computes)f(the)h(encryption)0
13737 y(algorithm.)1882 15242 y(W)-101 b(e)394 b(pro)-34
b(v)g(e)395 b(in)f(Theorem)h(4.12)g(that)g(if)g(encryption)g(sc)-34
b(hemes)395 b(exist,)g(then)h(so)e(do)h(unobfuscatable)i(encryp-)0
16747 y(tion)478 b(sc)-34 b(hemes)477 b(that)h(satisfy)g(the)f(same)g
(securit)-34 b(y)477 b(requiremen)-34 b(ts.)30838 16308
y Fl(8)32121 16747 y Ft(This)478 b(means)f(that)h(an)-34
b(y)478 b(de\014nition)g(of)g(an)0 18253 y(obfuscators)456
b(that)h(will)d(b)34 b(e)455 b(strong)h(enough)g(to)g(allo)-34
b(w)455 b(the)h(con)-34 b(v)g(ersion)455 b(of)g(priv)-67
b(ate-k)-34 b(ey)454 b(encryption)i(sc)-34 b(hemes)0
19758 y(in)g(to)337 b(public-k)-34 b(ey)337 b(encryption)f(sc)-34
b(hemes)337 b(men)-34 b(tioned)338 b(in)e(Section)h(1.1,)349
b(w)-34 b(ould)338 b(b)34 b(e)336 b(imp)34 b(ossible)337
b(to)f(meet)h(\(b)34 b(ecause)0 21264 y(there)404 b(exist)g
(unobfuscatable)i(encryption)e(sc)-34 b(hemes\).)25353
20824 y Fl(9)1882 22769 y Ft(W)-101 b(e)373 b(presen)-34
b(t)374 b(analogous)h(de\014nitions)g(for)e(unobfuscatable)j(signature)
e(sc)-34 b(hemes,)380 b(MA)-34 b(Cs,)380 b(and)375 b(pseudoran-)0
24275 y(dom)405 b(functions.)0 27109 y Fz(De\014nition)466
b(4.9)606 b Fr(A)350 b(signatur)-62 b(e)347 b(scheme)g
Ft(\()p Fo(G;)202 b(S)s(;)g(V)269 b Ft(\))350 b Fr(is)e(c)-62
b(al)62 b(le)-62 b(d)472 b Ft(unobfuscatable)352 b Fr(if)c(ther)-62
b(e)348 b(exists)g(a)h(PPT)g Fo(A)h Fr(such)0 28614 y(that)17715
30120 y Ft(Pr)14699 31394 y Fl(\()p Fa(SK)127 b Fj(;)p
Fa(VK)g Fl(\))18833 30883 y Fd(R)18645 31394 y FG( )p
Fj(G)p Fl(\(1)21155 31142 y Fc(k)21664 31394 y Fl(\))22030
30120 y Ft([)p Fo(A)p Ft(\()23966 29813 y Ff(g)23747
30120 y Fo(S)24490 30308 y Fa(SK)25937 30120 y Ft(\))337
b(=)f Fr(SK)177 b Ft(])336 b Fp(\025)h Ft(1)269 b Fp(\000)g
Ft(neg)17 b(\()p Fo(k)38 b Ft(\))0 33783 y Fr(wher)-62
b(e)3642 33476 y Ff(g)3423 33783 y Fo(S)4166 33971 y
Fa(SK)6046 33783 y Fr(is)433 b(any)g(cir)-62 b(cuit)432
b(which)g(c)-62 b(omputes)431 b(the)i(signatur)-62 b(e)431
b(function)i(with)g(signing)f(key)h(SK)176 b(.)0 36617
y Fz(De\014nition)466 b(4.10)606 b Fr(A)322 b(message)e(authentic)-62
b(ation)319 b(scheme)h Ft(\()p Fo(G;)202 b(S)s(;)g(V)269
b Ft(\))322 b Fr(is)g(c)-62 b(al)62 b(le)-62 b(d)445
b Ft(unobfuscatable)324 b Fr(if)e(ther)-62 b(e)321 b(exists)0
38122 y(a)433 b(PPT)i Fo(A)e Fr(such)f(that)18319 39627
y Ft(Pr)16848 40902 y Fj(K)17892 40390 y Fd(R)17704 40902
y FG( )p Fj(G)p Fl(\(1)20214 40650 y Fc(k)20723 40902
y Fl(\))21089 39627 y Ft([)p Fo(A)p Ft(\()23027 39321
y Ff(f)22806 39627 y Fo(S)23549 39815 y Fj(K)24461 39627
y Ft(\))337 b(=)f Fo(K)87 b Ft(])337 b Fp(\025)f Ft(1)270
b Fp(\000)f Ft(neg)17 b(\()p Fo(k)38 b Ft(\))0 43291
y Fr(wher)-62 b(e)3644 42984 y Ff(f)3423 43291 y Fo(S)4166
43479 y Fj(K)5511 43291 y Fr(is)433 b(any)g(cir)-62 b(cuit)432
b(which)g(c)-62 b(omputes)431 b(the)i(tagging)f(function)g(with)h
(tagging)f(key)g Fo(K)87 b Fr(.)0 46124 y Fz(De\014nition)466
b(4.11)606 b Fr(A)310 b(pseudor)-62 b(andom)306 b(function)j(ensemble)f
Fp(f)p Fo(h)29625 46312 y Fj(K)30537 46124 y Fp(g)31143
46371 y Fj(K)63 b FG(2f)p Fl(0)p Fj(;)p Fl(1)p FG(g)34767
46119 y Fy(\003)35614 46124 y Fr(is)309 b(c)-62 b(al)62
b(le)-62 b(d)433 b Ft(unobfuscatable)312 b Fr(if)d(ther)-62
b(e)0 47630 y(exists)432 b(a)h(p.p.t)f Fo(A)i Fr(such)e(that)18187
49135 y Ft(Pr)16612 50410 y Fj(K)17656 49898 y Fd(R)17468
50410 y FG( f)p Fl(0)p Fj(;)p Fl(1)p FG(g)20550 50158
y Fc(k)21061 49135 y Ft([)p Fo(A)p Ft(\()22862 48829
y Ff(g)22778 49135 y Fo(H)23786 49323 y Fj(K)24697 49135
y Ft(\))337 b(=)f Fo(K)87 b Ft(])337 b Fp(\025)f Ft(1)270
b Fp(\000)f Ft(neg)17 b(\()p Fo(k)38 b Ft(\))0 52799
y Fr(wher)-62 b(e)3507 52492 y Ff(g)3423 52799 y Fo(H)4431
52987 y Fj(K)5776 52799 y Fr(is)433 b(any)g(cir)-62 b(cuit)431
b(that)h(c)-62 b(omputes)431 b Fo(h)21636 52987 y Fj(K)22548
52799 y Fr(.)1882 55632 y Ft(One)493 b(implication)h(of)f(the)h
(existence)f(of)h(unobfuscatable)i(pseudorandom)f(function)g(ensem)-34
b(bles)494 b(is)f(that)0 57138 y(for)455 b(man)-34 b(y)455
b Fr(natur)-62 b(al)453 b Ft(proto)34 b(cols)455 b(that)h(are)e(secure)
g(in)h(the)g(random)g(oracle)f(mo)34 b(del)454 b(\(suc)-34
b(h)456 b(as)f(the)g(Fiat{Shamir)0 58643 y(authen)-34
b(tication)381 b(proto)34 b(col)379 b([FS87)q(]\),)384
b(one)379 b(can)h(\014nd)g(a)f(pseudorandom)i(function)g(ensem)-34
b(ble)379 b Fp(f)p Fo(h)44296 58840 y Fj(k)44866 58643
y Fp(g)45472 58890 y Fj(k)24 b FG(2f)p Fl(0)p Fj(;)p
Fl(1)p FG(g)48754 58638 y Fy(\003)49670 58643 y Ft(suc)-34
b(h)0 60149 y(that)374 b(if)f(the)h(random)g(oracle)e(is)h(replaced)g
(with)h Fr(any)479 b Ft(circuit)373 b(that)h(computes)g
Fo(h)37064 60346 y Fj(k)37633 60149 y Ft(,)379 b(the)374
b(proto)34 b(col)373 b(w)-34 b(ould)374 b(not)g(b)34
b(e)0 61654 y(secure.)p 0 62473 20800 45 v 1382 63190
a Fw(8)1843 63614 y Fx(Recall)289 b(that,)300 b(for)288
b(simplicit)-28 b(y)-85 b(,)301 b(w)-28 b(e)288 b(only)g(consider)h
(deterministic)h(encryption)f(sc)-28 b(hemes)289 b(here)g(and)f
(relaxed)h(notions)f(of)h(securit)-28 b(y)0 64831 y(that)342
b(are)f(consisten)-28 b(t)341 b(with)h(them)g(\(cf.,)g(F)-85
b(o)28 b(otnote)342 b(2\).)1382 65626 y Fw(9)1843 66049
y Fx(Of)469 b(course,)500 b(this)468 b(do)28 b(es)469
b(not)g(mean)f(that)h(public-k)-28 b(ey)470 b(encryption)f(sc)-28
b(hemes)468 b(do)g(not)h(exist,)501 b(nor)468 b(that)h(there)f(do)g
(not)h(exist)0 67267 y(priv)-57 b(ate-k)-28 b(ey)375
b(encryption)e(sc)-28 b(hemes)374 b(where)e(one)h(can)g(giv)-28
b(e)374 b(the)f(adv)-28 b(ersary)372 b(a)h(circuit)g(that)g(computes)h
(the)f(encryption)g(algorithm)0 68484 y(without)409 b(loss)h(of)f
(securit)-28 b(y)409 b(\(indeed,)428 b(an)-28 b(y)409
b(public-k)-28 b(ey)411 b(encryption)f(sc)-28 b(heme)410
b(is)f(in)h(particular)f(suc)-28 b(h)408 b(a)h(priv)-57
b(ate-k)-28 b(ey)412 b(encryption\).)0 69702 y(What)401
b(this)h(means)f(is)h(that)f(there)g(exists)i(no)e(general)g(purp)28
b(ose)401 b(w)-28 b(a)g(y)400 b(to)i(transform)e(a)h(priv)-57
b(ate)402 b(k)-28 b(ey)402 b(encryption)f(sc)-28 b(heme)402
b(in)-28 b(to)401 b(a)0 70919 y(public)342 b(k)-28 b(ey)342
b(encryption)g(b)-28 b(y)341 b(obfuscating)g(the)h(encryption)g
(algorithm.)25394 74521 y Ft(22)p eop
%%Page: 23 24
23 23 bop 0 1218 a Fz(Theorem)464 b(4.12)2039 b Fr(1.)606
b(If)433 b(signatur)-62 b(e)432 b(schemes)f(exist,)h(then)h(so)g(do)g
(unobfusc)-62 b(atable)429 b(signatur)-62 b(e)431 b(schemes.)1433
3719 y(2.)605 b(If)434 b(private-key)c(encryption)i(schemes)f(exist,)h
(then)h(so)g(do)g(unobfusc)-62 b(atable)429 b(encryption)i(schemes.)
1433 6221 y(3.)605 b(If)420 b(pseudor)-62 b(andom)417
b(function)h(ensembles)g(exist,)k(then)d(so)g(do)g(unobfusc)-62
b(atable)415 b(pseudor)-62 b(andom)416 b(function)3030
7727 y(ensembles.)1433 10228 y(4.)605 b(If)533 b(message)d(authentic)
-62 b(ation)528 b(schemes)i(exist,)556 b(then)531 b(so)h(do)f(unobfusc)
-62 b(atable)528 b(message)i(authentic)-62 b(ation)3030
11734 y(schemes.)0 14568 y Fz(Pro)39 b(of)448 b(Sk)-39
b(etc)g(h:)1430 b Ft(First)391 b(note)f(that)i(the)e(existence)g(of)g
(an)-34 b(y)391 b(one)g(of)f(these)h(primitiv)-34 b(es)389
b(implies)h(the)h(existence)0 16073 y(of)529 b(one-w)-34
b(a)g(y)530 b(functions)g([IL89].)912 b(Therefore,)560
b(Theorem)529 b(4.2)f(giv)-34 b(es)529 b(us)g(a)g(totally)g
(unobfuscatable)i(function)0 17579 y(ensem)-34 b(ble)404
b Fp(H)348 b Ft(=)337 b Fp(fH)9440 17776 y Fj(k)10008
17579 y Fp(g)p Ft(.)1882 19084 y(No)-34 b(w,)457 b(w)-34
b(e)446 b(shall)g(sk)-34 b(etc)g(h)447 b(the)g(construction)g(of)f(the)
h(unobfuscatable)h(signature)f(sc)-34 b(heme.)664 b(All)445
b(other)i(con-)0 20589 y(structions)402 b(are)e(similar.)536
b(T)-101 b(ak)-34 b(e)401 b(an)g(existing)g(signature)g(sc)-34
b(heme)400 b(\()p Fo(G;)202 b(S)s(;)g(V)269 b Ft(\))401
b(\(where)g Fo(G)g Ft(is)g(the)g(k)-34 b(ey)400 b(generation)0
22095 y(algorithm,)583 b Fo(S)617 b Ft(the)548 b(signing)g(algorithm,)
583 b(and)548 b Fo(V)817 b Ft(the)548 b(v)-34 b(eri\014cation)547
b(algorithm\).)969 b(De\014ne)547 b(the)h(new)g(sc)-34
b(heme)0 23600 y(\()p Fo(G)1424 23161 y FG(0)1735 23600
y Fo(;)202 b(S)3087 23161 y FG(0)3397 23600 y Fo(;)g(V)4913
23161 y FG(0)5223 23600 y Ft(\))405 b(as)f(follo)-34
b(ws:)1882 25106 y(The)578 b(generator)g Fo(G)11068 24666
y FG(0)11956 25106 y Ft(on)g(input)h(1)17825 24666 y
Fj(k)18972 25106 y Ft(uses)f(the)g(generator)g Fo(G)g
Ft(to)g(generate)f(signing)i(and)f(v)-34 b(erifying)577
b(k)-34 b(eys)0 26611 y(\()p Fr(SK)177 b Fo(;)202 b Fr(VK)176
b Ft(\))5880 25924 y Fw(R)5567 26611 y Fp( )287 b Fo(G)p
Ft(\(1)9096 26171 y Fj(k)9665 26611 y Ft(\).)618 b(It)430
b(then)h(samples)g(a)f(circuit)g Fo(f)25929 25924 y Fw(R)25616
26611 y Fp( )287 b(H)28139 26808 y Fj(`)28579 26611 y
Ft(,)436 b(where)430 b Fo(`)381 b Ft(=)f Fp(j)p Fr(SK)176
b Fp(j)p Ft(.)617 b(The)431 b(new)f(signing)h(k)-34 b(ey)430
b Fr(SK)51689 26134 y FG(0)0 28117 y Ft(is)404 b(\()p
Fr(SK)177 b Fo(;)202 b(f)130 b Ft(\))405 b(while)f(the)g(v)-34
b(eri\014cation)404 b(k)-34 b(ey)404 b Fr(VK)21334 27639
y FG(0)22049 28117 y Ft(is)f(the)i(same)f(as)g Fr(VK)176
b Ft(.)1882 29622 y(W)-101 b(e)403 b(can)i(no)-34 b(w)405
b(de\014ne)15099 31128 y Fo(S)15912 30627 y FG(0)15842
31436 y Fa(SK)128 b Fj(;f)18100 31128 y Ft(\()p Fo(m)p
Ft(\))20376 30441 y Fl(def)20554 31128 y Ft(=)447 b(\()p
Fo(S)23158 31316 y Fa(SK)24605 31128 y Ft(\()p Fo(m)p
Ft(\))p Fo(;)202 b(f)130 b Ft(\()p Fo(m)p Ft(\))p Fo(;)202
b Fr(SK)448 b Fp(\010)270 b Fo(\031)43 b Ft(\()p Fo(f)130
b Ft(\)\))p Fo(;)0 33353 y Ft(where)404 b Fo(\031)448
b Ft(is)403 b(the)i(function)g(from)g(the)f(unobfuscatabilit)-34
b(y)407 b(condition)e(in)f(De\014nition)g(4.1.)18491
36312 y Fo(V)19467 35811 y FG(0)19198 36611 y Fa(VK)20815
36312 y Ft(\()p Fo(m;)202 b Ft(\()p Fo(\034)70 b(;)202
b(x)p Ft(\)\))26404 35624 y Fl(def)26582 36312 y Ft(=)447
b Fo(V)28679 36500 y Fa(VK)30296 36312 y Ft(\()p Fo(m;)202
b(\034)137 b Ft(\))0 39035 y(W)-101 b(e)468 b(claim)f(that)i(\()p
Fo(G)9562 38595 y FG(0)9873 39035 y Fo(;)202 b(S)11225
38595 y FG(0)11536 39035 y Fo(;)g(V)13051 38595 y FG(0)13361
39035 y Ft(\))469 b(is)f(an)g(unobfuscatable,)487 b(y)-34
b(et)468 b(secure,)483 b(signature)469 b(sc)-34 b(heme.)731
b(Clearly)-101 b(,)482 b(giv)-34 b(en)468 b(an)-34 b(y)0
40540 y(circuit)550 b(that)j(computes)e Fo(S)13048 40100
y FG(0)12978 40909 y Fa(SK)128 b Fj(;f)15236 40540 y
Ft(,)588 b(one)551 b(can)g(obtain)h Fr(SK)544 b Fp(\010)367
b Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))553 b(and)f(a)f(circuit)f(that)
j(computes)e(the)h(same)0 42046 y(function)412 b(as)f
Fo(f)130 b Ft(.)560 b(P)-34 b(ossession)412 b(of)f(the)g(latter)g
(enables)g(one)g(to)h(reconstruct)f(the)g(original)f(circuit)h
Fo(f)541 b Ft(itself,)413 b(from)0 43551 y(whic)-34 b(h)405
b Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))406 b(and)f(then)g
Fr(SK)581 b Ft(can)404 b(b)34 b(e)404 b(computed.)1882
45057 y(T)-101 b(o)408 b(see)f(that)h(sc)-34 b(heme)408
b(\()p Fo(G)13838 44617 y FG(0)14149 45057 y Fo(;)202
b(S)15501 44617 y FG(0)15811 45057 y Fo(;)g(V)17326 44617
y FG(0)17637 45057 y Ft(\))408 b(retains)f(the)h(securit)-34
b(y)407 b(of)h(the)g(sc)-34 b(heme)408 b(\()p Fo(G;)202
b(S)s(;)g(V)269 b Ft(\),)408 b(observ)-34 b(e)407 b(that)i(b)34
b(eing)0 46562 y(giv)-34 b(en)485 b(oracle)f(access)h(to)g
Fo(S)12871 46122 y FG(0)12801 46931 y Fa(SK)128 b Fj(;f)15544
46562 y Ft(is)485 b(equiv)-67 b(alen)-34 b(t)485 b(to)g(b)34
b(eing)485 b(giv)-34 b(en)486 b(oracle)e(access)g(to)i
Fo(S)40394 46750 y Fa(SK)42326 46562 y Ft(and)g Fo(f)130
b Ft(,)506 b(along)485 b(with)0 48067 y(b)34 b(eing)542
b(giv)-34 b(en)543 b(the)g(string)f Fo(\031)43 b Ft(\()p
Fo(f)130 b Ft(\))364 b Fp(\010)e Fr(SK)176 b Ft(.)954
b(Using)542 b(the)h(facts)g(that)h Fo(\031)43 b Ft(\()p
Fo(f)130 b Ft(\))545 b(is)d(indistinguishable)i(from)e(random)0
49573 y(giv)-34 b(en)391 b(oracle)g(access)g(to)h Fo(f)522
b Ft(and)392 b(that)h Fo(f)522 b Ft(is)391 b(c)-34 b(hosen)392
b(indep)34 b(enden)-34 b(tly)393 b(of)f Fr(SK)176 b Ft(,)394
b(it)d(can)h(b)34 b(e)391 b(easily)g(sho)-34 b(wn)393
b(that)g(the)0 51078 y(presence)404 b(of)g Fo(f)535 b
Ft(and)405 b Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))271
b Fp(\010)e Fr(SK)581 b Ft(do)34 b(es)404 b(not)h(help)f(an)h(adv)-34
b(ersary)403 b(break)h(the)h(signature)f(sc)-34 b(heme.)1882
52584 y(The)321 b(construction)h(of)f(an)g(unobfuscatable)h(encryption)
f(sc)-34 b(heme)321 b(and)g(pseudorandom)i(function)f(ensem)-34
b(ble)0 54089 y(is)337 b(similar.)515 b(The)338 b(only)f(detail)g(is)g
(that)h(when)g(w)-34 b(e)338 b(construct)g(the)f(pseudorandom)i
(function)g(ensem)-34 b(ble,)350 b(w)-34 b(e)338 b(need)0
55595 y(to)485 b(observ)-34 b(e)484 b(that)i(Theorem)e(4.2)g(can)h(b)34
b(e)484 b(mo)34 b(di\014ed)485 b(to)g(giv)-34 b(e)483
b Fp(H)496 b Ft(whic)-34 b(h)485 b(is)g(also)f(a)g(family)g(of)h
(pseudorandom)0 57100 y(functions.)527 b(\(T)-101 b(o)365
b(do)f(this,)372 b(all)364 b(places)g(where)g(the)g(functions)i
Fo(f)495 b Ft(in)364 b Fp(H)375 b Ft(w)-34 b(ere)364
b(de\014ned)h(to)g(b)34 b(e)364 b(zero)f(should)i(instead)0
58606 y(b)34 b(e)404 b(replaced)f(with)i(v)-67 b(alues)404
b(of)g(a)g(pseudorandom)i(function.\))22658 b Fb(2)0
61852 y Fn(4.4)1495 b(Obfuscating)500 b(restricted)g(circuit)f(classes)
0 64140 y Ft(Giv)-34 b(en)390 b(our)f(imp)34 b(ossibilit)-34
b(y)390 b(results)g(for)f(obfuscating)j(general)d(circuits,)i(one)f(ma)
-34 b(y)390 b(ask)f(whether)i(it)e(is)h(easier)f(to)0
65645 y(obfuscate)433 b(computationally)g(restricted)e(classes)h(of)g
(circuits.)621 b(Here)431 b(w)-34 b(e)432 b(argue)g(that)h(this)g(is)e
(unlik)-34 b(ely)432 b(for)g(all)0 67151 y(but)405 b(v)-34
b(ery)404 b(w)-34 b(eak)404 b(mo)34 b(dels)404 b(of)g(computation.)
25394 74521 y(23)p eop
%%Page: 24 25
24 24 bop 0 1218 a Fz(Theorem)464 b(4.13)606 b Fr(If)416
b(factoring)e(Blum)i(inte)-62 b(gers)414 b(is)i(\\har)-62
b(d")27918 778 y Fl(10)29328 1218 y Fr(then)415 b(ther)-62
b(e)415 b(is)g(a)h(family)e Fp(H)42019 1415 y Fj(k)43004
1218 y Fr(of)h(unobfusc)-62 b(atable)0 2723 y(functions)372
b(such)g(that)g(every)g Fo(f)14652 2036 y Fd(R)14329
2723 y Fp( )139 b(H)16704 2920 y Fj(k)17646 2723 y Fr(is)373
b(c)-62 b(omputable)370 b(by)j(a)g(c)-62 b(onstant-depth)370
b(thr)-62 b(eshold)370 b(cir)-62 b(cuit)372 b(of)h(size)f
Ft(p)34 b(oly)16 b(\()p Fo(k)38 b Ft(\))0 4229 y Fr(\(i.e.,)432
b(in)i Fz(TC)6438 4411 y Fh(0)7035 4229 y Fr(\).)0 6878
y Fz(Pro)39 b(of)349 b(Sk)-39 b(etc)g(h:)1189 b Ft(Naor)304
b(and)h(Reingold)g([NR97])f(sho)-34 b(w)g(ed)306 b(that)f(under)g(the)f
(stated)h(assumptions,)326 b(there)304 b(exists)0 8383
y(a)435 b(family)g(of)h(pseudorandom)h(functions)g(computable)g(in)e
Fz(TC)29553 8565 y Fh(0)30150 8383 y Ft(.)633 b(Th)-34
b(us,)444 b(w)-34 b(e)436 b(simply)f(need)h(to)g(c)-34
b(hec)g(k)435 b(that)i(w)-34 b(e)0 9889 y(can)375 b(build)h(our)f
(unobfuscatable)i(functions)f(from)g(suc)-34 b(h)376
b(a)f(family)f(without)j(a)e(substan)-34 b(tial)377 b(increase)d(in)h
(depth.)0 11394 y(Recall)522 b(that)h(the)g(unobfuscatable)i(function)f
(ensem)-34 b(ble)522 b Fp(H)28330 11591 y Fj(k)29421
11394 y Ft(constructed)i(in)e(the)h(pro)34 b(of)523 b(of)g(Theorem)f
(3.11)0 12900 y(consists)341 b(of)g(functions)h(of)f(the)f(form)h
Fo(C)17888 13097 y Fj(\013;\014)19383 12900 y Ft(#)p
Fo(D)374 b Ft(or)340 b Fo(Z)24019 13097 y Fj(k)24588
12900 y Ft(#)p Fo(D)34 b Ft(,)352 b(where)341 b Fo(D)373
b Ft(is)340 b(from)h(the)g(family)f Fp(D)42700 13097
y Fj(\013;\014)44535 12900 y Ft(of)h(Lemma)e(3.6.)0 14405
y(It)603 b(is)g(easy)g(to)g(see)g(that)h Fo(C)13326 14602
y Fj(\013;\014)15424 14405 y Ft(and)g Fo(Z)18807 14602
y Fj(k)19979 14405 y Ft(are)f(in)g Fz(TC)25792 14587
y Fh(0)26389 14405 y Ft(,)653 b(so)603 b(w)-34 b(e)604
b(only)f(need)g(to)g(c)-34 b(hec)g(k)603 b(that)i Fp(D)45842
14602 y Fj(\013;\014)47939 14405 y Ft(consists)0 15911
y(of)552 b(circuits)f(in)g Fz(TC)9466 16093 y Fh(0)10063
15911 y Ft(.)980 b(The)552 b(computational)h(complexit)-34
b(y)551 b(of)h(circuits)f(in)g(the)h(family)f Fp(D)42999
16108 y Fj(\013;\014)45045 15911 y Ft(is)g(dominated)0
17416 y(b)-34 b(y)586 b(p)34 b(erforming)585 b(encryptions)h(and)g
(decryptions)f(in)h(a)f(priv)-67 b(ate-k)-34 b(ey)584
b(encryption)i(sc)-34 b(heme)585 b(\(Enc)p Fo(;)202 b
Ft(Dec\))585 b(and)0 18921 y(ev)-67 b(aluating)546 b(a)g(pseudorandom)i
(function)g Fo(f)20835 19125 y Fj(K)21691 18873 y Fy(0)22591
18921 y Ft(whic)-34 b(h)547 b(is)f(used)h(to)g(derandomize)f(the)g
(probabilistic)h(circuit)0 20427 y Fo(D)1004 20624 y
Fj(K)10 b(;\013;\014)3563 20427 y Ft(.)730 b(If)468 b(w)-34
b(e)468 b(use)g(the)h(Naor{Reingold)f(pseudorandom)i(functions)f(b)34
b(oth)469 b(for)f Fo(f)39279 20631 y Fj(K)40135 20378
y Fy(0)40957 20427 y Ft(and)g(to)h(construct)g(the)0
21932 y(encryption)437 b(sc)-34 b(heme)437 b(\(in)g(the)g(usual)g(w)-34
b(a)g(y)-101 b(,)446 b(setting)437 b(Enc)26270 22120
y Fj(K)27181 21932 y Ft(\()p Fo(b)p Ft(\))392 b(=)f(\()p
Fo(r)-34 b(;)202 b(f)32485 22120 y Fj(K)33397 21932 y
Ft(\()p Fo(r)34 b Ft(\))292 b Fp(\010)f Fo(b)p Ft(\)\),)445
b(then)438 b(the)f(resulting)g(circuit)0 23438 y(is)404
b(in)g Fz(TC)4610 23620 y Fh(0)5207 23438 y Ft(.)45550
b Fb(2)0 26655 y Fn(4.5)1495 b(Relativization)0 28943
y Ft(In)391 b(this)g(section,)i(w)-34 b(e)392 b(discuss)f(whether)g
(our)g(results)g(relativize.)532 b(T)-101 b(o)392 b(do)f(this,)j(w)-34
b(e)391 b(m)-34 b(ust)392 b(clarify)e(the)h(de\014nition)0
30448 y(of)362 b(an)g(obfuscator)i(relativ)-34 b(e)360
b(to)j(an)f(oracle)f Fo(F)505 b Ft(:)337 b Fp(f)p Ft(0)p
Fo(;)202 b Ft(1)p Fp(g)24575 30009 y FG(\003)25437 30448
y Fp(!)337 b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)29949
30009 y FG(\003)30475 30448 y Ft(.)524 b(What)363 b(w)-34
b(e)362 b(mean)g(is)f(that)j(all)d(algorithms)h(in)0
31954 y(the)435 b(de\014nition,)443 b(including)436 b(the)f(one)g(b)34
b(eing)434 b(obfuscated)j(and)e(including)h(the)f(adv)-34
b(ersary)-101 b(,)442 b(ha)-34 b(v)g(e)435 b(oracle)f(access)0
33459 y(to)350 b Fo(F)168 b Ft(.)521 b(F)-101 b(or)349
b(a)g(circuit,)360 b(this)350 b(means)f(that)i(the)f(circuit)e(can)i
(ha)-34 b(v)g(e)350 b(gates)f(for)h(ev)-67 b(aluating)349
b Fo(F)168 b Ft(.)521 b(W)-101 b(e)349 b(\014x)g(an)h(enco)34
b(ding)0 34965 y(of)433 b(\(oracle\))g(circuits)f(as)h(binary)g
(strings)h(suc)-34 b(h)434 b(that)g(a)f(circuit)f(describ)34
b(ed)432 b(b)-34 b(y)434 b(a)f(string)g(of)g(length)h
Fo(s)f Ft(can)g(only)0 36470 y(mak)-34 b(e)404 b(oracle)f(queries)g(of)
i(total)f(length)h(at)g(most)f Fo(s)p Ft(.)1882 37976
y(By)g(insp)34 b(ection,)404 b(our)h(initial)f(\(easy\))h(imp)34
b(ossibilit)-34 b(y)404 b(results)h(hold)g(relativ)-34
b(e)404 b(to)h(an)-34 b(y)405 b(oracle,)e(as)i(the)g(in)-34
b(v)g(olv)g(e)0 39481 y(only)404 b(sim)-34 b(ulation)405
b(and)g(diagonalization.)0 42130 y Fz(Prop)39 b(osition)465
b(4.14)606 b Fr(Pr)-62 b(op)g(osition)509 b(3.4)g(\(imp)-62
b(ossibility)506 b(of)k(2-cir)-62 b(cuit)507 b(obfusc)-62
b(ators\))506 b(and)k(The)-62 b(or)g(em)510 b(3.5)f(\(im-)0
43636 y(p)-62 b(ossibility)430 b(of)j(TM)g(obfusc)-62
b(ators\))430 b(hold)i(r)-62 b(elative)431 b(to)i(any)g(or)-62
b(acle.)1882 46285 y Ft(In)-34 b(terestingly)-101 b(,)404
b(ho)-34 b(w)g(ev)g(er,)404 b(our)g(main)h(imp)34 b(ossibilit)-34
b(y)404 b(results)g(do)g(not)h(relativize.)0 48934 y
Fz(Prop)39 b(osition)465 b(4.15)606 b Fr(Ther)-62 b(e)523
b(is)g(an)g(or)-62 b(acle)522 b(r)-62 b(elative)521 b(to)i(which)f
(e\016cient)g(cir)-62 b(cuit)521 b(obfusc)-62 b(ators)520
b(exist.)825 b(Thus,)0 50440 y(The)-62 b(or)g(em)433
b(3.8,3.11,)d(and)j(Cor)-62 b(ol)62 b(lary)432 b(3.10)g(do)h(not)g(r)
-62 b(elativize.)1882 53089 y Ft(This)368 b(can)f(b)34
b(e)367 b(view)-34 b(ed)367 b(b)34 b(oth)368 b(as)g(evidence)e(that)j
(these)e(results)g(are)g(non)-34 b(trivial,)375 b(and)368
b(as)f(\(further\))i(evidence)0 54594 y(that)405 b(relativization)e(is)
h(not)h(a)f(go)34 b(o)g(d)404 b(indication)h(of)f(what)i(w)-34
b(e)404 b(can)g(pro)-34 b(v)g(e.)0 57243 y Fz(Pro)39
b(of)645 b(Sk)-39 b(etc)g(h:)2146 b Ft(The)562 b(oracle)f
Fo(F)767 b Ft(=)19971 56334 y Ff(S)20981 57602 y Fj(k)21752
57243 y Fo(F)22531 57440 y Fj(k)23662 57243 y Ft(will)561
b(consist)h(of)g(t)-34 b(w)g(o)563 b(parts)f Fo(F)38282
57440 y Fj(k)39451 57243 y Ft(=)599 b Fp(O)41958 57440
y Fj(k)42526 57243 y Ft(#E)44361 57440 y Fj(k)44930 57243
y Ft(,)i(where)561 b Fp(O)50495 57440 y Fj(k)51663 57243
y Ft(:)0 58749 y Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)2963
58309 y Fj(k)3688 58749 y Fp(\002)156 b(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)7750 58309 y Fj(k)8655 58749 y Fp(!)337
b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)13167 58309 y Fl(6)p
Fj(k)14206 58749 y Ft(,)359 b(and)348 b(E)18027 58946
y Fj(k)18932 58749 y Ft(:)337 b Fp(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)22569 58309 y Fl(6)p Fj(k)23764 58749 y
Fp(\002)156 b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)27826
58309 y Fj(k)28731 58749 y Fp(!)337 b(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)33243 58309 y Fj(k)33812 58749 y Ft(.)520
b Fp(O)35634 58946 y Fj(k)36550 58749 y Ft(is)347 b(simply)g(a)h
(uniformly)f(random)0 60254 y(injectiv)-34 b(e)487 b(function)i(of)e
(the)h(giv)-34 b(en)487 b(parameters.)788 b(E)24433 60451
y Fj(k)25002 60254 y Ft(\()p Fo(x;)202 b(y)43 b Ft(\))488
b(is)f(de\014ned)h(as)f(follo)-34 b(ws:)706 b(If)487
b(there)g(exists)g(a)g(\()p Fo(C)19 b(;)202 b(r)34 b
Ft(\))0 61760 y(suc)-34 b(h)400 b(that)g Fp(O)6315 61957
y Fj(k)6884 61760 y Ft(\()p Fo(C)19 b(;)202 b(r)34 b
Ft(\))338 b(=)e Fo(x)p Ft(,)399 b(then)h(E)16458 61957
y Fj(k)17027 61760 y Ft(\()p Fo(x;)202 b(y)43 b Ft(\))338
b(=)e Fo(C)22408 61320 y Fj(F)23190 61760 y Ft(\()p Fo(y)43
b Ft(\))400 b(\(where)g Fo(C)486 b Ft(is)399 b(view)-34
b(ed)399 b(as)g(the)h(description)f(of)g(a)h(circuit\).)0
63265 y(Otherwise,)420 b(E)6912 63462 y Fj(k)7481 63265
y Ft(\()p Fo(x;)202 b(y)43 b Ft(\))359 b(=)f Fp(?)p Ft(.)577
b(Note)418 b(that)g(this)g(de\014nition)g(of)f Fo(F)29340
63462 y Fj(k)30327 63265 y Ft(is)f(not)i(circular,)h(b)34
b(ecause)417 b Fo(C)504 b Ft(can)417 b(only)g(mak)-34
b(e)0 64771 y(oracle)403 b(queries)g(of)i(size)e(at)h(most)h
Fp(j)p Fo(C)87 b Fp(j)336 b Ft(=)h Fo(k)38 b Ft(,)404
b(and)h(hence)f(can)g(only)g(query)f Fo(F)35125 64974
y Fj(k)35639 64722 y Fy(0)36397 64771 y Ft(for)h Fo(k)38921
64331 y FG(0)39568 64771 y Fp(\024)337 b Fo(k)38 b(=)p
Ft(2.)1882 66276 y(No)-34 b(w)579 b(w)-34 b(e)579 b(can)g(view)f
Fo(x)627 b Ft(=)g Fp(O)15997 66473 y Fj(k)16565 66276
y Ft(\()p Fo(C)19 b(;)202 b(r)34 b Ft(\))579 b(as)g(an)g(obfuscation)h
(of)f Fo(C)665 b Ft(using)579 b(coin)g(tosses)f Fo(r)34
b Ft(.)1061 b(This)579 b(satis\014es)0 67782 y(the)460
b(syn)-34 b(tactic)461 b(requiremen)-34 b(ts)459 b(of)i(obfuscation,)
475 b(since)459 b Fp(j)p Fo(x)p Fp(j)429 b Ft(=)g Fo(O)34
b Ft(\()p Fp(j)p Fo(C)87 b Fp(j)p Ft(\))460 b(and)g(the)g(E)38440
67979 y Fj(k)39469 67782 y Ft(allo)-34 b(ws)460 b(one)g(to)g(e\016cien)
-34 b(tly)p 0 68842 20800 45 v 976 69559 a Fw(10)1843
69982 y Fx(This)349 b(result)g(is)g(also)g(implied)i(if)e(the)g
(Decisional)i(Di\016e{Hellman)g(problem)e(is)g(\\hard";)k(see)c([NR97])
g(for)g(precise)g(statemen)-28 b(ts)0 71200 y(of)341
b(these)h(assumptions.)25394 74521 y Ft(24)p eop
%%Page: 25 26
25 25 bop 0 1218 a Ft(ev)-67 b(aluate)334 b Fo(C)87 b
Ft(\()p Fo(y)43 b Ft(\))336 b(giv)-34 b(en)335 b(just)g
Fo(x)g Ft(and)g Fo(y)43 b Ft(.)516 b(\(T)-101 b(ec)-34
b(hnically)-101 b(,)348 b(w)-34 b(e)335 b(should)h(de\014ne)f(the)g
(obfuscation)h(of)f Fo(C)422 b Ft(to)335 b(b)34 b(e)334
b(a)h(circuit)0 2723 y(whic)-34 b(h)405 b(has)g Fo(x)e
Ft(hardwired)i(in)f(and)h(mak)-34 b(es)404 b(an)h(oracle)e(query)g(to)i
(E)30737 2920 y Fj(k)31305 2723 y Ft(.\))1882 4229 y(So)445
b(w)-34 b(e)445 b(only)g(need)g(to)g(pro)-34 b(v)g(e)445
b(the)g(virtual)f(blac)-34 b(k-b)34 b(o)-34 b(x)445 b(prop)34
b(ert)-34 b(y)-101 b(.)660 b(By)444 b(a)h(union)h(b)34
b(ound)446 b(o)-34 b(v)g(er)444 b(p)34 b(olynomial-)0
5734 y(time)392 b(adv)-34 b(ersaries)393 b Fo(A)f Ft(of)h(description)g
(size)f(smaller)g(than)i Fo(k)38 b(=)p Ft(2)392 b(and)i(circuits)e
Fo(C)480 b Ft(of)393 b(size)e Fo(k)38 b Ft(,)395 b(it)e(su\016ces)g(to)
g(pro)-34 b(v)g(e)0 7240 y(the)405 b(follo)-34 b(wing)404
b(claim.)10371 6800 y Fl(11)0 10073 y Fz(Claim)464 b(4.16)606
b Fr(F)-93 b(or)494 b(every)e(PPT)j Fo(A)e Fr(ther)-62
b(e)493 b(exists)f(a)h(PPT)i Fo(S)564 b Fr(such)492 b(that)g(for)h
(every)g(cir)-62 b(cuit)491 b Fo(C)581 b Fr(of)493 b(size)f
Fo(k)38 b Fr(,)509 b(the)0 11579 y(fol)62 b(lowing)432
b(holds)g(with)h(pr)-62 b(ob)g(ability)430 b(at)j(le)-62
b(ast)431 b Ft(1)270 b Fp(\000)f Ft(2)23618 11139 y FG(\000)p
Fl(2)p Fj(k)25823 11579 y Fr(over)432 b Fo(F)168 b Fr(:)10356
13143 y Ff(\014)10356 13870 y(\014)10356 14597 y(\014)10356
15325 y(\014)10356 16052 y(\014)12132 15264 y Ft(Pr)10760
16539 y Fj(r)11399 16027 y Fd(R)11211 16539 y FG( f)p
Fl(0)p Fj(;)p Fl(1)p FG(g)14293 16287 y Fc(k)15006 14282
y Ff(\002)15511 15264 y Fo(A)16420 14764 y Fj(F)17201
15264 y Ft(\()p Fp(O)18637 15461 y Fj(k)19206 15264 y
Ft(\()p Fo(C)19 b(;)202 b(r)34 b Ft(\)\))338 b(=)f(1)24848
14282 y Ff(\003)25622 15264 y Fp(\000)269 b Ft(Pr)28336
13919 y Ff(h)28908 15264 y Fo(S)29721 14764 y Fj(F)26
b(;C)31393 15264 y Ft(\(1)32470 14764 y Fj(k)33039 15264
y Ft(\))337 b(=)g(1)35733 13919 y Ff(i)36305 13143 y(\014)36305
13870 y(\014)36305 14597 y(\014)36305 15325 y(\014)36305
16052 y(\014)37046 15264 y Fp(\024)f Ft(2)38931 14764
y FG(\000)p Fl(\012\()p Fj(k)24 b Fl(\))1882 19583 y
Ft(Fix)324 b(a)g(PPT)h Fo(A)p Ft(.)512 b(W)-101 b(e)324
b(de\014ne)h(the)g(sim)-34 b(ulator)325 b Fo(S)395 b
Ft(as)324 b(follo)-34 b(ws.)513 b Fo(S)30018 19143 y
Fj(F)26 b(;C)31689 19583 y Ft(\(1)32766 19143 y Fj(k)33336
19583 y Ft(\))325 b(c)-34 b(ho)34 b(oses)324 b Fo(x)39491
18896 y Fw(R)39178 19583 y Fp( )110 b(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)43463 19143 y Fl(6)p Fj(k)44827 19583 y
Ft(and)325 b(sim)-34 b(ulates)0 21088 y Fo(A)909 20648
y Fj(F)1691 21088 y Ft(\()p Fo(x)p Ft(\),)485 b(using)470
b(its)f(o)-34 b(wn)471 b Fo(F)168 b Ft(-oracle)469 b(to)g(answ)-34
b(er)470 b Fo(A)p Ft('s)f(oracle)f(queries,)485 b Fr(exc)-62
b(ept)467 b Fo(A)p Ft('s)i(queries)f(to)i(E)44925 21292
y Fj(k)45439 21040 y Fy(0)46261 21088 y Ft(for)f Fo(k)48850
20648 y FG(0)49606 21088 y Fp(\025)445 b Fo(k)38 b Ft(.)0
22594 y(On)404 b Fo(A)p Ft('s)g(query)g(\()p Fo(x)8683
22154 y FG(0)8993 22594 y Fo(;)202 b(y)10169 22154 y
FG(0)10480 22594 y Ft(\))405 b(to)f(E)13662 22797 y Fj(k)14176
22545 y Fy(0)14529 22594 y Ft(,)g Fo(S)474 b Ft(feeds)405
b Fo(A)f Ft(the)g(resp)34 b(onse)404 b Fo(z)457 b Ft(computed)405
b(as)g(follo)-34 b(ws:)1481 25095 y(1.)606 b(If)404 b
Fo(x)4935 24656 y FG(0)5582 25095 y Ft(=)337 b Fo(x)p
Ft(,)403 b(then)i(set)g Fo(z)389 b Ft(=)337 b Fo(C)87
b Ft(\()p Fo(y)17243 24656 y FG(0)17555 25095 y Ft(\))404
b(\(computed)i(using)e(oracle)f(access)h(to)g Fo(C)87
b Ft(\).)1481 27597 y(2.)606 b(Else)593 b(if)h Fo(x)7796
27157 y FG(0)8759 27597 y Ft(=)653 b Fp(O)11320 27801
y Fj(k)11834 27549 y Fy(0)12187 27597 y Ft(\()p Fo(C)13611
27157 y FG(0)13922 27597 y Fo(;)202 b(r)15042 27157 y
FG(0)15351 27597 y Ft(\))594 b(for)g(some)g(previous)f(query)g(\()p
Fo(C)31686 27157 y FG(0)31997 27597 y Fo(;)202 b(r)33117
27157 y FG(0)33427 27597 y Ft(\))594 b(to)g(the)g Fp(O)39405
27801 y Fj(k)39919 27549 y Fy(0)40272 27597 y Ft(-oracle,)640
b(then)595 b(set)f Fo(z)705 b Ft(=)3030 29103 y(\()p
Fo(C)4454 28663 y FG(0)4765 29103 y Ft(\))5236 28663
y Fj(F)6018 29103 y Ft(\()p Fo(y)7126 28663 y FG(0)7438
29103 y Ft(\))404 b(\(computed)i(recursiv)-34 b(ely)402
b(using)j(these)f(same)g(rules\).)1481 31604 y(3.)606
b(Else)404 b(set)g Fo(z)390 b Ft(=)336 b Fp(?)p Ft(.)0
34106 y(F)-101 b(rom)401 b(the)g(fact)h(that)g(a)f(circuit)f(of)i(size)
e Fo(s)h Ft(can)g(only)g(mak)-34 b(e)400 b(oracle)g(queries)g(of)i
(total)f(length)h Fo(s)p Ft(,)f(it)g(follo)-34 b(ws)402
b(that)0 35612 y(the)396 b(recursiv)-34 b(e)394 b(ev)-67
b(aluation)396 b(of)g(\()p Fo(C)15747 35172 y FG(0)16058
35612 y Ft(\))16529 35172 y Fj(F)17311 35612 y Ft(\()p
Fo(y)43 b Ft(\))397 b(only)e(incurs)h(a)f(p)34 b(olynomial)395
b(o)-34 b(v)g(erhead)396 b(in)g(running)g(time.)536 b(Also)395
b(note)0 37117 y(that)405 b Fo(S)474 b Ft(nev)-34 b(er)404
b(queries)f(the)i(E)14076 37321 y Fj(k)14590 37069 y
Fy(0)15347 37117 y Ft(oracle)e(for)h Fo(k)21376 36677
y FG(0)22024 37117 y Fp(\025)336 b Fo(k)38 b Ft(.)1882
38622 y(Let)398 b(us)h(denote)f(the)h(execution)f(of)h(the)g(ab)34
b(o)-34 b(v)g(e)398 b(sim)-34 b(ulation)400 b(for)e(a)g(particular)g
Fo(x)g Ft(b)-34 b(y)399 b Fo(S)41798 38183 y Fj(F)26
b(;C)43470 38622 y Ft(\()p Fo(x)p Ft(\).)537 b(Notice)398
b(that)0 40128 y(when)508 b Fo(x)h Ft(=)g Fp(O)6887 40325
y Fj(k)7456 40128 y Ft(\()p Fo(C)19 b(;)202 b(r)34 b
Ft(\))508 b(for)f(some)h Fo(r)34 b Ft(,)532 b(then)508
b Fo(S)21137 39688 y Fj(F)26 b(;C)22809 40128 y Ft(\()p
Fo(x)p Ft(\))508 b(and)g Fo(A)28321 39688 y Fj(F)29103
40128 y Ft(\()p Fo(x)p Ft(\))g(ha)-34 b(v)g(e)508 b(exactly)e(the)i
(same)f(b)34 b(eha)-34 b(vior)507 b Fr(unless)0 41633
y Ft(the)451 b(ab)34 b(o)-34 b(v)g(e)451 b(sim)-34 b(ulation)451
b(pro)34 b(duces)451 b(some)g(query)f(\()p Fo(x)24366
41194 y FG(0)24676 41633 y Fo(;)202 b(y)25852 41194 y
FG(0)26163 41633 y Ft(\))451 b(suc)-34 b(h)452 b(that)g
Fo(x)33232 41194 y FG(0)33956 41633 y Fp(2)414 b Ft(Image\()p
Fp(O)39813 41837 y Fj(k)40327 41585 y Fy(0)40680 41633
y Ft(\),)463 b Fo(x)42644 41194 y FG(0)43368 41633 y
Fp(6)p Ft(=)414 b Fo(x)p Ft(,)462 b(and)451 b Fo(x)49313
41194 y FG(0)50074 41633 y Ft(w)-34 b(as)0 43139 y(not)441
b(obtained)f(b)-34 b(y)441 b(a)e(previous)h(query)f(to)h
Fp(O)20723 43343 y Fj(k)21237 43090 y Fy(0)21590 43139
y Ft(.)646 b(Since)439 b Fp(O)473 b Ft(is)440 b(a)g(random)g
(length-tripling)h(function,)449 b(it)440 b(follo)-34
b(ws)0 44644 y(that)412 b(the)g(latter)f(happ)34 b(ens)412
b(with)g(probabilit)-34 b(y)412 b(at)f(most)h(p)34 b(oly)16
b(\()p Fo(k)38 b Ft(\))275 b Fp(\001)f Ft(2)31647 44204
y Fl(2)p Fj(k)32686 44644 y Fo(=)p Ft(2)33898 44204 y
Fl(6)p Fj(k)34938 44644 y Ft(,)412 b(tak)-34 b(en)412
b(o)-34 b(v)g(er)411 b(the)g(c)-34 b(hoice)411 b(of)g
Fo(F)580 b Ft(and)0 46150 y(a)435 b(random)i Fo(r)469
b Ft(\(recall)434 b(that)j Fo(x)388 b Ft(=)h Fp(O)16311
46347 y Fj(k)16880 46150 y Ft(\()p Fo(C)19 b(;)202 b(r)34
b Ft(\)\).)20635 45710 y Fl(12)22265 46150 y Ft(Th)-34
b(us,)445 b(with)436 b(probabilit)-34 b(y)436 b(at)g(least)f(1)290
b Fp(\000)g Ft(2)41848 45710 y FG(\000)p Fl(3)p Fj(k)44055
46150 y Ft(o)-34 b(v)g(er)435 b(the)h(c)-34 b(hoice)0
47655 y(of)404 b Fo(F)168 b Ft(,)405 b Fo(S)3882 47215
y Fj(F)26 b(;C)5554 47655 y Ft(\()p Fp(O)6990 47852 y
Fj(k)7559 47655 y Ft(\()p Fo(C)19 b(;)202 b(r)34 b Ft(\)\))338
b(=)e Fo(A)13503 47215 y Fj(F)14285 47655 y Ft(\()p Fp(O)15721
47852 y Fj(k)16290 47655 y Ft(\()p Fo(C)19 b(;)202 b(r)34
b Ft(\)\))405 b(for)f(all)g(but)h(a)f(2)27490 47215 y
FG(\000)p Fl(\012\()p Fj(k)24 b Fl(\))30606 47655 y Ft(fraction)405
b(of)f Fo(r)34 b Ft('s.)1882 49161 y(Th)-34 b(us,)405
b(pro)-34 b(ving)405 b(Claim)f(4.16)g(reduces)g(to)g(sho)-34
b(wing)406 b(that:)8654 50725 y Ff(\014)8654 51452 y(\014)8654
52179 y(\014)8654 52907 y(\014)8654 53634 y(\014)10429
52846 y Ft(Pr)9058 54124 y Fj(r)9686 53613 y Fw(R)9509
54124 y FG( )o(f)p Fl(0)p Fj(;)p Fl(1)p FG(g)12590 53872
y Fc(k)13303 51864 y Ff(\002)13808 52846 y Fo(S)14621
52346 y Fj(F)26 b(;C)16293 52846 y Ft(\()p Fp(O)17729
53043 y Fj(k)18298 52846 y Ft(\()p Fo(C)19 b(;)202 b(r)34
b Ft(\)\))338 b(=)e(1)23939 51864 y Ff(\003)24713 52846
y Fp(\000)1884 b Ft(Pr)25926 54124 y Fj(x)26632 53613
y Fw(R)26456 54124 y FG( )o(f)p Fl(0)p Fj(;)p Fl(1)p
FG(g)29537 53872 y Fw(6)p Fc(k)30655 51864 y Ff(\002)31160
52846 y Fo(S)31973 52346 y Fj(F)26 b(;C)33645 52846 y
Ft(\()p Fo(x)p Ft(\))337 b(=)g(1)37503 51864 y Ff(\003)38008
50725 y(\014)38008 51452 y(\014)38008 52179 y(\014)38008
52907 y(\014)38008 53634 y(\014)38749 52846 y Fp(\024)f
Ft(2)40634 52346 y FG(\000)p Fl(\012\()p Fj(k)24 b Fl(\))0
56850 y Ft(with)405 b(high)g(probabilit)-34 b(y)404 b(\(sa)-34
b(y)-101 b(,)405 b(1)269 b Fp(\000)g Ft(2)17148 56410
y Fl(3)p Fj(k)18187 56850 y Ft(\))405 b(o)-34 b(v)g(er)404
b(the)g(c)-34 b(hoice)404 b(of)h Fo(F)168 b Ft(.)1882
58687 y(In)323 b(other)g(w)-34 b(ords,)339 b(w)-34 b(e)324
b(need)f(to)g(sho)-34 b(w)324 b(that)g(the)f(function)h
Fo(G)p Ft(\()p Fo(r)34 b Ft(\))30725 58000 y Fl(def)30903
58687 y Ft(=)284 b Fp(O)33095 58884 y Fj(k)33664 58687
y Ft(\()p Fo(C)19 b(;)202 b(r)34 b Ft(\))323 b(is)g(a)g(pseudorandom)h
(generator)0 60193 y(against)369 b Fo(S)70 b Ft(.)527
b(Since)369 b Fo(G)f Ft(is)h(a)f(random)i(function)g(from)f
Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)27350 59753 y Fj(k)28255
60193 y Fp(!)337 b(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)32767
59753 y Fl(6)p Fj(k)33806 60193 y Ft(,)376 b(this)369
b(w)-34 b(ould)370 b(b)34 b(e)368 b(ob)-34 b(vious)370
b(w)-34 b(ere)368 b(it)h(not)0 61698 y(for)447 b(the)g(fact)g(that)h
Fo(S)517 b Ft(has)448 b(oracle)d(access)h(to)i Fo(F)615
b Ft(\(whic)-34 b(h)448 b(is)f(correlated)f(with)h Fo(G)p
Ft(\).)667 b(Recall,)457 b(ho)-34 b(w)g(ev)g(er,)458
b(that)448 b(w)-34 b(e)0 63204 y(made)357 b(sure)f(that)h
Fo(S)427 b Ft(do)34 b(es)356 b(not)h(query)f(the)h(E)20435
63407 y Fj(k)20949 63155 y Fy(0)21302 63204 y Ft(-oracle)e(for)i(an)-34
b(y)356 b Fo(k)29881 62764 y FG(0)30529 63204 y Fp(\025)336
b Fo(k)38 b Ft(.)523 b(This)357 b(enables)f(us)h(to)g(use)f(the)h
(follo)-34 b(wing)0 64709 y(lemma,)403 b(pro)-34 b(v)g(en)405
b(in)f(App)34 b(endix)405 b(B.)p 0 65764 20800 45 v 976
66481 a Fw(11)1843 66904 y Fx(Note)393 b(that)f(w)-28
b(e)392 b(are)g(only)h(pro)-28 b(ving)392 b(the)h(virtual)g(blac)-28
b(k-b)28 b(o)-28 b(x)393 b(prop)28 b(ert)-28 b(y)392
b(against)h(adv)-28 b(ersaries)392 b(of)g(\\b)28 b(ounded)392
b(non)-28 b(uniformit)g(y)-85 b(,")0 68122 y(whic)-28
b(h)451 b(in)g(particular)h(includes)g(all)g(uniform)f(PPT)g(adv)-28
b(ersaries.)786 b(Presumably)451 b(it)h(can)g(also)g(b)28
b(e)452 b(pro)-28 b(v)g(en)450 b(against)i(non)-28 b(uniform)0
69339 y(adv)g(ersaries,)342 b(but)f(w)-28 b(e)341 b(stic)-28
b(k)342 b(to)f(uniform)g(adv)-28 b(ersaries)341 b(for)g(simplicit)-28
b(y)-85 b(.)976 70134 y Fw(12)1843 70557 y Fx(T)g(ec)-28
b(hnically)-85 b(,)342 b(this)g(probabilit)-28 b(y)342
b(\(and)f(later)h(ones)f(in)h(the)g(pro)28 b(of)78 b(\))341
b(should)g(also)g(b)28 b(e)342 b(tak)-28 b(en)342 b(o)-28
b(v)g(er)341 b(the)h(coin)f(tosses)h(of)f Fi(A)p Fx(/)p
Fi(S)57 b Fx(.)25394 74521 y Ft(25)p eop
%%Page: 26 27
26 26 bop 0 1218 a Fz(Lemma)464 b(4.17)606 b Fr(Ther)-62
b(e)475 b(is)g(a)g(c)-62 b(onstant)474 b Fo(\016)459
b(>)413 b Ft(0)475 b Fr(such)f(that)h(the)f(fol)62 b(lowing)474
b(holds)g(for)h(al)62 b(l)475 b(su\016ciently)e(lar)-62
b(ge)474 b Fo(K)0 2723 y Fr(and)525 b(any)h Fo(L)503
b Fp(\025)h Fo(K)8753 2283 y Fl(2)9279 2723 y Fr(.)834
b(L)-62 b(et)525 b Fo(D)559 b Fr(b)-62 b(e)525 b(an)h(algorithm)d(that)
i(makes)f(at)h(most)g Fo(K)35183 2283 y Fj(\016)36214
2723 y Fr(or)-62 b(acle)524 b(queries)f(and)j(let)f Fo(G)g
Fr(b)-62 b(e)525 b(a)0 4229 y(r)-62 b(andom)433 b(inje)-62
b(ctive)431 b(function)i Fo(G)336 b Ft(:)h([)p Fo(K)87
b Ft(])336 b Fp(!)h Ft([)p Fo(L)p Ft(])p Fr(.)557 b(Then)433
b(with)g(pr)-62 b(ob)g(ability)430 b(at)j(le)-62 b(ast)432
b Ft(1)269 b Fp(\000)g Ft(2)40589 3789 y FG(\000)p Fj(K)42177
3476 y Fc(\016)43121 4229 y Fr(over)432 b Fo(G)p Fr(,)12708
5860 y Ff(\014)12708 6587 y(\014)12708 7315 y(\014)12708
8042 y(\014)13730 7618 y Ft(Pr)13112 8484 y Fj(x)p FG(2)p
Fl([)p Fj(K)63 b Fl(])15850 6636 y Ff(\002)16355 7618
y Fo(D)17393 7117 y Fj(G)18181 7618 y Ft(\()p Fo(G)p
Ft(\()p Fo(x)p Ft(\)\))338 b(=)e(1)23934 6636 y Ff(\003)24709
7618 y Fp(\000)762 b Ft(Pr)25921 8484 y Fj(y)32 b FG(2)p
Fl([)p Fj(L)p Fl(])28410 6636 y Ff(\002)28915 7618 y
Fo(D)29953 7117 y Fj(G)30740 7618 y Ft(\()p Fo(y)43 b
Ft(\))339 b(=)d(1)34543 6636 y Ff(\003)35048 5860 y(\014)35048
6587 y(\014)35048 7315 y(\014)35048 8042 y(\014)35789
7618 y Fp(\024)37709 6798 y Ft(1)p 37201 7339 1622 49
v 37201 8449 a Fo(K)38317 8099 y Fj(\016)38955 7618 y
Fo(:)1882 11320 y Ft(Let)397 b(us)h(see)e(ho)-34 b(w)399
b(Lemma)e(4.17)g(implies)g(what)h(w)-34 b(e)398 b(w)-34
b(an)g(t.)538 b(Let)397 b Fo(K)424 b Ft(=)336 b(2)34783
10880 y Fj(k)35749 11320 y Ft(and)399 b(asso)34 b(ciate)396
b([)p Fo(K)87 b Ft(])397 b(with)h Fp(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)51094 10880 y Fj(k)51663 11320 y Ft(.)0
12826 y(W)-101 b(e)399 b(\014x)i(all)e(v)-67 b(alues)399
b(of)i Fp(O)11423 13029 y Fj(k)11937 12777 y Fy(0)12690
12826 y Ft(for)f(all)g Fo(k)16890 12386 y FG(0)17537
12826 y Fp(6)p Ft(=)337 b Fo(k)438 b Ft(and)401 b(E)23064
13029 y Fj(k)23578 12777 y Fy(0)24331 12826 y Ft(for)f(all)f
Fo(k)28530 12386 y FG(0)29178 12826 y Fo(<)336 b(k)38
b Ft(.)537 b(W)-101 b(e)400 b(also)g(\014x)g(the)h(v)-67
b(alues)399 b(of)h Fp(O)46254 13023 y Fj(k)46823 12826
y Ft(\()p Fo(C)48247 12386 y FG(0)48558 12826 y Fo(;)202
b(r)34 b Ft(\))400 b(for)0 14663 y(all)456 b Fo(C)2689
14223 y FG(0)3425 14663 y Fp(6)p Ft(=)424 b Fo(C)87 b
Ft(,)470 b(and)458 b(view)e Fo(G)p Ft(\()p Fo(r)34 b
Ft(\))14590 13976 y Fl(def)14768 14663 y Ft(=)482 b Fp(O)17158
14860 y Fj(k)17727 14663 y Ft(\()p Fo(C)19 b(;)202 b(r)34
b Ft(\))458 b(as)f(a)g(random)h(injectiv)-34 b(e)456
b(function)j(from)e([)p Fo(K)87 b Ft(])456 b(to)i(the)f(remaining)0
16168 y Fo(L)337 b Ft(=)f Fo(K)3557 15728 y Fl(6)4150
16168 y Fp(\000)67 b Ft(\()p Fo(K)154 b Fp(\000)67 b
Ft(1\))g Fp(\001)g Fo(K)389 b Ft(elemen)-34 b(ts)303
b(of)h Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)19888 15728
y Fl(6)p Fj(k)20927 16168 y Ft(.)505 b(The)303 b(only)g(oracle)e
(queries)h(of)i Fo(S)373 b Ft(that)304 b(v)-67 b(ary)301
b(with)j(the)f(c)-34 b(hoice)303 b(of)g Fo(G)0 17674
y Ft(are)c(queries)f(to)h Fp(O)8239 17871 y Fj(k)9107
17674 y Ft(at)h(p)34 b(oin)-34 b(ts)300 b(of)f(the)h(form)f(\()p
Fo(C)19 b(;)202 b(r)34 b Ft(\),)321 b(whic)-34 b(h)300
b(is)f(equiv)-67 b(alen)-34 b(t)298 b(to)i(queries)e(to)i
Fo(G)p Ft(.)503 b(Th)-34 b(us,)321 b(Lemma)299 b(4.17)0
19179 y(implies)431 b(that)h(the)g(output)i(of)e Fo(G)f
Ft(is)g(indistinguishable)i(from)f(the)g(uniform)g(distribution)h(on)f
(some)f(subset)i(of)0 20685 y Fp(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)2963 20245 y Fl(6)p Fj(k)4371 20685 y Ft(of)370
b(size)e Fo(L)p Ft(.)527 b(Since)369 b(the)g(latter)g(has)h
(statistical)f(di\013erence)g(\()p Fo(K)32721 20245 y
Fl(6)33447 20685 y Fp(\000)199 b Fo(L)p Ft(\))p Fo(=K)37607
20245 y Fl(6)38470 20685 y Fo(<)336 b Ft(1)p Fo(=K)42077
20245 y Fl(4)42972 20685 y Ft(from)370 b(the)f(uniform)0
22190 y(distribution)452 b(on)f Fp(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)11349 21750 y Fl(6)p Fj(k)12389 22190 y
Ft(,)462 b(w)-34 b(e)451 b(conclude)g(that)h Fo(G)e Ft(is)h
Fo(")p Ft(-pseudorandom)i(\(for)e Fo(")415 b Ft(=)f(1)p
Fo(=K)41536 21750 y Fj(\016)42341 22190 y Ft(+)301 b(1)p
Fo(=K)45913 21750 y Fl(4)46853 22190 y Ft(=)414 b(2)48816
21750 y FG(\000)p Fl(\012\()p Fj(k)24 b Fl(\))51529 22190
y Ft(\))0 23817 y(against)405 b Fo(S)474 b Ft(with)405
b(probabilit)-34 b(y)405 b(at)f(least)g(1)270 b Fp(\000)f
Ft(2)21369 23378 y FG(\000)p Fj(K)22957 23065 y Fc(\016)23803
23817 y Fo(>)336 b Ft(1)270 b Fp(\000)f Ft(2)27776 23378
y FG(\000)p Fl(3)p Fj(k)29547 23817 y Ft(,)404 b(as)g(desired.)15267
b Fb(2)1882 26651 y Ft(While)597 b(our)h(result)g(do)34
b(es)598 b(not)g(relativize)e(in)i(the)g(usual)h(sense,)646
b(the)598 b(pro)34 b(of)598 b(do)34 b(es)598 b(w)-34
b(ork)598 b(for)g(a)g(sligh)-34 b(tly)0 28157 y(di\013eren)g(t)354
b(form)g(of)h(relativization,)362 b(whic)-34 b(h)355
b(w)-34 b(e)355 b(refer)e(to)h(as)g Fr(b)-62 b(ounde)g(d)385
b(r)-62 b(elativization)351 b Ft(\(and)k(is)e(ho)-34
b(w)355 b(the)g(Random)0 29662 y(Oracle)524 b(Mo)34 b(del)525
b(is)g(sometimes)h(in)-34 b(terpreted)526 b(in)f(cryptograph)-34
b(y)-101 b(.\))904 b(In)525 b Fr(b)-62 b(ounde)g(d)543
b(r)-62 b(elativization)p Ft(,)552 b(an)526 b(oracle)e(is)0
31168 y(a)477 b Fr(\014nite)g Ft(function)h(with)g(\014xed)g(input)g
(length)g(\(p)34 b(olynomially)476 b(related)h(to)g(the)h(securit)-34
b(y)477 b(parameter)g Fo(k)38 b Ft(\),)495 b(and)0 32673
y(all)569 b(algorithms/circuits)h(in)g(the)h(proto)34
b(col)570 b(can)g(ha)-34 b(v)g(e)570 b(running)h(time)f(larger)f(than)j
(this)e(length)h(\(but)g(still)0 34179 y(p)34 b(olynomial)490
b(in)h Fo(k)38 b Ft(\).)799 b(In)491 b(particular,)512
b(in)491 b(the)g(con)-34 b(text)491 b(of)h(obfuscation,)513
b(this)492 b(means)f(that)h(the)f(circuit)f(to)h(b)34
b(e)0 35684 y(obfuscated)406 b(can)e(ha)-34 b(v)g(e)405
b(size)e(p)34 b(olynomial)403 b(in)i(this)f(length.)0
38518 y Fz(Prop)39 b(osition)465 b(4.18)606 b Fr(The)-62
b(or)g(ems)406 b(3.11)g(and)h(3.8)f(\(one-way)g(functions)f(imply)h
(unobfusc)-62 b(atable)403 b(functions)i(and)0 40023
y(imp)-62 b(ossibility)593 b(of)k(cir)-62 b(cuit)596
b(obfusc)-62 b(ators\),)633 b(and)597 b(Cor)-62 b(ol)62
b(lary)596 b(3.10)g(\(unc)-62 b(onditional)595 b(imp)-62
b(ossibility)593 b(of)k(e\016cient)0 41529 y(cir)-62
b(cuit)432 b(obfusc)-62 b(ators\))429 b(hold)j(under)h(b)-62
b(ounde)g(d)431 b(r)-62 b(elativization)430 b(\(for)j(any)g(or)-62
b(acle\).)0 44363 y Fz(Pro)39 b(of)560 b(Sk)-39 b(etc)g(h:)1826
b Ft(The)488 b(only)f(mo)34 b(di\014cation)488 b(needed)g(in)f(the)h
(construction)g(is)f(to)h(deal)f(with)h(oracle)f(gates)0
45868 y(in)519 b(the)h(Hom)f(algorithm)h(in)f(the)g(pro)34
b(of)520 b(of)f(Lemma)g(3.6.)883 b(Let's)519 b(call)f(sa)-34
b(y)520 b(the)f(oracle)f Fo(F)688 b Ft(has)520 b(input)g(length)0
47374 y Fo(`)d Ft(and)h(output)h(length)f(1)f(\(without)i(loss)e(of)h
(generalit)-34 b(y\).)877 b(W)-101 b(e)517 b(augmen)-34
b(t)518 b(the)g(Hom)40387 47562 y Fj(K)41816 47374 y
Ft(to)f(also)g(tak)-34 b(e)518 b(inputs)0 48879 y(of)447
b(the)g(form)g(\()p Fo(c)7457 49061 y Fl(1)7982 48879
y Fo(;)202 b(:)g(:)g(:)f(;)h(c)11201 49076 y Fj(`)11640
48879 y Fo(;)g Fm(oracle)r Ft(\))447 b(\(where)g(\()p
Fo(c)21930 49061 y Fl(1)22455 48879 y Fo(;)202 b(:)g(:)g(:)f(;)h(c)
25674 49076 y Fj(`)26113 48879 y Ft(\))447 b(are)f(ciphertexts\),)457
b(on)447 b(whic)-34 b(h)448 b(it)e(naturally)h(outputs)0
50384 y(Enc)2037 50572 y Fj(K)2948 50384 y Ft(\()p Fo(F)168
b Ft(\(Dec)6842 50572 y Fj(K)7754 50384 y Ft(\()p Fo(c)8750
50566 y Fl(1)9276 50384 y Ft(\))p Fo(;)202 b Ft(Dec)12289
50572 y Fj(K)13200 50384 y Ft(\()p Fo(c)14196 50566 y
Fl(2)14722 50384 y Ft(\))p Fo(;)g(:)g(:)g(:)f(;)h Ft(Dec)19891
50572 y Fj(K)20802 50384 y Ft(\()p Fo(c)21798 50581 y
Fj(`)22238 50384 y Ft(\)\)\).)505 b(The)303 b(rest)f(of)h(the)g(pro)34
b(of)303 b(pro)34 b(ceeds)301 b(essen)-34 b(tially)302
b(unc)-34 b(hanged.)51094 51890 y Fb(2)0 55709 y Fs(5)1793
b(On)598 b(a)f(Complexit)-50 b(y)600 b(Analogue)e(of)g(Rice's)g
(Theorem)0 58415 y Ft(Rice's)424 b(Theorem)i(asserts)f(that)i(the)e
(only)g(prop)34 b(erties)425 b(of)h(partial)f(recursiv)-34
b(e)424 b(functions)j(that)g(can)e(b)34 b(e)425 b(decided)0
59920 y(from)362 b(their)g(represen)-34 b(tations)363
b(as)f(T)-101 b(uring)363 b(mac)-34 b(hines)363 b(are)e(trivial.)524
b(T)-101 b(o)362 b(state)h(this)f(precisely)-101 b(,)369
b(w)-34 b(e)362 b(denote)h(b)-34 b(y)363 b([)p Fo(M)132
b Ft(])0 61426 y(the)346 b(\(p)34 b(ossibly)345 b(partial\))h(function)
g(that)h(the)f(T)-101 b(uring)345 b(Mac)-34 b(hine)346
b Fo(M)477 b Ft(computes.)520 b(Similarly)-101 b(,)356
b(for)345 b([)p Fo(C)87 b Ft(])344 b(denotes)i(the)0
62931 y(function)405 b(computed)h(b)-34 b(y)404 b(a)g(circuit)g
Fo(C)87 b Ft(.)0 65765 y Fz(Rice's)545 b(Theorem)970
b Fr(L)-62 b(et)498 b Fo(L)452 b Fp(\022)g(f)p Ft(0)p
Fo(;)202 b Ft(1)p Fp(g)18390 65325 y FG(\003)19414 65765
y Fr(b)-62 b(e)497 b(a)g(language)e(such)h(that)g(for)h(any)g
Fo(M)65 b(;)202 b(M)39971 65325 y FG(0)40779 65765 y
Fr(,)513 b Ft([)p Fo(M)132 b Ft(])452 b Fp(\021)g Ft([)p
Fo(M)47138 65325 y FG(0)47449 65765 y Ft(])497 b Fr(implies)0
67271 y(that)453 b Fo(M)505 b Fp(2)374 b Fo(L)747 b Fp(\()-202
b(\))748 b Fo(M)11212 66831 y FG(0)11896 67271 y Fp(2)374
b Fo(L)p Fr(.)619 b(If)454 b Fo(L)g Fr(is)f(de)-62 b(cidable,)455
b(then)f Fo(L)g Fr(is)f(trivial)f(in)j(the)d(sense)h(that)g(either)f
Fo(L)374 b Ft(=)f Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)51474
66831 y FG(\003)0 68776 y Fr(or)433 b Fo(L)337 b Ft(=)f
Fp(;)p Fr(.)25394 74521 y Ft(26)p eop
%%Page: 27 28
27 27 bop 1882 1218 a Ft(The)481 b(di\016cult)-34 b(y)480
b(of)h(problems)f(suc)-34 b(h)481 b(as)g Fq(SA)-106 b(T)479
b Ft(suggest)i(that)g(p)34 b(erhaps)481 b(Rice's)f(theorem)g(can)g(b)34
b(e)480 b(\\scaled-)0 2723 y(do)-34 b(wn")563 b(and)f(that)h(deciding)e
(prop)34 b(erties)561 b(of)h(\014nite)g(functions)h(from)e(their)h
(descriptions)f(as)h(circuits)f(is)g(in-)0 4229 y(tractable.)1882
5734 y(Simply)404 b(replacing)g(the)h(w)-34 b(ord)405
b(\\T)-101 b(uring)405 b(mac)-34 b(hine")405 b(with)h(\\circuit")e(and)
h(\\decidable")f(with)h(\\p)34 b(olynomial)0 7240 y(time")472
b(do)34 b(es)472 b(not)g(w)-34 b(ork.)743 b(A)472 b(coun)-34
b(terexample)472 b(is)g(the)g(language)g Fo(L)450 b Ft(=)g
Fp(f)p Fo(C)536 b Fp(2)450 b(f)p Ft(0)p Fo(;)202 b Ft(1)p
Fp(g)39697 6800 y FG(\003)40673 7240 y Fp(j)449 b Fo(C)87
b Ft(\(0\))451 b(=)f(0)p Fp(g)472 b Ft(that)h(can)0 8745
y(b)34 b(e)465 b(decided)h(in)g(p)34 b(olynomial)465
b(time,)481 b(ev)-34 b(en)465 b(though)j([)p Fo(C)87
b Ft(])438 b Fp(\021)i Ft([)p Fo(C)28855 8305 y FG(0)29165
8745 y Ft(])465 b(implies)g(\()q Fo(C)526 b Fp(2)440
b Fo(L)878 b Fp(\()-202 b(\))879 b Fo(C)43012 8305 y
FG(0)43762 8745 y Fp(2)440 b Fo(L)p Ft(\),)481 b(and)466
b(b)34 b(oth)0 10250 y Fo(L)493 b Fp(6)p Ft(=)f Fp(f)p
Ft(0)p Fo(;)202 b Ft(1)p Fp(g)5716 9811 y FG(\003)6740
10250 y Ft(and)498 b Fo(L)493 b Fp(6)p Ft(=)g Fp(;)p
Ft(.)819 b(Y)-101 b(et,)521 b(there)498 b(is)f(a)h(sense)g(in)g(whic)
-34 b(h)499 b Fo(L)e Ft(is)h(trivial)e(|)i(to)g(decide)g(whether)g
Fo(C)580 b Fp(2)493 b Fo(L)0 11756 y Ft(e\016cien)-34
b(tly)467 b(one)h(do)34 b(es)468 b(not)h(need)f(to)g(use)g
Fo(C)555 b Ft(itself,)484 b(but)469 b(rather)f(one)g(can)g(do)g(with)h
(oracle)e(access)g(to)i Fo(C)554 b Ft(only)-101 b(.)0
13261 y(This)405 b(motiv)-67 b(ates)404 b(the)g(follo)-34
b(wing)405 b(conjecture:)0 15854 y Fz(Conjecture)465
b(5.1)f(\(Scaled-do)-39 b(wn)465 b(Rice's)g(Theorem\))606
b Fr(L)-62 b(et)492 b Fo(L)443 b Fp(\022)g(f)p Ft(0)p
Fo(;)202 b Ft(1)p Fp(g)36423 15414 y FG(\003)37441 15854
y Fr(b)-62 b(e)492 b(a)g(language)e(such)h(that)g(for)0
17359 y(cir)-62 b(cuits)471 b Fo(C)19 b(;)202 b(C)6707
16919 y FG(0)7018 17359 y Fr(,)482 b Ft([)p Fo(C)87 b
Ft(])408 b Fp(\021)f Ft([)p Fo(C)12547 16919 y FG(0)12858
17359 y Ft(])472 b Fr(implies)g(that)f Fo(C)495 b Fp(2)408
b Fo(L)816 b Fp(\()-202 b(\))816 b Fo(C)28584 16919 y
FG(0)29303 17359 y Fp(2)408 b Fo(L)p Fr(.)675 b(If)473
b Fo(L)408 b Fp(2)g Fz(BPP)p Fr(,)482 b(then)473 b Fo(L)g
Fr(is)f(trivial)f(in)i(the)0 18865 y(sense)433 b(that)f(ther)-62
b(e)432 b(exists)g(a)h(PPT)i Fo(S)503 b Fr(such)432 b(that)17594
21552 y Fo(C)423 b Fp(2)337 b Fo(L)g Fp(\))g Ft(Pr)o([)p
Fo(S)25188 21052 y Fl([)p Fj(C)63 b Fl(])26499 21552
y Ft(\(1)27576 21052 y FG(j)p Fj(C)g FG(j)28888 21552
y Ft(\))337 b(=)g(1])f Fo(>)33667 20732 y Ft(2)p 33667
21274 607 49 v 33667 22384 a(3)17594 25072 y Fo(C)423
b Fp(62)337 b Fo(L)g Fp(\))g Ft(Pr)o([)p Fo(S)25188 24571
y Fl([)p Fj(C)63 b Fl(])26499 25072 y Ft(\(1)27576 24571
y FG(j)p Fj(C)g FG(j)28888 25072 y Ft(\))337 b(=)g(0])f
Fo(>)33667 24252 y Ft(2)p 33667 24793 V 33667 25903 a(3)1882
28010 y(W)-101 b(e)382 b(no)-34 b(w)385 b(consider)d(a)i
(generalization)e(of)i(this)f(conjecture)g(to)h Fr(pr)-62
b(omise)413 b(pr)-62 b(oblems)381 b Ft([ESY84],)386 b
Fr(i.e.,)413 b Ft(decision)0 29515 y(problems)280 b(restricted)e(to)i
(some)f(subset)i(of)e(strings.)498 b(F)-101 b(ormally)g(,)303
b(a)279 b(promise)g(problem)h(\005)f(is)g(a)h(pair)f(\005)337
b(=)f(\(\005)48374 29703 y Fj(Y)49184 29515 y Fo(;)202
b Ft(\005)50632 29703 y Fj(N)51529 29515 y Ft(\))0 31021
y(of)406 b(disjoin)-34 b(t)406 b(sets)g(of)f(strings,)h(corresp)34
b(onding)405 b(to)h Fq(yes)f Ft(and)h Fq(no)g Ft(instances,)g(resp)34
b(ectiv)-34 b(ely)-101 b(.)540 b(The)406 b(generalization)0
32526 y(of)479 b(Conjecture)h(5.1)f(w)-34 b(e)479 b(seek)f(is)h(the)g
(follo)-34 b(wing,)498 b(where)479 b Fz(BPP)f Ft(is)h(the)g
(generalization)f(of)i Fz(BPP)e Ft(to)h(promise)0 34032
y(problems:)0 36624 y Fz(Conjecture)465 b(5.2)606 b Fr(L)-62
b(et)468 b Ft(\005)400 b(=)g(\(\005)15703 36812 y Fj(Y)16514
36624 y Fo(;)202 b Ft(\005)17962 36812 y Fj(N)18858 36624
y Ft(\))469 b Fr(b)-62 b(e)468 b(a)g(pr)-62 b(omise)467
b(pr)-62 b(oblem)468 b(such)e(that)i(for)g(cir)-62 b(cuits)466
b Fo(C)19 b(;)202 b(C)45533 36184 y FG(0)45844 36624
y Fr(,)477 b Ft([)p Fo(C)87 b Ft(])400 b Fp(\021)g Ft([)p
Fo(C)51353 36184 y FG(0)51663 36624 y Ft(])0 38129 y
Fr(implies)482 b(that)h(b)-62 b(oth)482 b Fo(C)516 b
Fp(2)428 b Ft(\005)12878 38317 y Fj(Y)14544 38129 y Fp(\()-202
b(\))857 b Fo(C)18576 37690 y FG(0)19315 38129 y Fp(2)428
b Ft(\005)21460 38317 y Fj(Y)22754 38129 y Fr(and)484
b Fo(C)515 b Fp(2)428 b Ft(\005)28685 38317 y Fj(N)30438
38129 y Fp(\()-202 b(\))857 b Fo(C)34470 37690 y FG(0)35209
38129 y Fp(2)428 b Ft(\005)37354 38317 y Fj(N)38251 38129
y Fr(.)708 b(If)485 b Ft(\005)428 b Fp(2)g Fz(BPP)o Fr(,)497
b(then)483 b Ft(\005)i Fr(is)0 39635 y(trivial)432 b(in)i(the)e(sense)h
(that)f(ther)-62 b(e)432 b(exists)g(a)h(PPT)i Fo(S)503
b Fr(such)432 b(that)17147 42323 y Fo(C)423 b Fp(2)337
b Ft(\005)20490 42511 y Fj(Y)21637 42323 y Fp(\))g Ft(Pr)o([)p
Fo(S)25635 41822 y Fl([)p Fj(C)63 b Fl(])26946 42323
y Ft(\(1)28023 41822 y FG(j)p Fj(C)g FG(j)29335 42323
y Ft(\))337 b(=)g(1])f Fo(>)34114 41503 y Ft(2)p 34114
42044 V 34114 43154 a(3)17103 45842 y Fo(C)424 b Fp(2)337
b Ft(\005)20447 46030 y Fj(N)21680 45842 y Fp(\))g Ft(Pr[)p
Fo(S)25679 45342 y Fl([)p Fj(C)63 b Fl(])26990 45842
y Ft(\(1)28067 45342 y FG(j)p Fj(C)g FG(j)29378 45842
y Ft(\))338 b(=)e(0])g Fo(>)34158 45022 y Ft(2)p 34158
45563 V 34158 46674 a(3)1882 48713 y(Our)404 b(construction)h(of)f
(unobfuscatable)j(functions)e(implies)f(that)h(the)g(latter)f
(conjecture)g(is)g(false.)0 51305 y Fz(Theorem)464 b(5.3)606
b Fr(If)434 b(one-way)e(functions)g(exist,)h(then)g(Conje)-62
b(ctur)g(e)431 b(5.2)i(is)g(false.)0 53897 y Fz(Pro)39
b(of)626 b(Sk)-39 b(etc)g(h:)2075 b Ft(Let)545 b Fp(H)583
b Ft(=)571 b Fp(fH)17545 54094 y Fj(k)18114 53897 y Fp(g)18720
54094 y Fj(k)24 b FG(2)p Fe(N)21101 53897 y Ft(b)34 b(e)544
b(the)i(unobfuscatable)h(function)g(ensem)-34 b(ble)545
b(giv)-34 b(en)545 b(b)-34 b(y)545 b(Theo-)0 55403 y(rem)445
b(3.11,)454 b(and)446 b(let)f Fo(\031)j Ft(:)11486 54494
y Ff(S)12496 55761 y Fj(k)13267 55403 y Ft(Supp)r(\()p
Fp(H)17456 55600 y Fj(k)18025 55403 y Ft(\))405 b Fp(!)g(f)p
Ft(0)p Fo(;)202 b Ft(1)p Fp(g)445 b Ft(b)34 b(e)445 b(the)g(prop)34
b(ert)-34 b(y)445 b(guaran)-34 b(teed)447 b(b)-34 b(y)445
b(the)h(unobfuscatabilit)-34 b(y)0 56908 y(condition.)1882
58414 y(Consider)404 b(the)h(follo)-34 b(wing)405 b(promise)f(problem)g
(\005)337 b(=)f(\(\005)27292 58602 y Fj(Y)28102 58414
y Fo(;)202 b Ft(\005)29550 58602 y Fj(N)30447 58414 y
Ft(\):)12460 61925 y(\005)13369 62113 y Fj(Y)15286 61925
y Ft(=)17336 59852 y Ff(\()18312 61925 y Fo(C)424 b Ft(:)336
b([)p Fo(C)87 b Ft(])337 b Fp(2)23383 60773 y Ff([)23800
63408 y Fj(k)24932 61925 y Ft(Supp)r(\()p Fp(H)29121
62122 y Fj(k)29690 61925 y Ft(\))404 b(and)h Fo(\031)43
b Ft(\([)p Fo(C)87 b Ft(]\))337 b(=)g(1)38448 59852 y
Ff(\))12374 66094 y Ft(\005)13283 66282 y Fj(N)15286
66094 y Ft(=)17336 64021 y Ff(\()18312 66094 y Fo(C)424
b Ft(:)336 b([)p Fo(C)87 b Ft(])337 b Fp(2)23383 64942
y Ff([)23800 67577 y Fj(k)24932 66094 y Ft(Supp)r(\()p
Fp(H)29121 66291 y Fj(k)29690 66094 y Ft(\))404 b(and)h
Fo(\031)43 b Ft(\([)p Fo(C)87 b Ft(]\))337 b(=)g(0)38448
64021 y Ff(\))0 69694 y Ft(\005)457 b Fp(2)f Fz(BPP)475
b Ft(b)34 b(ecause)476 b Fo(\031)43 b Ft(\()p Fo(f)130
b Ft(\))478 b(is)e(easy)f(to)i(compute)f(with)h(access)f(to)g(a)g
(circuit)f(that)i(computes)g Fo(f)130 b Ft(.)755 b(But)476
b(since)0 71200 y Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\))406
b(is)e(hard)h(to)f(compute)h(with)g(blac)-34 b(k-b)34
b(o)-34 b(x)404 b(access)g(to)g Fo(f)130 b Ft(,)405 b(no)f
Fo(S)474 b Ft(satisfying)405 b(Conjecture)g(5.2)f(can)g(exist.)1620
b Fb(2)25394 74521 y Ft(27)p eop
%%Page: 28 29
28 28 bop 1882 1218 a Ft(It)472 b(is)f(an)h(in)-34 b(teresting)472
b(problem)g(to)g(w)-34 b(eak)g(en)473 b(or)e(ev)-34 b(en)472
b(remo)-34 b(v)g(e)471 b(the)h(h)-34 b(yp)34 b(othesis)473
b(that)g(one-w)-34 b(a)g(y)472 b(functions)0 2723 y(exist.)529
b(Reasons)378 b(to)f(b)34 b(eliev)-34 b(e)376 b(that)i(this)g(ma)-34
b(y)377 b(b)34 b(e)376 b(p)34 b(ossible)377 b(are:)525
b(1.)k(The)378 b(conjecture)f(is)g(only)f(ab)34 b(out)378
b(w)-34 b(orst)378 b(case)0 4229 y(complexit)-34 b(y)274
b(and)h(not)f(a)-34 b(v)g(erage)274 b(case,)300 b(and)275
b(2.)495 b(The)274 b(conjectures)h(imply)e(some)i(sort)f(of)g
(computational)i(di\016cult)-34 b(y)-101 b(.)0 5734 y(F)g(or)490
b(instance,)511 b(if)490 b Fz(NP)479 b Fp(\022)g Fz(BPP)489
b Ft(then)h(b)34 b(oth)491 b(conjectures)f(are)f(false,)511
b(as)489 b Fq(Cir)-27 b(cuit)550 b(Sa)-81 b(tisfiability)518
b Ft(is)489 b(not)0 7240 y(decidable)412 b(using)g(blac)-34
b(k-b)34 b(o)-34 b(x)412 b(access.)561 b(\(Using)413
b(blac)-34 b(k-b)34 b(o)-34 b(x)411 b(access,)i(one)f(cannot)h
(distinguish)g(a)f(circuit)f(that)i(is)0 8745 y(satis\014ed)328
b(on)f(exactly)f(one)h(randomly)g(c)-34 b(hosen)327 b(input)h(from)f
(an)g(unsatis\014able)i(circuit.\))512 b(So)327 b(if)g(w)-34
b(e)327 b(could)g(w)-34 b(eak)g(en)0 10250 y(the)405
b(h)-34 b(yp)34 b(othesis)405 b(of)f(Theorem)g(5.3)g(to)h
Fz(NP)336 b Fp(6\022)h Fz(BPP)o Ft(,)404 b(Conjecture)h(5.2)e(w)-34
b(ould)406 b(b)34 b(e)404 b(false)g Fr(unc)-62 b(onditional)62
b(ly)p Ft(.)1882 11756 y(W)-101 b(e)557 b(ha)-34 b(v)g(e)559
b(sho)-34 b(wn)559 b(that)g(in)f(the)g(con)-34 b(text)559
b(of)f(complexit)-34 b(y)-101 b(,)596 b(the)558 b(generalization)f(of)h
(Scaled-do)-34 b(wn)560 b(Rice's)0 13261 y(Theorem)364
b(\(Conjecture)h(5.1\))e(to)h(promise)g(problems)f(\(i.e.,)371
b(Conjecture)364 b(5.2\))g(fails.)525 b(When)364 b(trying)f(to)h
(\014nd)h(out)0 14767 y(what)488 b(this)f(implies)e(ab)34
b(out)488 b(Conjecture)f(5.1)f(itself,)507 b(one)487
b(migh)-34 b(t)487 b(try)f(to)h(get)g(in)-34 b(tuition)487
b(from)g(what)h(happ)34 b(ens)0 16272 y(in)472 b(the)g(con)-34
b(text)473 b(of)f(computabilit)-34 b(y)-101 b(.)743 b(This)473
b(direction)f(is)f(pursued)i(in)f(App)34 b(endix)473
b(A.)742 b(It)472 b(turns)h(out)f(that)h(the)0 17778
y(results)510 b(in)f(this)h(con)-34 b(text)510 b(are)f(inconclusiv)-34
b(e.)855 b(W)-101 b(e)509 b(explore)f(three)i(w)-34 b(a)g(ys)510
b(to)g(generalize)e(Rice's)h(Theorem)h(to)0 19283 y(promise)429
b(problems.)614 b(The)429 b(\014rst,)436 b(naiv)-34 b(e)429
b(approac)-34 b(h)430 b(fails,)435 b(and)430 b(there)f(are)g(t)-34
b(w)g(o)430 b(non-naiv)-34 b(e)430 b(generalizations,)435
b(of)0 20789 y(whic)-34 b(h)405 b(one)f(succeeds)g(and)h(one)f(fails.)
1882 22294 y(What)k(do)h(our)f(results)g(sa)-34 b(y)409
b(ab)34 b(out)409 b(the)f(claim)f(\\the)i(b)34 b(est)408
b(thing)h(y)-34 b(ou)409 b(can)f(do)h(with)f(a)h(circuit/program)e(is)0
23800 y(run)416 b(it"?)574 b(T)-101 b(o)416 b(answ)-34
b(er)417 b(this)f(question,)i(w)-34 b(e)417 b(m)-34 b(ust)417
b(\014rst)f(in)-34 b(terpret)416 b(this)g(sen)-34 b(tence)416
b(in)g(a)g(more)f(formal)g(w)-34 b(a)g(y)-101 b(.)574
b(The)0 25305 y(in)-34 b(terpretation)426 b(w)-34 b(e)426
b(suggest)g(is)f(\\deciding)g(an)-34 b(y)425 b(non-trivial)g(seman)-34
b(tic)426 b(prop)34 b(ert)-34 b(y)425 b(of)h(circuits)e(is)h(in)-34
b(tractable")0 26811 y(where)331 b(\\non)-34 b(trivial")332
b(is)f(de\014ned)h(ab)34 b(o)-34 b(v)g(e)332 b(and)g(b)-34
b(y)332 b(\\seman)-34 b(tic)332 b(prop)34 b(ert)-34 b(y")332
b(w)-34 b(e)331 b(mean)h(a)f(prop)34 b(ert)-34 b(y)332
b(of)g(the)f Fr(function)0 28316 y Ft(that)450 b(the)g(circuit)f
(computes,)461 b(rather)449 b(than)i(a)e(prop)34 b(ert)-34
b(y)449 b(of)h(the)g(particular)f(circuit.)673 b(This)450
b(in)-34 b(terpretation)450 b(is)0 29822 y(expressed)404
b(in)g(Conjectures)h(5.1)f(and)h(5.2.)1882 31327 y(Since)358
b(w)-34 b(e)359 b(ha)-34 b(v)g(en't)359 b(dispro)-34
b(v)g(ed)359 b(Conjecture)g(5.1,)367 b(ho)-34 b(w)359
b(can)g(w)-34 b(e)358 b(sa)-34 b(y)359 b(that)g(obfuscation)h(is)e(imp)
34 b(ossible?)523 b(The)0 32832 y(answ)-34 b(er)418 b(is)g(that)g
(obfuscation)i(needs)d(m)-34 b(uc)g(h)419 b(more)e(than)i(Conjecture)g
(5.1.)578 b(Informally)-101 b(,)420 b(Conjecture)f(5.1)e(only)0
34338 y(sa)-34 b(ys)355 b(that)g(for)g(ev)-34 b(ery)353
b(non)-34 b(trivial)355 b(prop)34 b(ert)-34 b(y)355 b(\()p
Fr(i.e.,)386 b Ft(one)355 b(whic)-34 b(h)355 b(cannot)h(b)34
b(e)354 b(decided)g(with)i(oracle)d(access\),)364 b(there)0
35843 y(exist)344 b(circuits)g(from)h(whic)-34 b(h)345
b(it)g(is)f(hard)i(to)e(decide)h(the)g(prop)34 b(ert)-34
b(y)-101 b(.)518 b(Obfuscation,)358 b(on)345 b(the)g(other)g(hand,)357
b(requires)0 37349 y(that)450 b(for)f(ev)-34 b(ery)448
b(non)-34 b(trivial)449 b(prop)34 b(ert)-34 b(y)449 b
Fr(and)475 b(every)e(function)g Fo(f)580 b Ft(\(for)450
b(whic)-34 b(h)449 b(the)h(prop)34 b(ert)-34 b(y)449
b(is)g(hard)g(to)g(decide)0 38854 y(giv)-34 b(en)354
b(oracle)g(access\),)363 b(there)355 b(exist)e(circuits)h
Fr(that)387 b(c)-62 b(ompute)385 b(the)i(function)f Fo(f)485
b Ft(from)355 b(whic)-34 b(h)355 b(it)g(is)f(hard)h(to)f(decide)0
40360 y(the)395 b(prop)34 b(ert)-34 b(y)-101 b(.)535
b(Still,)396 b(it)e(ma)-34 b(y)395 b(b)34 b(e)394 b(within)h(reac)-34
b(h)395 b(to)f(also)h(dispro)-34 b(v)g(e)394 b(Conjecture)i(5.1,)f(and)
h(w)-34 b(e)394 b(lea)-34 b(v)g(e)394 b(this)h(as)g(an)0
41865 y(op)34 b(en)404 b(problem.)0 45684 y Fs(6)1793
b(Obfuscating)598 b(Sampling)h(Algorithms)0 48390 y Ft(In)340
b(our)g(in)-34 b(v)g(estigation)340 b(of)g(obfuscators)i(th)-34
b(us)341 b(far,)352 b(w)-34 b(e)340 b(ha)-34 b(v)g(e)341
b(in)-34 b(terpreted)340 b(the)g(\\functionalit)-34 b(y")341
b(of)f(a)g(program)g(as)0 49896 y(b)34 b(eing)281 b(the)g(function)i
(it)d(computes.)499 b(Ho)-34 b(w)g(ev)g(er,)305 b(sometimes)281
b(one)g(is)g(in)-34 b(terested)282 b(in)f(other)g(asp)34
b(ects)281 b(of)g(a)g(program's)0 51401 y(b)34 b(eha)-34
b(vior,)348 b(and)336 b(in)f(suc)-34 b(h)336 b(cases)e(a)h(corresp)34
b(onding)335 b(c)-34 b(hange)336 b(should)g(b)34 b(e)335
b(made)g(to)g(the)h(de\014nition)g(of)f(obfuscators.)0
52907 y(In)343 b(this)g(section,)354 b(w)-34 b(e)343
b(consider)f(programs)h(that)h(are)e Fr(sampling)375
b(algorithms)p Ft(,)352 b(i.e.)517 b(are)342 b(probabilistic)h
(algorithms)0 54412 y(that)420 b(tak)-34 b(e)418 b(no)h(input)h
(\(other)f(than)g(p)34 b(ossibly)419 b(a)f(length)h(parameter\),)j(and)
e(pro)34 b(duce)418 b(an)h(output)h(according)f(to)0
55917 y(some)404 b(desired)g(distribution.)1882 57423
y(F)-101 b(or)416 b(simplicit)-34 b(y)-101 b(,)418 b(w)-34
b(e)417 b(only)f(w)-34 b(ork)416 b(with)h(sampling)g(algorithms)f(giv)
-34 b(en)416 b(b)-34 b(y)416 b(circuits)g(|)g(a)g(circuit)g
Fo(C)503 b Ft(with)417 b Fo(m)0 58928 y Ft(input)348
b(gates)g(and)f Fo(n)g Ft(output)i(gates)e(can)g(b)34
b(e)347 b(view)-34 b(ed)347 b(as)g(a)g(sampling)h(algorithm)f(for)g
(the)h(distribution)g Fp(hh)p Fo(C)87 b Fp(ii)349 b Ft(on)0
60434 y Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)2963 59994
y Fj(n)3954 60434 y Ft(obtained)366 b(b)-34 b(y)366 b(ev)-67
b(aluating)364 b Fo(C)452 b Ft(on)366 b Fo(m)f Ft(uniform)h(and)f
(indep)34 b(enden)-34 b(t)367 b(random)f(bits.)526 b(If)365
b Fo(A)g Ft(is)f(an)i(algorithm)0 61939 y(and)479 b Fo(C)565
b Ft(is)478 b(a)h(circuit,)495 b(w)-34 b(e)479 b(write)f
Fo(A)16386 61499 y FG(hh)p Fj(C)63 b FG(ii)19117 61939
y Ft(to)478 b(indicate)h(that)g Fo(A)f Ft(has)h Fr(sampling)500
b(ac)-62 b(c)g(ess)476 b Ft(to)j Fo(C)87 b Ft(.)761 b(That)479
b(is,)497 b Fo(A)478 b Ft(can)0 63445 y(obtain,)423 b(on)c(request,)i
(indep)34 b(enden)-34 b(t)421 b(and)e(uniform)g(random)g(samples)g
(from)g(the)g(distribution)h(de\014ned)f(b)-34 b(y)419
b Fo(C)87 b Ft(.)0 64950 y(The)405 b(natural)f(analogue)h(of)f(the)h
(de\014nition)g(of)g(circuit)e(obfuscators)i(to)g(sampling)g
(algorithms)f(follo)-34 b(ws.)0 67784 y Fz(De\014nition)466
b(6.1)f(\(sampling)g(obfuscator\))606 b Fr(A)427 b(pr)-62
b(ob)g(abilistic)422 b(algorithm)j Fp(O)460 b Fr(is)426
b(a)h Ft(sampling)397 b(obfuscator)428 b Fr(if,)0 69290
y(for)433 b(some)g(p)-62 b(olynomial)431 b Fo(p)p Fr(,)i(the)g(fol)62
b(lowing)432 b(thr)-62 b(e)g(e)432 b(c)-62 b(onditions)431
b(hold:)25394 74521 y Ft(28)p eop
%%Page: 29 30
29 29 bop 1818 1218 a Fp(\017)606 b Fr(\(functionality\))479
b(F)-93 b(or)483 b(every)e(cir)-62 b(cuit)481 b Fo(C)87
b Fr(,)495 b Fp(O)34 b Ft(\()p Fo(C)87 b Ft(\))483 b
Fr(is)g(a)f(cir)-62 b(cuit)481 b(that)g(samples)g(the)g(same)h
(distribution)e(as)3030 2723 y Fo(C)87 b Fr(.)1818 5225
y Fp(\017)606 b Fr(\(p)-62 b(olynomial)340 b(slowdown\))i(Ther)-62
b(e)342 b(is)g(a)g(p)-62 b(olynomial)341 b Fo(p)h Fr(such)f(that)h(for)
g(every)f(cir)-62 b(cuit)341 b Fo(C)87 b Fr(,)360 b Fp(jO)34
b Ft(\()p Fo(C)87 b Ft(\))p Fp(j)337 b(\024)f Fo(p)p
Ft(\()p Fp(j)p Fo(C)87 b Fp(j)p Ft(\))p Fr(.)1818 7727
y Fp(\017)606 b Fr(\(\\virtual)432 b(black)g(b)-62 b(ox")434
b(pr)-62 b(op)g(erty\))432 b(F)-93 b(or)434 b(any)g(PPT)i
Fo(A)p Fr(,)e(ther)-62 b(e)433 b(is)h(a)g(PPT)h Fo(S)505
b Fr(and)434 b(a)g(ne)-62 b(gligible)432 b(function)h
Fo(\013)3030 9232 y Fr(such)f(that)g(for)h(al)62 b(l)433
b(cir)-62 b(cuits)431 b Fo(C)13751 10796 y Ff(\014)13751
11523 y(\014)13751 12251 y(\014)14155 12190 y Ft(Pr)201
b([)p Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(C)87 b Ft(\)\))338
b(=)e(1])270 b Fp(\000)f Ft(Pr)26281 10845 y Ff(h)26854
12190 y Fo(S)27667 11690 y FG(hh)p Fj(C)63 b FG(ii)29919
12190 y Ft(\(1)30996 11690 y FG(j)p Fj(C)g FG(j)32308
12190 y Ft(\))337 b(=)f(1)35001 10845 y Ff(i)35574 10796
y(\014)35574 11523 y(\014)35574 12251 y(\014)36315 12190
y Fp(\024)g Fo(\013)t Ft(\()p Fp(j)p Fo(C)87 b Fp(j)p
Ft(\))p Fo(:)0 15646 y Fr(We)433 b(say)f(that)g Fp(O)468
b Fr(is)433 b Ft(e\016cien)-34 b(t)434 b Fr(if)f(it)g(runs)g(in)h(p)-62
b(olynomial)431 b(time.)1882 18480 y Ft(W)-101 b(e)350
b(do)i(not)f(kno)-34 b(w)352 b(whether)g(this)g(de\014nition)g(is)f
(imp)34 b(ossible)350 b(to)i(meet,)361 b(but)352 b(w)-34
b(e)351 b(can)g(rule)g(out)h(the)f(follo)-34 b(wing)0
19985 y(\(seemingly\))404 b(stronger)g(de\014nition.)0
22819 y Fz(De\014nition)466 b(6.2)f(\(strong)g(sampling)g(obfuscator\))
606 b Fr(A)502 b Ft(strong)478 b(sampling)h(obfuscator)503
b Fr(is)e(de\014ne)-62 b(d)501 b(in)h(the)0 24325 y(same)399
b(way)g(as)f(a)i(sampling)e(obfusc)-62 b(ator,)402 b(exp)-62
b(e)g(ct)398 b(that)g(the)h(\\virtual)e(black)g(b)-62
b(ox")399 b(pr)-62 b(op)g(erty)397 b(is)i(r)-62 b(eplac)g(e)g(d)397
b(with)i(the)0 25830 y(fol)62 b(lowing.)1818 28332 y
Fp(\017)606 b Fr(\(\\virtual)537 b(black)h(b)-62 b(ox")539
b(pr)-62 b(op)g(erty\))537 b(F)-93 b(or)539 b(any)h(PPT)g
Fo(A)p Fr(,)566 b(ther)-62 b(e)539 b(is)g(a)g(PPT)h Fo(S)610
b Fr(such)538 b(that)g(the)h(ensembles)3030 29837 y Fp(f)p
Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(C)87 b Ft(\)\))p
Fp(g)8987 30025 y Fj(C)10320 29837 y Fr(and)543 b Fp(f)p
Fo(S)14203 29398 y FG(hh)p Fj(C)63 b FG(ii)16455 29837
y Ft(\(1)17532 29398 y FG(j)p Fj(C)g FG(j)18844 29837
y Ft(\))p Fp(g)19921 30025 y Fj(C)21253 29837 y Fr(ar)-62
b(e)542 b(c)-62 b(omputational)62 b(ly)539 b(indistinguishable.)880
b(That)542 b(is,)569 b(for)543 b(every)3030 31343 y(PPT)435
b Fo(D)34 b Fr(,)432 b(ther)-62 b(e)433 b(is)g(a)g(ne)-62
b(gligible)431 b(function)h Fo(\013)439 b Fr(such)431
b(that)10346 32907 y Ff(\014)10346 33634 y(\014)10346
34361 y(\014)10750 34301 y Ft(Pr)202 b([)p Fo(D)34 b
Ft(\()p Fo(C)19 b(;)202 b(A)p Ft(\()p Fp(O)34 b Ft(\()p
Fo(C)87 b Ft(\)\)\))338 b(=)e(1])270 b Fp(\000)f Ft(Pr)26281
32955 y Ff(h)26854 34301 y Fo(D)34 b Ft(\()p Fo(C)19
b(;)202 b(S)30600 33800 y FG(hh)p Fj(C)63 b FG(ii)32852
34301 y Ft(\(1)33929 33800 y FG(j)p Fj(C)g FG(j)35241
34301 y Ft(\)\))337 b(=)g(1)38406 32955 y Ff(i)38978
32907 y(\014)38978 33634 y(\014)38978 34361 y(\014)39719
34301 y Fp(\024)f Fo(\013)t Ft(\()p Fp(j)p Fo(C)87 b
Fp(j)p Ft(\))p Fo(:)0 37757 y Fz(Prop)39 b(osition)465
b(6.3)607 b Fr(If)433 b(one-way)g(functions)f(exist,)g(then)h(str)-62
b(ong)432 b(sampling)h(obfusc)-62 b(ators)429 b(do)k(not)g(exist.)0
40591 y Fz(Pro)39 b(of)378 b(Sk)-39 b(etc)g(h:)1259 b
Ft(If)329 b(one-w)-34 b(a)g(y)331 b(functions)f(exist,)344
b(then)330 b(there)f(exist)g(message)g(authen)-34 b(tication)331
b(co)34 b(des)329 b(\(MA)-34 b(Cs\))0 42096 y(that)387
b(are)e(existen)-34 b(tially)385 b(unforgeable)h(under)g(c)-34
b(hosen)387 b(message)e(attac)-34 b(k.)533 b(Let)385
b(T)-101 b(ag)38099 42387 y Fj(K)39395 42096 y Ft(denote)387
b(the)f(tagging)g(\(i.e.,)0 43602 y(signing\))421 b(algorithm)g(for)f
(suc)-34 b(h)421 b(a)g(MA)-34 b(C)421 b(with)g(k)-34
b(ey)420 b Fo(K)87 b Ft(,)424 b(and)d(de\014ne)g(a)g(circuit)e
Fo(C)37507 43790 y Fj(K)38419 43602 y Ft(\()p Fo(x)p
Ft(\))364 b(=)g(\()p Fo(x;)202 b Ft(T)-101 b(ag)45431
43893 y Fj(K)46343 43602 y Ft(\()p Fo(x)p Ft(\)\).)588
b(That)0 45107 y(is,)363 b(the)354 b(distribution)h(sampled)f(b)-34
b(y)354 b Fo(C)17280 45295 y Fj(K)18545 45107 y Ft(is)g(simply)f(a)h
(random)g(message)f(together)h(with)h(its)f(tag.)522
b(No)-34 b(w)354 b(supp)34 b(ose)0 46613 y(there)351
b(exists)h(a)f(sampling)h(obfuscator)h Fp(O)34 b Ft(,)361
b(and)353 b(consider)e(the)h(PPT)g(adv)-34 b(ersary)351
b Fo(A)g Ft(de\014ned)i(b)-34 b(y)352 b Fo(A)p Ft(\()p
Fo(C)87 b Ft(\))337 b(=)g Fo(C)87 b Ft(.)521 b(By)0 48118
y(the)302 b(de\014nition)i(of)e(a)g(sampling)h(obfuscator,)323
b(there)302 b(exists)g(a)g(PPT)g(sim)-34 b(ulator)303
b Fo(S)372 b Ft(whic)-34 b(h,)323 b(when)303 b(giving)e(sampling)0
49624 y(access)333 b(to)g Fp(hh)p Fo(C)6730 49812 y Fj(K)7643
49624 y Fp(ii)p Ft(,)348 b(pro)34 b(duces)333 b(an)h(output)h
(computationally)f(indistinguishable)h(from)e Fo(A)p
Ft(\()p Fp(O)34 b Ft(\()p Fo(C)44473 49812 y Fj(K)45385
49624 y Ft(\)\))338 b(=)e Fp(O)34 b Ft(\()p Fo(C)50280
49812 y Fj(K)51192 49624 y Ft(\).)0 51129 y(That)324
b(is,)338 b(after)322 b(receiving)f(the)i(tags)g(of)f(p)34
b(olynomially)322 b(man)-34 b(y)323 b(random)g(messages,)338
b Fo(S)393 b Ft(pro)34 b(duces)322 b(a)g(circuit)g(whic)-34
b(h)0 52635 y(is)411 b(indistinguishable)h(from)f(one)g(whic)-34
b(h)411 b(generates)g(random)h(messages)f(with)g(its)g(tags.)559
b(This)412 b(will)e(con)-34 b(tradict)0 54140 y(the)405
b(securit)-34 b(y)403 b(of)i(the)f(MA)-34 b(C.)1882 55645
y(Let)340 b Fo(q)380 b Ft(=)337 b Fo(q)43 b Ft(\()p Fp(j)p
Fo(K)87 b Fp(j)p Ft(\))341 b(b)34 b(e)341 b(a)g(p)34
b(olynomial)340 b(b)34 b(ound)342 b(on)f(the)g(n)-34
b(um)g(b)34 b(er)342 b(of)f(samples)g(receiv)-34 b(ed)339
b(from)i Fp(hh)p Fo(C)45225 55833 y Fj(K)46137 55645
y Fp(ii)h Ft(obtained)0 57151 y(b)-34 b(y)399 b Fo(S)70
b Ft(,)400 b(and)f(consider)f(a)h(distinguisher)g Fo(D)432
b Ft(whic)-34 b(h)400 b(do)34 b(es)398 b(the)h(follo)-34
b(wing)400 b(on)f(input)h(\()p Fo(C)39515 57339 y Fj(K)40427
57151 y Fo(;)202 b(C)41919 56711 y FG(0)42229 57151 y
Ft(\):)536 b(Reco)-34 b(v)g(er)398 b(the)h(k)-34 b(ey)0
58656 y Fo(K)506 b Ft(from)420 b Fo(C)5282 58844 y Fj(K)6194
58656 y Ft(.)583 b(Obtain)420 b Fo(q)323 b Ft(+)280 b(1)419
b(random)h(samples)f(\()p Fo(x)24516 58838 y Fl(1)25042
58656 y Fo(;)202 b(y)26175 58838 y Fl(1)26701 58656 y
Ft(\))p Fo(;)g(:)g(:)g(:)f(;)h Ft(\()p Fo(x)31030 58838
y Fj(q)32 b Fl(+1)32740 58656 y Fo(;)202 b(y)33873 58838
y Fj(q)32 b Fl(+1)35582 58656 y Ft(\))420 b(from)f Fo(C)40306
58217 y FG(0)40617 58656 y Ft(.)584 b(Output)421 b(1)e(if)g(the)h
Fo(x)50810 58838 y Fj(i)51185 58656 y Ft('s)0 60162 y(are)404
b(all)f(distinct)i(and)g Fo(y)11042 60344 y Fj(i)11754
60162 y Ft(=)337 b(T)-101 b(ag)15037 60453 y Fj(K)15949
60162 y Ft(\()p Fo(x)17113 60344 y Fj(i)17489 60162 y
Ft(\))404 b(for)g(all)g Fo(i)p Ft(.)1882 61667 y(Clearly)-101
b(,)590 b Fo(D)c Ft(outputs)556 b(1)e(with)g(high)h(probabilit)-34
b(y)554 b(on)g(input)h(\()p Fo(C)32522 61855 y Fj(K)33434
61667 y Fo(;)202 b(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(C)37689
61855 y Fj(K)38601 61667 y Ft(\)\)\).)988 b(\(The)555
b(only)e(reason)h(it)0 63173 y(migh)-34 b(t)409 b(fail)e(to)h(output)i
(1)e(is)f(that)i(the)f Fo(x)18641 63355 y Fj(i)19017
63173 y Ft('s)f(migh)-34 b(t)409 b(not)f(all)g(b)34 b(e)407
b(distinct,)i(whic)-34 b(h)409 b(happ)34 b(ens)409 b(with)f(exp)34
b(onen)-34 b(tially)0 64678 y(small)502 b(probabilit)-34
b(y)-101 b(.\))836 b(On)503 b(the)g(other)g(hand,)528
b(the)504 b(securit)-34 b(y)502 b(of)h(the)h(MA)-34 b(C)503
b(implies)g(that)g Fo(D)536 b Ft(outputs)505 b(1)e(with)0
66184 y(negligible)423 b(probabilit)-34 b(y)424 b(on)g(input)g(\()p
Fo(C)17916 66372 y Fj(K)18828 66184 y Fo(;)202 b(S)20180
65744 y FG(hh)p Fj(C)21583 65893 y Fc(K)22365 65744 y
FG(ii)23152 66184 y Ft(\(1)24229 65744 y FG(j)p Fj(K)63
b FG(j)25664 66184 y Ft(\)\))424 b(\(o)-34 b(v)g(er)424
b(the)g(c)-34 b(hoice)423 b(of)h Fo(K)510 b Ft(and)425
b(the)e(coin)h(tosses)g(of)g(all)0 67689 y(algorithms\).)703
b(The)459 b(reason)f(is)h(that,)473 b(whenev)-34 b(er)459
b Fo(D)491 b Ft(outputs)461 b(1,)472 b(the)459 b(circuit)f(output)i(b)
-34 b(y)459 b Fo(S)529 b Ft(has)459 b(generated)g(a)0
69195 y(v)-67 b(alid)403 b(message-tag)i(pair)f(not)h(receiv)-34
b(ed)402 b(from)j(the)f Fp(hh)p Fo(C)25737 69383 y Fj(K)26650
69195 y Fp(ii)p Ft(-oracle.)19659 b Fb(2)25394 74521
y Ft(29)p eop
%%Page: 30 31
30 30 bop 1882 1218 a Ft(F)-101 b(or)460 b(sampling)g(obfuscators)i(in)
e(the)g(sense)h(of)f(De\014nition)h(6.1,)473 b(w)-34
b(e)461 b(do)f(not)h(kno)-34 b(w)461 b(ho)-34 b(w)461
b(to)g(pro)-34 b(v)g(e)460 b(imp)34 b(os-)0 2723 y(sibilit)-34
b(y)-101 b(.)591 b(In)-34 b(terestingly)-101 b(,)426
b(w)-34 b(e)422 b(can)g(sho)-34 b(w)423 b(that)g(they)e(imply)h(the)g
(non)-34 b(trivialit)g(y)422 b(of)g Fz(SZK)p Ft(,)k(the)c(class)f(of)h
(promise)0 4229 y(problems)404 b(p)34 b(ossessing)405
b(statistical)f(zero-kno)-34 b(wledge)404 b(pro)34 b(ofs.)0
6927 y Fz(Prop)39 b(osition)465 b(6.4)607 b Fr(If)433
b(e\016cient)f(sampling)g(obfusc)-62 b(ators)430 b(exist,)i(then)h
Fz(SZK)337 b Fp(6)p Ft(=)f Fz(BPP)o Fr(.)0 9624 y Fz(Pro)39
b(of:)1454 b Ft(It)399 b(is)g(kno)-34 b(wn)400 b(that)h(the)e(follo)-34
b(wing)400 b(promise)f(problem)g(\005)337 b(=)f(\(\005)34533
9812 y Fj(Y)35344 9624 y Fo(;)202 b Ft(\005)36792 9812
y Fj(N)37688 9624 y Ft(\))399 b(is)g(in)h Fz(SZK)e Ft([SV97)q(])h
(\(and)h(in)0 11130 y(fact)405 b(has)f(a)g(nonin)-34
b(teractiv)g(e)405 b(p)34 b(erfect)404 b(zero-kno)-34
b(wledge)404 b(pro)34 b(of)404 b(system)g([DDPY98,)g(GSV99)q(]\):)17324
13717 y(\005)18233 13905 y Fj(Y)20149 13717 y Ft(=)1107
b Fp(f)p Fo(C)424 b Ft(:)336 b Fp(hh)p Fo(C)87 b Fp(ii)338
b Ft(=)f Fo(U)30051 13899 y Fj(n)30677 13717 y Fp(g)17237
15555 y Ft(\005)18146 15743 y Fj(N)20149 15555 y Ft(=)1107
b Fp(f)p Fo(C)424 b Ft(:)336 b Fp(j)p Ft(Supp)r(\()p
Fo(C)87 b Ft(\))p Fp(j)337 b(\024)f Ft(2)32253 15054
y Fj(n=)p Fl(2)33820 15555 y Fp(g)p Fo(;)0 18142 y Ft(where)423
b Fo(n)g Ft(denotes)h(the)g(output)h(length)f(of)g(the)g(circuit)e
Fo(C)511 b Ft(and)424 b Fo(U)30781 18324 y Fj(n)31830
18142 y Ft(is)f(the)h(uniform)g(distribution)g(on)g Fp(f)p
Ft(0)p Fo(;)202 b Ft(1)p Fp(g)51037 17702 y Fj(n)51663
18142 y Ft(.)0 19647 y(No)-34 b(w)592 b(supp)34 b(ose)592
b(that)g(an)g(e\016cien)-34 b(t)591 b(sampling)h(obfuscator)g
Fp(O)625 b Ft(exists.)1100 b(Since,)637 b(analogous)592
b(to)g(Lemma)e(3.9,)0 21153 y(suc)-34 b(h)605 b(obfuscators)g(imply)e
(the)h(existence)f(of)h(one-w)-34 b(a)g(y)604 b(functions,)655
b(there)604 b(also)f(exists)h(a)f(length-doubling)0 22658
y(pseudorandom)315 b(generator)e Fo(G)g Ft([HILL99)n(].)508
b(Let)313 b Fo(G)23195 22840 y Fj(n)24158 22658 y Ft(:)336
b Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p Fp(g)27794 22218 y
Fj(n=)p Fl(2)29698 22658 y Fp(!)337 b(f)p Ft(0)p Fo(;)202
b Ft(1)p Fp(g)34210 22218 y Fj(n)35149 22658 y Ft(denote)314
b(the)f(circuit)f(that)i(ev)-67 b(aluates)0 24164 y Fo(G)404
b Ft(on)h(inputs)g(of)f(length)h Fo(n=)p Ft(2.)1882 25669
y(No)-34 b(w,)529 b(b)-34 b(y)504 b(the)g(de\014nition)h(of)g
(pseudorandom)g(generators)f(and)g(a)g(h)-34 b(ybrid)505
b(argumen)-34 b(t,)529 b(sampling)505 b(access)0 27175
y(to)466 b Fp(hh)p Fo(G)3438 27357 y Fj(n)4065 27175
y Fp(ii)h Ft(is)e(indistinguishable)i(from)f(sampling)h(access)e(to)h
Fo(U)30251 27357 y Fj(n)30877 27175 y Ft(.)724 b(Th)-34
b(us,)482 b(b)-34 b(y)466 b(the)h(de\014nition)f(of)h(a)e(sampling)0
28680 y(obfuscator,)562 b Fp(O)34 b Ft(\()p Fo(G)8819
28862 y Fj(n)9446 28680 y Ft(\))530 b(is)g(computationally)g
(indistinguishable)i(from)e Fp(O)34 b Ft(\()p Fo(U)35629
28862 y Fj(n)36255 28680 y Ft(\),)561 b(where)530 b(b)-34
b(y)530 b Fo(U)43892 28862 y Fj(n)45048 28680 y Ft(w)-34
b(e)530 b(mean)g(the)0 30186 y(trivial)369 b(circuit)h(that)i(samples)e
(uniformly)h(from)f Fo(U)23548 30368 y Fj(n)24174 30186
y Ft(.)527 b(By)370 b(functionalit)-34 b(y)-101 b(,)378
b Fp(O)34 b Ft(\()p Fo(U)36518 30368 y Fj(n)37144 30186
y Ft(\))371 b(is)f(alw)-34 b(a)g(ys)371 b(a)f Fq(yes)h
Ft(instance)g(of)0 31691 y(\005)404 b(and)h Fp(O)34 b
Ft(\()p Fo(G)6093 31873 y Fj(n)6719 31691 y Ft(\))405
b(is)f(alw)-34 b(a)g(ys)404 b(a)h Fq(no)f Ft(instance.)539
b(It)404 b(follo)-34 b(ws)405 b(that)g(\005)472 b Fo(=)-741
b Fp(2)337 b Fz(BPP)o Ft(.)p 51114 31691 886 886 v 0
34389 a Fz(Remark)464 b(6.5)606 b Ft(By)474 b(using)492
b Fq(St)-81 b(a)g(tistical)535 b(Difference)p Ft(,)494
b(the)475 b(complete)f(problem)h(for)f Fz(SZK)h Ft(from)f([SV97)q(],)0
35894 y(in)367 b(place)g(of)g(the)g(promise)g(problem)g(\005,)375
b(the)367 b(ab)34 b(o)-34 b(v)g(e)367 b(prop)34 b(osition)368
b(can)f(b)34 b(e)367 b(extended)g(to)h(the)f(natural)h(de\014nition)0
37400 y(of)451 b Fr(appr)-62 b(oximate)473 b(sampling)i(obfusc)-62
b(ators)p Ft(,)457 b(in)450 b(whic)-34 b(h)452 b Fp(O)34
b Ft(\()p Fo(C)87 b Ft(\))450 b(only)g(needs)h(to)g(sample)f(a)g
(distribution)i(of)e(small)0 38905 y(statistical)404
b(di\013erence)g(from)g(that)h(of)g Fo(C)87 b Ft(.)0
42703 y Fs(7)1793 b(W)-149 b(eak)-50 b(er)599 b(Notions)f(of)f
(Obfuscation)0 45409 y Ft(Our)350 b(imp)34 b(ossibilit)-34
b(y)350 b(results)h(rule)f(out)h(the)g(standard,)362
b(\\virtual)351 b(blac)-34 b(k)350 b(b)34 b(o)-34 b(x")350
b(notion)i(of)f(obfuscators)g(as)g(imp)34 b(os-)0 46914
y(sible,)407 b(along)h(with)g(sev)-34 b(eral)407 b(of)h(its)f
(applications.)549 b(Ho)-34 b(w)g(ev)g(er,)408 b(it)g(do)34
b(es)407 b(not)h(mean)g(that)g(there)g(is)f(no)h(metho)34
b(d)408 b(of)0 48420 y(making)387 b(programs)h(\\unin)-34
b(telligible")388 b(in)f(some)h(meaningful)g(and)g(precise)f(sense.)533
b(Suc)-34 b(h)388 b(a)g(metho)34 b(d)388 b(could)f(still)0
49925 y(pro)-34 b(v)g(e)415 b(useful)h(for)f(soft)-34
b(w)g(are)417 b(protection.)572 b(In)415 b(this)h(section,)h(w)-34
b(e)416 b(suggest)f(t)-34 b(w)g(o)417 b(w)-34 b(eak)g(er)415
b(de\014nitions)h(of)g(obfusca-)0 51431 y(tors)369 b(that)h(a)-34
b(v)g(oid)369 b(the)g(\\virtual)g(blac)-34 b(k)368 b(b)34
b(o)-34 b(x")369 b(paradigm)h(\(and)g(hence)e(are)g(not)i(ruled)f(out)g
(b)-34 b(y)369 b(our)g(imp)34 b(ossibilit)-34 b(y)0 52936
y(pro)34 b(of)94 b(\).)1882 54442 y(The)419 b(w)-34 b(eak)g(er)419
b(de\014nition)h(asks)e(that)i(if)f(t)-34 b(w)g(o)420
b(circuits)f(compute)g(the)g(same)g(function,)424 b(then)419
b(their)g(obfusca-)0 55947 y(tions)405 b(should)g(b)34
b(e)404 b(indistinguishable.)539 b(F)-101 b(or)405 b(simplicit)-34
b(y)-101 b(,)403 b(w)-34 b(e)404 b(only)g(consider)g(the)h(circuit)e(v)
-34 b(ersion)404 b(here.)0 58645 y Fz(De\014nition)466
b(7.1)f(\(indistinguishabilit)-39 b(y)468 b(obfuscator\))606
b Fr(A)-31 b(n)374 b Ft(indistinguishabilit)-34 b(y)340
b(obfuscator)375 b Fr(is)e(de\014ne)-62 b(d)0 60151 y(in)386
b(the)f(same)g(way)g(as)g(a)h(cir)-62 b(cuit)384 b(obfusc)-62
b(ator,)392 b(exc)-62 b(ept)384 b(that)g(the)h(\\virtual)e(black)h(b)
-62 b(ox")385 b(pr)-62 b(op)g(erty)384 b(is)h(r)-62 b(eplac)g(e)g(d)384
b(with)0 61656 y(the)432 b(fol)62 b(lowing:)1818 64045
y Fp(\017)606 b Fr(\(indistinguishability\))461 b(F)-93
b(or)466 b(any)g(PPT)i Fo(A)p Fr(,)474 b(ther)-62 b(e)465
b(is)h(a)h(ne)-62 b(gligible)464 b(function)h Fo(\013)471
b Fr(such)465 b(that)g(for)h(any)g(two)3030 65550 y(cir)-62
b(cuits)432 b Fo(C)8187 65732 y Fl(1)8713 65550 y Fo(;)202
b(C)10118 65732 y Fl(2)11077 65550 y Fr(which)433 b(c)-62
b(ompute)431 b(the)h(same)h(function)f(and)i(ar)-62 b(e)432
b(of)h(the)g(same)f(size)h Fo(k)38 b Fr(,)16905 68137
y Fp(j)p Ft(Pr)201 b([)p Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p
Fo(C)22796 68319 y Fl(1)23323 68137 y Ft(\)\)])269 b
Fp(\000)h Ft(Pr)201 b([)p Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p
Fo(C)31638 68319 y Fl(2)32165 68137 y Ft(\)\)])p Fp(j)336
b(\024)h Fo(\013)t Ft(\()p Fo(k)38 b Ft(\))p Fo(:)1882
71200 y Ft(Some)379 b(\(v)-34 b(ery)378 b(sligh)-34 b(t\))380
b(hop)34 b(e)379 b(that)h(this)f(de\014nition)h(is)e(ac)-34
b(hiev)-67 b(able)378 b(comes)h(from)g(the)g(follo)-34
b(wing)379 b(observ)-67 b(ation.)25394 74521 y(30)p eop
%%Page: 31 32
31 31 bop 0 1218 a Fz(Prop)39 b(osition)465 b(7.2)607
b Fr(\(Ine\016cient\))432 b(indistinguishability)c(obfusc)-62
b(ators)429 b(exist.)0 3778 y Fz(Pro)39 b(of:)1276 b
Ft(Let)336 b Fp(O)34 b Ft(\()p Fo(C)87 b Ft(\))336 b(b)34
b(e)335 b(the)h(lexicographically)e(\014rst)i(circuit)f(of)h(size)f
Fp(j)p Fo(C)87 b Fp(j)335 b Ft(that)i(computes)g(the)f(same)g(function)
0 5284 y(as)404 b Fo(C)87 b Ft(.)p 51114 5284 886 886
v 1882 7844 a(While)603 b(it)h(w)-34 b(ould)606 b(b)34
b(e)603 b(v)-34 b(ery)604 b(in)-34 b(teresting)604 b(to)h(construct)g
(ev)-34 b(en)603 b(indistinguishabilit)-34 b(y)606 b(obfuscators,)655
b(their)0 9350 y(usefulness)516 b(is)f(limited)g(b)-34
b(y)516 b(the)g(fact)g(that)g(they)g(pro)-34 b(vide)515
b(no)h Fr(a)535 b(priori)514 b Ft(guaran)-34 b(tees)516
b(ab)34 b(out)517 b(obfuscations)g(of)0 10855 y(circuits)452
b Fo(C)5167 11037 y Fl(1)6146 10855 y Ft(and)i Fo(C)9418
11037 y Fl(2)10397 10855 y Ft(that)g(compute)g(di\013eren)-34
b(t)453 b(functions.)687 b(Ho)-34 b(w)g(ev)g(er,)465
b(it)453 b(turns)h(out)g(that,)466 b(if)453 b Fp(O)487
b Ft(is)452 b(e\016cien)-34 b(t,)0 12361 y(then)532 b(it)g(is)f(\\comp)
34 b(etitiv)-34 b(e")531 b(with)h(resp)34 b(ect)531 b(to)g(an)-34
b(y)532 b(pair)f(of)h(circuits.)920 b(That)532 b(is,)563
b(no)531 b Fp(O)41588 11921 y FG(0)42430 12361 y Ft(mak)-34
b(es)531 b Fo(C)47066 12543 y Fl(1)48124 12361 y Ft(and)h
Fo(C)51474 12543 y Fl(2)0 13866 y Ft(m)-34 b(uc)g(h)455
b(more)e(indistinguishable)j(than)f Fp(O)487 b Ft(do)34
b(es.)688 b(T)-101 b(o)454 b(state)g(this)h(precisely)-101
b(,)464 b(for)454 b(a)g(circuit)f Fo(C)541 b Ft(of)454
b(size)f(at)h(most)0 15372 y Fo(k)38 b Ft(,)451 b(w)-34
b(e)441 b(de\014ne)h(P)-34 b(ad)8888 15569 y Fj(k)9457
15372 y Ft(\()p Fo(C)87 b Ft(\))442 b(to)g(b)34 b(e)441
b(a)g(trivial)g(padding)h(of)g Fo(C)528 b Ft(to)442 b(size)f
Fo(k)38 b Ft(.)650 b(F)-101 b(eeding)442 b(P)-34 b(ad)39265
15569 y Fj(k)39833 15372 y Ft(\()p Fo(C)87 b Ft(\))443
b(instead)f(of)f Fo(C)529 b Ft(to)442 b(an)0 16877 y(obfuscator)410
b(can)f(b)34 b(e)409 b(though)-34 b(t)411 b(of)f(as)f(increasing)g(the)
g(\\securit)-34 b(y)409 b(parameter")g(from)g Fp(j)p
Fo(C)87 b Fp(j)408 b Ft(to)i Fo(k)38 b Ft(.)553 b(\(W)-101
b(e)409 b(c)-34 b(hose)409 b(not)0 18382 y(to)501 b(explicitly)e(in)-34
b(tro)34 b(duce)502 b(a)e(securit)-34 b(y)501 b(parameter)g(in)-34
b(to)501 b(the)g(de\014nition)h(of)f(obfuscators)i(to)e(a)-34
b(v)g(oid)501 b(the)g(extra)0 19888 y(notation.\))587
b(F)-101 b(or)420 b(the)g(pro)34 b(of,)424 b(w)-34 b(e)420
b(also)f(need)h(to)g(imp)34 b(ose)420 b(a)f(tec)-34 b(hnical,)424
b(but)d(natural,)j(constrain)-34 b(t)421 b(that)g(the)f(size)0
21393 y(of)404 b Fp(O)2379 20954 y FG(0)2690 21393 y
Ft(\()p Fo(C)87 b Ft(\))405 b(only)e(dep)34 b(ends)405
b(on)g(the)f(size)g(of)g Fo(C)87 b Ft(.)0 23954 y Fz(Prop)39
b(osition)465 b(7.3)607 b Fr(Supp)-62 b(ose)343 b Fp(O)380
b Fr(is)346 b(an)g(e\016cient)e(indistinguishability)d(obfusc)-62
b(ator.)524 b(L)-62 b(et)346 b Fp(O)42706 23514 y FG(0)43362
23954 y Fr(b)-62 b(e)346 b(any)f(algorithm)0 25459 y(satisfying)543
b(the)h(syntactic)f(r)-62 b(e)g(quir)g(ements)544 b(of)g(obfusc)-62
b(ation,)570 b(also)544 b(satisfying)f(the)h(c)-62 b(ondition)544
b(that)g Fp(jO)47975 25020 y FG(0)48285 25459 y Ft(\()p
Fo(C)87 b Ft(\))p Fp(j)540 b Ft(=)0 26965 y Fo(q)43 b
Ft(\()p Fp(j)p Fo(C)87 b Fp(j)p Ft(\))470 b Fr(for)f(some)f(\014xe)-62
b(d)469 b(p)-62 b(olynomial)467 b Fo(q)43 b Fr(.)665
b(Then)470 b(for)e(any)h(PPT)h Fo(A)p Fr(,)478 b(ther)-62
b(e)468 b(exists)g(a)h(PPT)h Fo(A)42971 26525 y FG(0)43751
26965 y Fr(and)f(a)g(ne)-62 b(gligible)0 28470 y(function)432
b Fo(\013)439 b Fr(such)432 b(that)g(for)h(al)62 b(l)433
b(cir)-62 b(cuits)431 b Fo(C)20046 28652 y Fl(1)20572
28470 y Fr(,)i Fo(C)22243 28652 y Fl(2)23203 28470 y
Fr(of)g(size)f Fo(k)38 b Fr(,)10631 29890 y Ff(\014)10631
30617 y(\014)11035 30920 y Ft(Pr)12537 29938 y Ff(\002)13042
30920 y Fo(A)p Ft(\()p Fp(O)c Ft(\(P)-34 b(ad)17963 31167
y Fj(q)32 b Fl(\()p Fj(k)24 b Fl(\))19716 30920 y Ft(\()p
Fo(C)21053 31102 y Fl(1)21579 30920 y Ft(\)\))338 b(=)e(1)24744
29938 y Ff(\003)25519 30920 y Fp(\000)269 b Ft(Pr)28233
29938 y Ff(\002)28738 30920 y Fo(A)p Ft(\()p Fp(O)34
b Ft(\(P)-34 b(ad)33659 31167 y Fj(q)32 b Fl(\()p Fj(k)24
b Fl(\))35411 30920 y Ft(\()p Fo(C)36748 31102 y Fl(2)37275
30920 y Ft(\)\))337 b(=)g(1)40440 29938 y Ff(\003)40945
29890 y(\014)40945 30617 y(\014)11738 32758 y Fp(\024)13788
31727 y Ff(\014)13788 32455 y(\014)14192 32758 y Ft(Pr)15694
31776 y Ff(\002)16199 32758 y Fo(A)17108 32257 y FG(0)17418
32758 y Ft(\()p Fp(O)18888 32257 y FG(0)19199 32758 y
Ft(\()p Fo(C)20536 32940 y Fl(1)21063 32758 y Ft(\)\))g(=)f(1)24227
31776 y Ff(\003)25002 32758 y Fp(\000)269 b Ft(Pr)27716
31776 y Ff(\002)28221 32758 y Fo(A)29130 32257 y FG(0)29440
32758 y Ft(\()p Fp(O)30910 32257 y FG(0)31221 32758 y
Ft(\()p Fo(C)32558 32940 y Fl(2)33085 32758 y Ft(\)\))337
b(=)g(1)36250 31776 y Ff(\003)36755 31727 y(\014)36755
32455 y(\014)37428 32758 y Ft(+)269 b Fo(\013)t Ft(\()p
Fo(k)38 b Ft(\))p Fo(:)0 35771 y Fz(Pro)h(of:)1468 b
Ft(De\014ne)404 b Fo(A)9904 35331 y FG(0)10215 35771
y Ft(\()p Fo(C)87 b Ft(\))12380 35084 y Fl(def)12558
35771 y Ft(=)446 b Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(C)87
b Ft(\)\).)540 b(Then,)405 b(for)f(an)-34 b(y)405 b(circuit)e
Fo(C)31856 35953 y Fj(i)32636 35771 y Ft(of)i(size)e
Fo(k)38 b Ft(,)404 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)9898
37191 y Ff(\014)9898 37918 y(\014)10302 38221 y Ft(Pr)11803
37239 y Ff(\002)12308 38221 y Fo(A)p Ft(\()p Fp(O)34
b Ft(\(P)-34 b(ad)17230 38468 y Fj(q)32 b Fl(\()p Fj(k)24
b Fl(\))18982 38221 y Ft(\()p Fo(C)20319 38403 y Fj(i)20696
38221 y Ft(\)\)\))337 b(=)g(1)24332 37239 y Ff(\003)25106
38221 y Fp(\000)269 b Ft(Pr)27820 37239 y Ff(\002)28325
38221 y Fo(A)29234 37720 y FG(0)29545 38221 y Ft(\()p
Fp(O)31015 37720 y FG(0)31325 38221 y Ft(\()p Fo(C)32662
38403 y Fj(i)33039 38221 y Ft(\)\))337 b(=)g(1)36204
37239 y Ff(\003)36709 37191 y(\014)36709 37918 y(\014)11005
40058 y Ft(=)13054 39028 y Ff(\014)13054 39755 y(\014)13458
40058 y Ft(Pr)14960 39077 y Ff(\002)15465 40058 y Fo(A)p
Ft(\()p Fp(O)34 b Ft(\(P)-34 b(ad)20386 40305 y Fj(q)32
b Fl(\()p Fj(k)24 b Fl(\))22139 40058 y Ft(\()p Fo(C)23476
40240 y Fj(i)23852 40058 y Ft(\)\)\))338 b(=)f(1)27489
39077 y Ff(\003)28263 40058 y Fp(\000)269 b Ft(Pr)30977
39077 y Ff(\002)31482 40058 y Fo(A)p Ft(\()p Fp(O)34
b Ft(\()p Fp(O)35331 39558 y FG(0)35642 40058 y Ft(\()p
Fo(C)36979 40240 y Fj(i)37355 40058 y Ft(\)\)\))338 b(=)e(1)40991
39077 y Ff(\003)41496 39028 y(\014)41496 39755 y(\014)11005
41896 y Fp(\024)1106 b Ft(neg)17 b(\()p Fo(q)43 b Ft(\()p
Fo(k)38 b Ft(\)\))339 b(=)e(neg)17 b(\()p Fo(k)38 b Ft(\))p
Fo(;)0 44346 y Ft(where)377 b(the)g(inequalit)-34 b(y)377
b(is)f(b)34 b(ecause)377 b(P)-34 b(ad)18854 44592 y Fj(q)32
b Fl(\()p Fj(k)24 b Fl(\))20606 44346 y Ft(\()p Fo(C)21943
44528 y Fj(i)22320 44346 y Ft(\))377 b(and)g Fp(O)26496
43906 y FG(0)26807 44346 y Ft(\()p Fo(C)28144 44528 y
Fj(i)28520 44346 y Ft(\))g(are)f(t)-34 b(w)g(o)379 b(circuits)d(of)h
(size)f Fo(q)43 b Ft(\()p Fo(k)38 b Ft(\))378 b(whic)-34
b(h)378 b(compute)0 45851 y(the)405 b(same)f(function)h(and)g(b)34
b(ecause)404 b Fp(O)438 b Ft(is)403 b(an)i(indistinguishabilit)-34
b(y)405 b(obfuscator.)540 b(Th)-34 b(us,)7640 47271 y
Ff(\014)7640 47998 y(\014)8044 48301 y Ft(Pr)9545 47319
y Ff(\002)10050 48301 y Fo(A)p Ft(\()p Fp(O)34 b Ft(\(P)-34
b(ad)14972 48548 y Fj(q)32 b Fl(\()p Fj(k)24 b Fl(\))16724
48301 y Ft(\()p Fo(C)18061 48483 y Fl(1)18588 48301 y
Ft(\)\))337 b(=)g(1)21753 47319 y Ff(\003)22527 48301
y Fp(\000)269 b Ft(Pr)25241 47319 y Ff(\002)25746 48301
y Fo(A)p Ft(\()p Fp(O)34 b Ft(\(P)-34 b(ad)30667 48548
y Fj(q)32 b Fl(\()p Fj(k)24 b Fl(\))32420 48301 y Ft(\()p
Fo(C)33757 48483 y Fl(2)34283 48301 y Ft(\)\)\))338 b(=)e(1)37919
47319 y Ff(\003)38425 47271 y(\014)38425 47998 y(\014)8746
50139 y Fp(\024)10796 49108 y Ff(\014)10796 49836 y(\014)11200
50139 y Ft(Pr)12702 49157 y Ff(\002)13207 50139 y Fo(A)p
Ft(\()p Fp(O)34 b Ft(\(P)-34 b(ad)18128 50385 y Fj(q)32
b Fl(\()p Fj(k)24 b Fl(\))19881 50139 y Ft(\()p Fo(C)21218
50321 y Fl(1)21744 50139 y Ft(\)\))338 b(=)e(1)24909
49157 y Ff(\003)25684 50139 y Fp(\000)269 b Ft(Pr)28397
49157 y Ff(\002)28903 50139 y Fo(A)29812 49638 y FG(0)30122
50139 y Ft(\()p Fp(O)31592 49638 y FG(0)31903 50139 y
Ft(\()p Fo(C)33240 50321 y Fl(1)33766 50139 y Ft(\)\))338
b(=)e(1)36931 49157 y Ff(\003)37436 49108 y(\014)37436
49836 y(\014)10796 51976 y Ft(+)11941 50946 y Ff(\014)11941
51673 y(\014)12345 51976 y Ft(Pr)13847 50994 y Ff(\002)14352
51976 y Fo(A)15261 51476 y FG(0)15571 51976 y Ft(\()p
Fp(O)17041 51476 y FG(0)17352 51976 y Ft(\()p Fo(C)18689
52158 y Fl(1)19216 51976 y Ft(\)\))h(=)f(1)22380 50994
y Ff(\003)23155 51976 y Fp(\000)269 b Ft(Pr)25869 50994
y Ff(\002)26374 51976 y Fo(A)27283 51476 y FG(0)27593
51976 y Ft(\()p Fp(O)29063 51476 y FG(0)29374 51976 y
Ft(\()p Fo(C)30711 52158 y Fl(2)31238 51976 y Ft(\)\))337
b(=)f(1)34402 50994 y Ff(\003)34908 50946 y(\014)34908
51673 y(\014)10796 53814 y Ft(+)11941 52783 y Ff(\014)11941
53511 y(\014)12345 53814 y Ft(Pr)13847 52832 y Ff(\002)14352
53814 y Fo(A)15261 53313 y FG(0)15571 53814 y Ft(\()p
Fp(O)17041 53313 y FG(0)17352 53814 y Ft(\()p Fo(C)18689
53996 y Fl(2)19216 53814 y Ft(\)\))h(=)f(1)22380 52832
y Ff(\003)23155 53814 y Fp(\000)269 b Ft(Pr)25869 52832
y Ff(\002)26374 53814 y Fo(A)p Ft(\()p Fp(O)34 b Ft(\(P)-34
b(ad)31295 54060 y Fj(q)32 b Fl(\()p Fj(k)24 b Fl(\))33048
53814 y Ft(\()p Fo(C)34385 53996 y Fl(2)34911 53814 y
Ft(\)\)\))338 b(=)e(1)38547 52832 y Ff(\003)39053 52783
y(\014)39053 53511 y(\014)8746 55651 y Fp(\024)1107 b
Ft(neg)17 b(\()p Fo(k)38 b Ft(\))271 b(+)15725 54621
y Ff(\014)15725 55348 y(\014)16129 55651 y Ft(Pr)17630
54669 y Ff(\002)18135 55651 y Fo(A)19044 55151 y FG(0)19355
55651 y Ft(\()p Fp(O)20825 55151 y FG(0)21136 55651 y
Ft(\()p Fo(C)22473 55833 y Fl(1)22999 55651 y Ft(\)\))338
b(=)e(1)26164 54669 y Ff(\003)26939 55651 y Fp(\000)269
b Ft(Pr)29652 54669 y Ff(\002)30157 55651 y Fo(A)31066
55151 y FG(0)31377 55651 y Ft(\()p Fp(O)32847 55151 y
FG(0)33158 55651 y Ft(\()p Fo(C)34495 55833 y Fl(2)35021
55651 y Ft(\)\))338 b(=)e(1)38186 54669 y Ff(\003)38691
54621 y(\014)38691 55348 y(\014)39365 55651 y Ft(+)269
b(neg)17 b(\()p Fo(k)38 b Ft(\))p Fo(:)p 51114 58101
V 0 60662 a Ft(Ev)-34 b(en)366 b(with)g(the)h(comp)34
b(etitiv)-34 b(eness)365 b(prop)34 b(ert)-34 b(y)-101
b(,)374 b(it)365 b(still)h(seems)f(imp)34 b(ortan)-34
b(t)367 b(to)f(ha)-34 b(v)g(e)367 b(explicit)e(guaran)-34
b(tees)366 b(on)g(the)0 62167 y(b)34 b(eha)-34 b(vior)468
b(of)g(an)h(obfuscator)g(on)g(circuits)e(that)i(compute)g(di\013eren)
-34 b(t)469 b(functions.)731 b(W)-101 b(e)468 b(no)-34
b(w)469 b(giv)-34 b(e)468 b(a)g(de\014nition)0 63673
y(that)509 b(pro)-34 b(vides)508 b(suc)-34 b(h)508 b(a)g(guaran)-34
b(tee,)534 b(while)507 b(still)h(a)-34 b(v)g(oiding)508
b(the)g(\\virtual)f(blac)-34 b(k)508 b(b)34 b(o)-34 b(x")508
b(paradigm.)849 b(Roughly)0 65178 y(sp)34 b(eaking,)410
b(it)g(sa)-34 b(ys)410 b(that)h(if)e(it)h(is)f(p)34 b(ossible)409
b(to)h(distinguish)h(the)f(obfuscations)i(of)d(a)h(pair)f(of)h
(circuits,)g(then)h(one)0 66684 y(can)428 b(\014nd)h(inputs)g(on)g
(whic)-34 b(h)428 b(they)g(di\013er)g(giv)-34 b(en)428
b(an)-34 b(y)428 b(pair)g(of)g(circuits)f(whic)-34 b(h)429
b(compute)g(the)f(same)g(functions.)0 68189 y(This)411
b(de\014nition)h(is)f(motiv)-67 b(ated)410 b(in)h(part)g(b)-34
b(y)412 b(our)e(imp)34 b(ossibilit)-34 b(y)411 b(pro)34
b(of,)412 b(in)f(whic)-34 b(h)412 b(the)f(functions)h(constructed)0
69694 y(are)424 b(suc)-34 b(h)425 b(that)g(it)g(is)f(p)34
b(ossible)424 b(to)h(e\016cien)-34 b(tly)423 b(\014nd)j(inputs)f(on)g
(whic)-34 b(h)425 b(they)g(di\013er)f(giv)-34 b(en)424
b(an)-34 b(y)425 b(pair)f(of)h(circuits)0 71200 y(computing)405
b(them.)25394 74521 y(31)p eop
%%Page: 32 33
32 32 bop 0 1218 a Fz(De\014nition)466 b(7.4)f(\(di\013ering-inputs)i
(obfuscator\))606 b Fr(A)-31 b(n)534 b Ft(di\013ering-inputs)515
b(obfuscator)535 b Fr(is)e(de\014ne)-62 b(d)533 b(in)h(the)0
2723 y(same)553 b(way)h(as)f(an)i(indistinguishability)548
b(obfusc)-62 b(ator,)581 b(exc)-62 b(ept)552 b(that)h(the)g
(\\indistinguishability")548 b(pr)-62 b(op)g(erty)552
b(is)0 4229 y(r)-62 b(eplac)g(e)g(d)431 b(with)i(the)f(fol)62
b(lowing:)1818 6600 y Fp(\017)606 b Fr(\(di\013ering-inputs)437
b(pr)-62 b(op)g(erty\))438 b(F)-93 b(or)440 b(any)g(PPT)h
Fo(A)p Fr(,)h(ther)-62 b(e)439 b(is)g(a)h(pr)-62 b(ob)g(abilistic)436
b(algorithm)i Fo(A)44979 6161 y FG(0)45730 6600 y Fr(and)h(a)h(ne)-62
b(gli-)3030 8106 y(gible)434 b(function)g Fo(\013)441
b Fr(such)434 b(that)g(the)g(fol)62 b(lowing)434 b(holds.)562
b(Supp)-62 b(ose)433 b Fo(C)33289 8288 y Fl(1)34251 8106
y Fr(and)i Fo(C)37473 8288 y Fl(2)38435 8106 y Fr(ar)-62
b(e)435 b(cir)-62 b(cuits)433 b(of)i(size)f Fo(k)474
b Fr(such)3030 9611 y(that)13481 11117 y Fo(")14316 10430
y Fl(def)14494 11117 y Ft(=)447 b Fp(j)p Ft(Pr)201 b([)p
Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(C)21775 11299 y Fl(1)22302
11117 y Ft(\)\))337 b(=)f(1])270 b Fp(\000)f Ft(Pr)201
b([)p Fo(A)p Ft(\()p Fp(O)34 b Ft(\()p Fo(C)32839 11299
y Fl(2)33366 11117 y Ft(\)\))337 b(=)g(1])o Fp(j)g Fo(>)f(\013)t
Ft(\()p Fo(k)38 b Ft(\))p Fo(:)3030 13264 y Fr(Then,)502
b(for)488 b(any)g Fo(C)11952 12824 y FG(0)11865 13589
y Fl(1)12391 13264 y Fo(;)202 b(C)13883 12824 y FG(0)13796
13589 y Fl(2)14810 13264 y Fr(of)488 b(size)f Fo(k)527
b Fr(such)487 b(that)f Fo(C)26154 12824 y FG(0)26067
13603 y Fj(i)26954 13264 y Fr(c)-62 b(omputes)485 b(the)j(same)f
(function)g(as)h Fo(C)44790 13446 y Fj(i)45654 13264
y Fr(for)g Fo(i)436 b Ft(=)f(1)p Fo(;)202 b Ft(2)p Fr(,)3030
14769 y Fo(A)3939 14330 y FG(0)4250 14769 y Ft(\()p Fo(C)5674
14330 y FG(0)5587 15094 y Fl(1)6113 14769 y Fo(;)g(C)7605
14330 y FG(0)7518 15094 y Fl(2)8044 14769 y Ft(\))434
b Fr(outputs)d(an)j(input)e(on)i(which)e Fo(C)24126 14951
y Fl(1)25086 14769 y Fr(and)h Fo(C)28306 14951 y Fl(2)29266
14769 y Fr(di\013er)g(in)h(time)f Ft(p)34 b(oly)16 b(\()p
Fo(k)38 b(;)202 b Ft(1)p Fo(=)p Ft(\()p Fo(")271 b Fp(\000)e
Fo(\013)t Ft(\()p Fo(k)38 b Ft(\)\)\))p Fr(.)1882 17447
y Ft(This)496 b(de\014nition)h(is)e(indeed)h(stronger)f(than)i(that)g
(of)f(indistinguishabilit)-34 b(y)497 b(obfuscators,)519
b(b)34 b(ecause)495 b(if)h Fo(C)51474 17629 y Fl(1)0
18953 y Ft(and)360 b Fo(C)3178 19135 y Fl(2)4064 18953
y Ft(compute)g(the)g(same)g(function,)369 b(then)360
b Fo(A)22644 18513 y FG(0)23315 18953 y Ft(can)f(nev)-34
b(er)359 b(\014nd)i(an)f(input)h(on)e(whic)-34 b(h)361
b(they)f(di\013er)f(and)h(hence)0 20458 y Fo(")404 b
Ft(m)-34 b(ust)406 b(b)34 b(e)403 b(negligible.)0 24253
y Fs(8)1793 b(W)-149 b(atermarking)599 b(and)f(Obfuscation)0
26959 y Ft(Generally)506 b(sp)34 b(eaking,)532 b Fr(\(fr)-62
b(agile\))526 b(watermarking)505 b Ft(is)i(the)g(problem)g(of)h(em)-34
b(b)34 b(edding)508 b(a)f(message)g(in)g(an)g(ob)67 b(ject)0
28464 y(suc)-34 b(h)499 b(that)g(the)f(message)g(is)f(di\016cult)h(to)h
(remo)-34 b(v)g(e)497 b(without)j(\\ruining")e(the)g(ob)67
b(ject.)821 b(Most)498 b(of)g(the)g(w)-34 b(ork)499 b(on)0
29970 y(w)-34 b(atermarking)381 b(has)h(fo)34 b(cused)382
b(on)f(w)-34 b(atermarking)382 b Fr(p)-62 b(er)g(c)g(eptual)377
b Ft(ob)67 b(jects,)387 b Fr(e.g.,)410 b Ft(images)381
b(or)g(audio)h(\014les.)530 b(\(See)381 b(the)0 31475
y(surv)-34 b(eys)330 b([MMS)7451 31035 y Fl(+)8239 31475
y Ft(98,)g(P)-101 b(AK99].\))514 b(Here)330 b(w)-34 b(e)331
b(concen)-34 b(trate)331 b(on)g(w)-34 b(atermarking)330
b Fr(pr)-62 b(o)g(gr)g(ams)p Ft(,)344 b(as)330 b(in)h([CT00,)g(NSS99)q
(].)0 32981 y(A)433 b(w)-34 b(atermarking)433 b(sc)-34
b(heme)432 b(should)i(consist)f(of)g(a)f Fr(marking)g
Ft(algorithm)g(whic)-34 b(h)434 b(em)-34 b(b)34 b(eds)433
b(a)f(message)h Fo(m)g Ft(in)-34 b(to)433 b(a)0 34486
y(giv)-34 b(en)452 b(program,)464 b(and)452 b(an)h Fr(extr)-62
b(action)450 b Ft(algorithm)i(whic)-34 b(h)453 b(extracts)f(the)g
(message)g(from)g(a)g(mark)-34 b(ed)452 b(program.)0
35991 y(In)-34 b(tuitiv)g(ely)-101 b(,)404 b(the)g(follo)-34
b(wing)405 b(prop)34 b(erties)404 b(should)h(b)34 b(e)404
b(satis\014ed:)1818 38363 y Fp(\017)606 b Ft(\(functionalit)-34
b(y\))406 b(The)f(mark)-34 b(ed)404 b(program)g(computes)h(the)g(same)f
(function)h(as)g(the)f(original)g(program.)1818 40813
y Fp(\017)606 b Ft(\(meaningfulness\))406 b(Most)f(programs)f(are)g
(unmark)-34 b(ed.)1818 43263 y Fp(\017)606 b Ft(\(fragilit)-34
b(y\))322 b(It)f(is)g(infeasible)g(to)h(remo)-34 b(v)g(e)321
b(the)g(mark)g(from)h(the)f(program)h(without)h(\(substan)-34
b(tially\))323 b(c)-34 b(hang-)3030 44768 y(ing)405 b(its)f(b)34
b(eha)-34 b(vior.)1882 47140 y(There)363 b(are)f(a)h(v)-67
b(arious)362 b(heuristic)g(metho)34 b(ds)364 b(for)f(soft)-34
b(w)g(are)364 b(w)-34 b(atermarking)363 b(in)g(the)g(literature)g(\()p
Fr(cf.,)395 b Ft([CT00]\),)0 48646 y(but)290 b(as)f(with)h
(obfuscation,)313 b(there)289 b(has)g(b)34 b(een)289
b(little)f(rigorous)h(w)-34 b(ork)289 b(on)h(this)f(problem.)500
b(Here)288 b(w)-34 b(e)290 b(do)f(not)h(attempt)0 50151
y(to)321 b(pro)-34 b(vide)320 b(a)g(thorough)i(de\014nitional)f
(treatmen)-34 b(t)321 b(of)g(soft)-34 b(w)g(are)321 b(w)-34
b(atermarking,)337 b(but)322 b(rather)e(consider)g(a)g(couple)0
51657 y(of)415 b(w)-34 b(eak)415 b(formalizations)g(whic)-34
b(h)416 b(w)-34 b(e)415 b(relate)f(to)h(our)g(results)g(on)h
(obfuscation.)572 b(The)415 b(di\016cult)-34 b(y)415
b(in)g(formalizing)0 53162 y(w)-34 b(atermarking)412
b(comes,)g(of)f(course,)i(in)e(capturing)h(the)f(fragilit)-34
b(y)411 b(prop)34 b(ert)-34 b(y)-101 b(.)560 b(As)411
b(with)h(obfuscation,)i(it)d(is)g(easy)0 54668 y(to)396
b(remo)-34 b(v)g(e)396 b(a)f(w)-34 b(atermark)397 b(from)f(programs)g
(for)g(functions)h(that)g(are)e(\(exactly\))h(learnable)f(with)i(mem)
-34 b(b)34 b(ership)0 56173 y(queries)574 b(\(b)-34 b(y)576
b(using)f(the)h(learning)e(algorithm)h(to)h(generate)e(a)h(new)h
(program)f(\(for)g(the)g(function\))i(that)f(is)0 57678
y(indep)34 b(enden)-34 b(t)421 b(of)e(the)h(marking\).)583
b(A)419 b(natural)h(question)f(is)g(whether)h(learnable)e(functions)j
(are)d(the)i(only)f(ones)0 59184 y(that)405 b(cause)f(problems.)539
b(That)405 b(is,)f(can)g(the)h(follo)-34 b(wing)404 b(de\014nition)i(b)
34 b(e)404 b(satis\014ed?)0 61862 y Fz(De\014nition)466
b(8.1)f(\(soft)-39 b(w)g(are)464 b(w)-39 b(atermarking\))606
b Fr(A)309 b Ft(\(soft)-34 b(w)g(are\))272 b(w)-34 b(atermarking)270
b(sc)-34 b(heme)310 b Fr(is)f(a)g(p)-62 b(air)309 b(of)g(\(keye)-62
b(d\))0 63367 y(pr)g(ob)g(abilistic)429 b(algorithms)i
Ft(\(Mark)p Fo(;)202 b Ft(Extract\))434 b Fr(satisfying)d(the)h(fol)62
b(lowing)432 b(pr)-62 b(op)g(erties:)1818 65739 y Fp(\017)606
b Fr(\(functionality\))363 b(F)-93 b(or)366 b(every)f(cir)-62
b(cuit)364 b Fo(C)87 b Fr(,)380 b(key)365 b Fo(K)87 b
Fr(,)380 b(and)366 b(message)e Fo(m)p Fr(,)379 b(the)366
b(string)f Ft(Mark)42629 65927 y Fj(K)43540 65739 y Ft(\()p
Fo(C)19 b(;)202 b(m)p Ft(\))368 b Fr(describ)-62 b(es)3030
67245 y(a)434 b(cir)-62 b(cuit)431 b(that)h(c)-62 b(omputes)431
b(the)i(same)f(function)h(as)f Fo(C)87 b Fr(.)1818 69694
y Fp(\017)606 b Fr(\(p)-62 b(olynomial)310 b(slowdown\))h(Ther)-62
b(e)311 b(is)g(a)h(p)-62 b(olynomial)310 b Fo(p)i Fr(such)e(that)h(for)
g(every)f(cir)-62 b(cuit)310 b Fo(C)87 b Fr(,)337 b Fp(j)p
Ft(Mark)46041 69882 y Fj(K)46952 69694 y Ft(\()p Fo(C)19
b(;)202 b(m)p Ft(\))p Fp(j)338 b(\024)3030 71200 y Fo(p)p
Ft(\()p Fp(j)p Fo(C)87 b Fp(j)269 b Ft(+)g Fp(j)p Fo(m)p
Fp(j)g Ft(+)h Fp(j)p Fo(K)87 b Fp(j)p Ft(\))p Fr(.)25394
74521 y Ft(32)p eop
%%Page: 33 34
33 33 bop 1818 1218 a Fp(\017)606 b Fr(\(extr)-62 b(action\))431
b(F)-93 b(or)434 b(every)e(cir)-62 b(cuit)432 b Fo(C)87
b Fr(,)433 b(key)f Fo(K)87 b Fr(,)434 b(and)f(message)f
Fo(m)p Fr(,)h Ft(Extract)37719 1406 y Fj(K)38631 1218
y Ft(\(Mark)41934 1406 y Fj(K)42845 1218 y Ft(\()p Fo(C)19
b(;)202 b(m)p Ft(\)\))338 b(=)f Fo(m)p Fr(.)1818 3683
y Fp(\017)606 b Fr(\(me)-62 b(aningfulness\))431 b(F)-93
b(or)434 b(every)e(cir)-62 b(cuit)431 b Fo(C)87 b Fr(,)434
b Ft(Pr)24737 3871 y Fj(K)25851 3683 y Ft([Extract)30214
3871 y Fj(K)31126 3683 y Ft(\()p Fo(C)87 b Ft(\))337
b Fp(6)p Ft(=)g Fp(?)p Ft(])f(=)g(neg)17 b(\()p Fp(j)p
Fo(C)87 b Fp(j)p Ft(\))p Fr(.)1818 6148 y Fp(\017)606
b Fr(\(fr)-62 b(agility\))431 b(F)-93 b(or)433 b(every)f(PPT)j
Fo(A)p Fr(,)e(ther)-62 b(e)432 b(is)h(a)h(PPT)g Fo(S)504
b Fr(such)432 b(that)g(for)h(every)f Fo(C)520 b Fr(and)434
b Fo(m)14382 8761 y Ft(Pr)14604 9568 y Fj(K)15884 7779
y Ff(\002)16389 8761 y Fo(A)p Ft(\(Mark)20601 8949 y
Fj(K)21512 8761 y Ft(\()p Fo(C)19 b(;)202 b(m)p Ft(\)\))339
b(=)d Fo(C)27984 8261 y FG(0)28728 8761 y Fr(s.t.)557
b Fo(C)31881 8261 y FG(0)32528 8761 y Fp(\021)336 b Fo(C)521
b Fr(and)433 b Ft(Extract)41575 8949 y Fj(K)42487 8761
y Ft(\()p Fo(C)43911 8261 y FG(0)44222 8761 y Ft(\))337
b Fp(6)p Ft(=)f Fo(m)47373 7779 y Ff(\003)15489 11526
y Fp(\024)1107 b Ft(Pr)19040 10181 y Ff(h)19613 11526
y Fo(S)20426 11026 y Fj(C)21214 11526 y Ft(\(1)22291
11026 y FG(j)p Fj(C)63 b FG(j)23603 11526 y Ft(\))337
b(=)g Fo(C)26644 11026 y FG(0)27388 11526 y Fr(s.t.)556
b Fo(C)30540 11026 y FG(0)31187 11526 y Fp(\021)337 b
Fo(C)33420 10181 y Ff(i)34262 11526 y Ft(+)269 b(neg)17
b(\()p Fp(j)p Fo(C)87 b Fp(j)p Ft(\))p Fo(;)3030 14637
y Fr(wher)-62 b(e)400 b Fo(K)488 b Fr(is)400 b(uniformly)f(sele)-62
b(cte)g(d)397 b(in)k Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p
Fp(g)23452 14197 y Fl(max\()p FG(j)p Fj(C)63 b FG(j)p
Fj(;)p FG(j)p Fj(m)p FG(j)p Fl(\))28862 14637 y Fr(,)407
b(and)400 b Fo(C)32915 14197 y FG(0)33563 14637 y Fp(\021)336
b Fo(C)488 b Fr(me)-62 b(ans)400 b(that)f Fo(C)43278
14197 y FG(0)43989 14637 y Fr(and)h Fo(C)487 b Fr(c)-62
b(ompute)3030 16142 y(the)433 b(same)f(function.)0 18552
y(We)h(say)f(that)g(the)h(scheme)f(is)h Ft(e\016cien)-34
b(t)433 b Fr(if)h Ft(Mark)f Fr(and)g Ft(Extract)h Fr(run)f(in)h(p)-62
b(olynomial)432 b(time.)1882 21276 y Ft(By)368 b(using)i(our)f
(construction)h(of)g(totally)f(unobfuscatable)i(functions,)378
b(w)-34 b(e)369 b(can)g(pro)-34 b(v)g(e)370 b(that)g(this)g
(de\014nition)0 22782 y(is)404 b(imp)34 b(ossible)404
b(to)g(meet.)0 25506 y Fz(Theorem)464 b(8.2)606 b Fr(If)530
b(one-way)d(functions)h(exist,)552 b(then)528 b(no)h(watermarking)f
(scheme)f(in)j(the)d(sense)i(of)f(De\014ni-)0 27011 y(tion)433
b(8.1)g(exists.)0 29735 y Fz(Pro)39 b(of)637 b(Sk)-39
b(etc)g(h:)2116 b Ft(Consider)555 b(the)g(totally)g(unobfuscatable)i
(function)f(ensem)-34 b(ble)554 b(guaran)-34 b(teed)556
b(b)-34 b(y)555 b(Theo-)0 31240 y(rem)369 b(4.2.)527
b(No)370 b(matter)f(ho)-34 b(w)371 b(w)-34 b(e)370 b(try)g(to)g(pro)34
b(duce)369 b(a)h(mark)-34 b(ed)370 b(circuit)e(from)i
Fo(f)35919 30553 y Fw(R)35607 31240 y Fp( )200 b(H)12
b Ft(,)376 b(the)370 b(algorithm)f Fo(A)h Ft(giv)-34
b(en)370 b(b)-34 b(y)0 32746 y(the)411 b(unobfuscatabilit)-34
b(y)414 b(condition)d(in)g(De\014nition)h(4.2)f(can)g(reconstruct)g
(the)g(canonical)g(circuit)f Fo(f)130 b Ft(,)414 b(whic)-34
b(h)412 b(b)-34 b(y)0 34251 y(the)374 b(meaningfulness)h(prop)34
b(ert)-34 b(y)374 b(is)f(unmark)-34 b(ed)374 b(with)h(high)f
(probabilit)-34 b(y)-101 b(.)529 b(On)373 b(the)h(other)g(hand,)381
b(the)374 b(sim)-34 b(ulator,)0 35757 y(giv)g(en)339
b(just)i(oracle)d(access)h(to)g Fo(f)130 b Ft(,)353 b(will)339
b(b)34 b(e)339 b(unable)h(pro)34 b(duce)339 b(an)-34
b(y)340 b(circuit)f(computing)h(the)g(same)f(function)i(\(since)0
37262 y(if)404 b(it)g(could,)g(then)h(it)f(could)h(compute)f
Fo(\031)43 b Ft(\()p Fo(f)130 b Ft(\),)406 b(whic)-34
b(h)405 b(is)f(pseudorandom\).)16759 b Fb(2)0 39986 y
Fz(Corollary)464 b(8.3)606 b Ft(E\016cien)-34 b(t)408
b Fr(watermarking)f(schemes)e(in)j(the)f(sense)g(of)g(De\014nition)g
(8.1)g(do)g(not)g(exist)g(\(unc)-62 b(on-)0 41492 y(ditional)62
b(ly\).)1882 44216 y Ft(Giv)-34 b(en)518 b(these)g(imp)34
b(ossibilit)-34 b(y)518 b(results,)546 b(w)-34 b(e)518
b(are)f(led)h(to)g(seek)f(the)i(w)-34 b(eak)g(est)518
b(p)34 b(ossible)518 b(form)-34 b(ulation)519 b(of)f(the)0
45721 y(fragilit)-34 b(y)404 b(condition)h(|)f(that)h(the)g(an)-34
b(y)404 b(adv)-34 b(ersary)404 b Fr(o)-62 b(c)g(c)g(asional)62
b(ly)400 b Ft(fails)k(to)h(remo)-34 b(v)g(e)403 b(the)i(mark.)0
48445 y Fz(De\014nition)466 b(8.4)f(\(o)39 b(ccasional)463
b(w)-39 b(atermarking\))606 b Fr(A)-31 b(n)313 b Ft(o)34
b(ccasional)313 b Fr(softwar)-62 b(e)311 b(watermarking)h(scheme)g(is)g
(de-)0 49950 y(\014ne)-62 b(d)579 b(in)f(the)g(same)f(way)h(as)g
(De\014nition)g(8.1,)614 b(exc)-62 b(ept)576 b(that)h(the)h(fr)-62
b(agility)576 b(c)-62 b(ondition)577 b(is)g(r)-62 b(eplac)g(e)g(d)576
b(with)i(the)0 51456 y(fol)62 b(lowing:)1818 53866 y
Fp(\017)606 b Fr(F)-93 b(or)434 b(every)e(PPT)i Fo(A)p
Fr(,)g(ther)-62 b(e)432 b(exists)g(a)h(cir)-62 b(cuit)432
b Fo(C)521 b Fr(and)433 b(a)g(message)e Fo(m)j Fr(such)e(that)5703
56479 y Ft(Pr)5925 57286 y Fj(K)7204 55498 y Ff(\002)7710
56479 y Fo(A)p Ft(\(Mark)11922 56667 y Fj(K)12833 56479
y Ft(\()p Fo(C)19 b(;)202 b(m)p Ft(\)\))339 b(=)d Fo(C)19305
55979 y FG(0)20049 56479 y Fr(s.t.)557 b Fo(C)23202 55979
y FG(0)23849 56479 y Fp(\021)336 b Fo(C)521 b Fr(and)433
b Ft(Extract)32896 56667 y Fj(K)33808 56479 y Ft(\()p
Fo(C)35232 55979 y FG(0)35543 56479 y Ft(\))337 b Fp(6)p
Ft(=)f Fo(m)38694 55498 y Ff(\003)39536 56479 y Fp(\024)h
Ft(1)269 b Fp(\000)g Ft(1)p Fo(=)p Ft(p)34 b(oly)17 b(\()p
Fp(j)p Fo(C)87 b Fp(j)p Ft(\))p Fo(;)3030 59730 y Fr(wher)-62
b(e)433 b Fo(K)521 b Fr(is)433 b(uniformly)f(sele)-62
b(cte)g(d)431 b(in)j Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p
Fp(g)23651 59290 y Fl(max\()p FG(j)p Fj(C)63 b FG(j)p
Fj(;)p FG(j)p Fj(m)p FG(j)p Fl(\))29061 59730 y Fr(.)1882
62454 y Ft(In)-34 b(terestingly)-101 b(,)341 b(in)326
b(con)-34 b(trast)327 b(to)f(the)g(usual)h(in)-34 b(tuition,)342
b(this)327 b(w)-34 b(eak)326 b(notion)h(of)f(w)-34 b(atermarking)326
b(is)g(inconsisten)-34 b(t)0 63960 y(with)405 b(obfuscation)h(\(ev)-34
b(en)404 b(the)h(w)-34 b(eak)g(est)404 b(notion)h(w)-34
b(e)405 b(prop)34 b(osed)405 b(in)f(Section)g(7\).)0
66684 y Fz(Prop)39 b(osition)465 b(8.5)607 b Fr(Oc)-62
b(c)g(asional)428 b(softwar)-62 b(e)429 b(watermarking)g(schemes)g(and)
h(e\016cient)f(indistinguishability)c(ob-)0 68189 y(fusc)-62
b(ators)462 b(\(as)i(in)h(De\014nition)g(7.1\))e(c)-62
b(annot)464 b(b)-62 b(oth)464 b(exist.)650 b(\(A)-62
b(ctual)62 b(ly,)469 b(we)c(r)-62 b(e)g(quir)g(e)464
b(the)f(watermarking)h(scheme)0 69694 y(to)474 b(satisfy)d(the)j
(additional)d(natur)-62 b(al)473 b(c)-62 b(ondition)473
b(that)f Fp(j)p Ft(Mark)28312 69882 y Fj(K)29223 69694
y Ft(\()p Fo(C)19 b(;)202 b(m)p Ft(\))p Fp(j)412 b Ft(=)e
Fo(q)43 b Ft(\()p Fp(j)p Fo(C)87 b Fp(j)p Ft(\))475 b
Fr(for)f(some)f(\014xe)-62 b(d)474 b(p)-62 b(olynomial)0
71200 y Fo(q)477 b Fr(and)433 b(al)62 b(l)433 b Fp(j)p
Fo(C)87 b Fp(j)336 b Ft(=)h Fp(j)p Fo(m)p Fp(j)f Ft(=)h
Fp(j)p Fo(K)87 b Fp(j)p Fr(.\))25394 74521 y Ft(33)p
eop
%%Page: 34 35
34 34 bop 0 1218 a Fz(Pro)39 b(of:)2382 b Ft(W)-101 b(e)616
b(view)h(the)g(obfuscator)h Fp(O)650 b Ft(as)617 b(a)g(\\w)-34
b(atermark)617 b(remo)-34 b(v)g(er.")1176 b(By)616 b(functionalit)-34
b(y)618 b(of)f(w)-34 b(ater-)0 2723 y(marking)583 b(and)g(obfuscation,)
629 b(for)583 b(ev)-34 b(ery)582 b(circuit)g Fo(C)670
b Ft(and)583 b(k)-34 b(ey)583 b Fo(K)87 b Ft(,)627 b
Fp(O)34 b Ft(\(Mark)36651 2911 y Fj(K)37563 2723 y Ft(\()p
Fo(C)19 b(;)202 b Ft(1)40064 2283 y FG(j)p Fj(C)63 b
FG(j)41376 2723 y Ft(\)\))584 b(is)e(a)h(circuit)f(com-)0
4229 y(puting)452 b(the)f(same)f(function)i(as)f Fo(C)87
b Ft(.)677 b(Let)450 b Fo(C)20569 3789 y FG(0)21330 4229
y Ft(b)34 b(e)450 b(a)h(padding)h(of)e Fo(C)538 b Ft(to)451
b(the)g(same)f(length)h(as)g(Mark)46467 4417 y Fj(K)47379
4229 y Ft(\()p Fo(C)19 b(;)202 b Ft(1)49880 3789 y FG(j)p
Fj(C)63 b FG(j)51192 4229 y Ft(\).)0 5734 y(By)502 b(fragilit)-34
b(y)-101 b(,)528 b(Extract)10937 5922 y Fj(K)11848 5734
y Ft(\()p Fp(O)34 b Ft(\(Mark)16621 5922 y Fj(K)17532
5734 y Ft(\()p Fo(C)19 b(;)202 b Ft(1\)\)\))504 b(=)e(1)h(with)h
(nonnegligible)f(probabilit)-34 b(y)-101 b(.)836 b(By)502
b(meaningfulness,)0 7240 y(Extract)4027 7428 y Fj(K)4938
7240 y Ft(\()p Fp(O)34 b Ft(\()p Fo(C)7832 6800 y FG(0)8143
7240 y Ft(\)\))338 b(=)e(1)270 b(with)g(negligible)e(probabilit)-34
b(y)-101 b(.)495 b(Th)-34 b(us,)297 b(Extract)33307 7428
y Fj(K)34488 7240 y Ft(distinguishes)271 b Fp(O)34 b
Ft(\()p Fo(C)43935 6800 y FG(0)44245 7240 y Ft(\))270
b(and)g Fp(O)34 b Ft(\(Mark)51510 7428 y Fj(K)52421 7240
y Ft(\()p Fo(C)19 b(;)202 b Ft(1)54922 6800 y FG(j)p
Fj(C)63 b FG(j)56234 7240 y Ft(\)\),)0 8745 y(con)-34
b(tradicting)405 b(the)g(indistinguishabilit)-34 b(y)405
b(prop)34 b(ert)-34 b(y)404 b(of)h Fp(O)34 b Ft(.)p 51114
8745 886 886 v 1882 11563 a(Note)443 b(that)h(this)g(prop)34
b(osition)443 b(fails)g(if)g(w)-34 b(e)444 b(allo)-34
b(w)443 b(Mark)27877 11751 y Fj(K)28788 11563 y Ft(\()p
Fo(C)19 b(;)202 b(m)p Ft(\))444 b(to)g(instead)g(b)34
b(e)442 b(an)i Fr(appr)-62 b(oximate)467 b(imple-)0 13069
y(mentation)326 b Ft(of)h Fo(C)414 b Ft(in)326 b(the)i(sense)e(of)i
(De\014nition)f(4.3.)512 b(Indeed,)343 b(in)326 b(suc)-34
b(h)328 b(a)f(case)f(it)h(seems)g(that)h(obfuscators)g(w)-34
b(ould)0 14574 y(b)34 b(e)317 b(useful)h(in)f(constructing)i(w)-34
b(atermarking)317 b(sc)-34 b(hemes,)335 b(for)317 b(the)h(w)-34
b(atermark)318 b(could)f(b)34 b(e)317 b(em)-34 b(b)34
b(edded)318 b(b)-34 b(y)318 b(c)-34 b(hanging)0 16080
y(the)322 b(v)-67 b(alue)321 b(of)g(the)h(function)h(at)f(a)f(random)h
(input,)339 b(after)322 b(whic)-34 b(h)322 b(an)g(obfuscator)h(is)e
(used)h(to)g(\\hide")g(this)g(c)-34 b(hange.)0 17585
y(Note)499 b(that)h(appro)-34 b(ximation)500 b(ma)-34
b(y)500 b(also)e(b)34 b(e)499 b(relev)-67 b(an)-34 b(t)499
b(in)g(the)g(fragilit)-34 b(y)499 b(condition,)523 b(for)499
b(it)g(w)-34 b(ould)500 b(b)34 b(e)499 b(nice)g(to)0
19090 y(prev)-34 b(en)g(t)405 b(adv)-34 b(ersaries)403
b(from)i(pro)34 b(ducing)404 b(unmark)-34 b(ed)405 b(appro)-34
b(ximate)405 b(implemen)-34 b(tations)405 b(of)g(the)f(function.)1882
20596 y(As)467 b(with)h(obfuscation,)484 b(p)34 b(ositiv)-34
b(e)467 b(theoretical)f(results)h(ab)34 b(out)468 b(w)-34
b(atermarking)467 b(w)-34 b(ould)469 b(b)34 b(e)466 b(v)-34
b(ery)467 b(w)-34 b(elcome.)0 22101 y(One)356 b(approac)-34
b(h,)367 b(tak)-34 b(en)356 b(b)-34 b(y)357 b(Naccac)-34
b(he,)365 b(Shamir,)h(and)357 b(Stern)f([NSS99)q(],)365
b(is)356 b(to)h(\014nd)g(w)-34 b(atermarking)357 b(sc)-34
b(hemes)356 b(for)0 23607 y(sp)34 b(eci\014c)403 b(useful)i(families)f
(of)g(functions.)0 27423 y Fs(9)1793 b(Directions)600
b(for)d(F)-149 b(urther)600 b(W)-149 b(ork)0 30129 y
Ft(W)-101 b(e)435 b(ha)-34 b(v)g(e)436 b(sho)-34 b(wn)437
b(that)g(obfuscation,)444 b(as)436 b(it)f(is)h(t)-34
b(ypically)434 b(understo)34 b(o)g(d)437 b(\()p Fr(i.e.,)461
b Ft(satisfying)436 b(a)g(virtual)f(blac)-34 b(k-b)34
b(o)-34 b(x)0 31635 y(prop)34 b(ert)-34 b(y\),)345 b(is)330
b(imp)34 b(ossible.)513 b(Ho)-34 b(w)g(ev)g(er,)345 b(w)-34
b(e)330 b(view)g(it)g(as)g(an)g(imp)34 b(ortan)-34 b(t)331
b(researc)-34 b(h)329 b(direction)h(to)g(explore)f(whether)0
33140 y(there)471 b(are)g(alternativ)-34 b(e)471 b(senses)g(in)g(whic)
-34 b(h)473 b(programs)e(can)g(b)34 b(e)471 b(made)h(\\unin)-34
b(telligible.")740 b(These)471 b(include)g(\(but)0 34646
y(are)343 b(not)h(limited)f(to\))i(the)f(follo)-34 b(wing)344
b(notions)g(of)g(obfuscation)h(whic)-34 b(h)345 b(are)e(not)h(ruled)f
(out)i(b)-34 b(y)344 b(our)f(imp)34 b(ossibilit)-34 b(y)0
36151 y(results:)1818 38640 y Fp(\017)606 b Ft(Indistinguishabilit)-34
b(y)527 b(\(or)f(di\013ering-input\))h(obfuscators,)557
b(as)526 b(in)g(De\014nition)g(7.1)f(\(or)h(De\014nition)g(7.4,)3030
40145 y(resp)34 b(ectiv)-34 b(ely\).)1818 42642 y Fp(\017)606
b Ft(Sampling)405 b(obfuscators,)g(as)g(in)f(De\014nition)g(6.1.)1818
45138 y Fp(\017)606 b Ft(Obfuscators)339 b(that)f(only)f(ha)-34
b(v)g(e)338 b(to)f(appro)-34 b(ximately)337 b(preserv)-34
b(e)337 b(functionalit)-34 b(y)338 b(with)g(resp)34 b(ect)337
b(to)g(a)g Fr(sp)-62 b(e)g(ci\014e)g(d)3030 46644 y Ft(distribution)451
b(on)e(inputs,)461 b(suc)-34 b(h)450 b(as)f(the)h(uniform)f
(distribution.)675 b(\(In)449 b(Section)h(4.2,)459 b(w)-34
b(e)450 b(ha)-34 b(v)g(e)449 b(ruled)g(out)3030 48149
y(obfuscators)406 b(that)f(preserving)e(functionalit)-34
b(y)406 b(on)e Fr(e)-62 b(ach)403 b Ft(input)i(with)g(high)f
(probabilit)-34 b(y)-101 b(.\))1818 50646 y Fp(\017)606
b Ft(Obfuscators)323 b(for)e(a)h(restricted,)337 b(y)-34
b(et)321 b(still)g(non)-34 b(trivial,)338 b(class)321
b(of)g(functions.)513 b(By)320 b(Theorem)i(4.13,)337
b(an)-34 b(y)322 b(suc)-34 b(h)3030 52151 y(class)492
b(of)h(functions)g(should)h(not)f(con)-34 b(tain)493
b Fz(TC)25260 52333 y Fh(0)25857 52151 y Ft(.)803 b(That)493
b(lea)-34 b(v)g(es)492 b(only)g(v)-34 b(ery)491 b(w)-34
b(eak)493 b(complexit)-34 b(y)491 b(classes)3030 53657
y(\(e.g.,)401 b Fz(A)-39 b(C)8080 53839 y Fh(0)8678 53657
y Ft(,)402 b(read-once)f(branc)-34 b(hing)402 b(programs\),)h(but)f
(the)g(class)f(of)h(functions)h(need)e(not)h(b)34 b(e)401
b(restricted)3030 55162 y(only)378 b(b)-34 b(y)379 b(\\computational")g
(p)34 b(o)-34 b(w)g(er:)526 b(syn)-34 b(tactic)379 b(or)f(functional)h
(restrictions)e(ma)-34 b(y)379 b(o\013er)e(a)h(more)g(fruitful)3030
56668 y(a)-34 b(v)g(en)g(ue.)504 b(W)-101 b(e)296 b(note)i(that)g(the)g
(constructions)g(of)f([CMR98)q(])f(can)h(b)34 b(e)297
b(view)-34 b(ed)297 b(as)g(some)g(form)g(of)h(obfuscators)3030
58173 y(for)391 b(\\delta)g(functions")h(\(i.e.,)g(functions)f
Fo(f)468 b Ft(:)336 b Fp(f)p Ft(0)p Fo(;)202 b Ft(1)p
Fp(g)26971 57733 y Fj(n)27934 58173 y Fp(!)337 b(f)p
Ft(0)p Fo(;)202 b Ft(1)p Fp(g)390 b Ft(whic)-34 b(h)392
b(tak)-34 b(e)390 b(on)h(the)g(v)-67 b(alue)390 b(1)g(at)h(exactly)3030
59679 y(one)405 b(p)34 b(oin)-34 b(t)405 b(in)f Fp(f)p
Ft(0)p Fo(;)202 b Ft(1)p Fp(g)12795 59239 y Fj(n)13421
59679 y Ft(.\))1882 62167 y(In)560 b(addition)g(to)h(obfuscation,)600
b(related)559 b(problems)h(suc)-34 b(h)561 b(as)e(homomorphic)i
(encryption)f(and)g(soft)-34 b(w)g(are)0 63673 y(w)g(atermarking)369
b(are)e(also)h(little)g(understo)34 b(o)g(d.)527 b(F)-101
b(or)368 b(soft)-34 b(w)g(are)370 b(w)-34 b(atermarking,)375
b(ev)-34 b(en)368 b(\014nding)i(a)e(reasonable)g(for-)0
65178 y(malization)339 b(of)h(the)f(problem)h(\(whic)-34
b(h)340 b(is)f(not)h(ruled)g(out)g(b)-34 b(y)339 b(our)h
(constructions,)353 b(unlik)-34 b(e)339 b(De\014nition)h(8.1\))f(seems)
0 66684 y(to)347 b(b)34 b(e)346 b(c)-34 b(hallenging,)358
b(whereas)347 b(for)g(homomorphic)g(encryption,)358 b(the)347
b(de\014nitions)h(are)e(\(more\))h(straigh)-34 b(tforw)g(ard,)0
68189 y(but)405 b(existence)e(is)h(still)g(op)34 b(en.)1882
69694 y(Finally)-101 b(,)618 b(our)576 b(in)-34 b(v)g(estigation)577
b(of)f(complexit)-34 b(y-theoretic)575 b(analogues)i(of)f(Rice's)f
(theorem)h(has)h(left)f(op)34 b(en)0 71200 y(questions,)404
b(suc)-34 b(h)405 b(as)g(whether)f(Conjecture)h(5.1)f(holds.)25394
74521 y(34)p eop
%%Page: 35 36
35 35 bop 0 1218 a Fs(Ac)-50 b(kno)g(wledgmen)g(ts)0
3923 y Ft(W)-101 b(e)348 b(are)g(grateful)h(to)g(Luca)g(T)-101
b(revisan)348 b(for)h(collab)34 b(oration)348 b(at)h(an)g(early)f
(stage)h(of)g(this)g(researc)-34 b(h.)519 b(W)-101 b(e)348
b(also)h(thank)0 5429 y(Dan)438 b(Boneh,)445 b(Ran)438
b(Canetti,)446 b(Mic)-34 b(hael)437 b(Rabin,)446 b(Y)-101
b(aco)-34 b(v)437 b(Y)-101 b(acobi,)445 b(and)438 b(the)g(anon)-34
b(ymous)439 b(CR)-101 b(YPTO)438 b(review)-34 b(ers)0
6934 y(for)404 b(helpful)h(discussions)g(and)g(commen)-34
b(ts.)1882 8440 y(This)595 b(w)-34 b(ork)595 b(w)-34
b(as)595 b(partially)f(supp)34 b(orted)596 b(b)-34 b(y)595
b(the)g(follo)-34 b(wing)596 b(funds:)921 b(Oded)595
b(Goldreic)-34 b(h)594 b(w)-34 b(as)596 b(supp)34 b(orted)0
9945 y(b)-34 b(y)539 b(the)g(Minerv)-67 b(a)538 b(F)-101
b(oundation,)574 b(German)-34 b(y;)606 b(Salil)538 b(V)-101
b(adhan)540 b(\(at)g(the)f(time)f(at)h(MIT\))g(w)-34
b(as)540 b(supp)34 b(orted)540 b(b)-34 b(y)539 b(a)0
11451 y(DOD/NDSEG)575 b(Graduate)i(F)-101 b(ello)-34
b(wship)576 b(and)g(an)g(NSF)g(Mathematical)g(Sciences)f(P)-34
b(ostdo)34 b(ctoral)576 b(Researc)-34 b(h)0 12956 y(F)-101
b(ello)-34 b(wship.)0 16775 y Fs(References)0 19481 y
Ft([BGI)2585 19041 y Fl(+)3371 19481 y Ft(01])1254 b(Boaz)455
b(Barak,)467 b(Oded)456 b(Goldreic)-34 b(h,)468 b(Russell)456
b(Impagaliazzo,)467 b(Stev)-34 b(en)456 b(Rudic)-34 b(h,)469
b(Amit)456 b(Sahai,)468 b(Salil)6174 20987 y(V)-101 b(adhan,)482
b(and)467 b(Ke)e(Y)-101 b(ang.)721 b(On)465 b(the)h(\(im\)p)34
b(ossibilit)-34 b(y)466 b(of)g(obfuscating)i(programs.)720
b(In)466 b(J.)f(Kilian,)6174 22492 y(editor,)573 b Fr(A)-62
b(dvanc)g(es)556 b(in)i(Cryptolo)-62 b(gy|CR)-93 b(YPTO)557
b('01)p Ft(,)571 b(Lecture)539 b(Notes)g(in)h(Computer)g(Science.)6174
23998 y(Springer-V)-101 b(erlag,)404 b(2001,)g(August)h(2001.)538
b(T)-101 b(o)404 b(app)34 b(ear.)0 26499 y([BR93])2537
b(Mihir)411 b(Bellare)f(and)i(Phillip)e(Roga)-34 b(w)g(a)g(y)-101
b(.)560 b(Random)413 b(oracles)d(are)g(practical:)553
b(A)411 b(paradigm)h(for)f(de-)6174 28005 y(signing)371
b(e\016cien)-34 b(t)370 b(proto)34 b(cols.)482 b(In)371
b Fr(Pr)-62 b(o)g(c)g(e)g(e)g(dings)400 b(of)i(the)f(First)h(A)-31
b(nnual)402 b(Confer)-62 b(enc)g(e)402 b(on)g(Computer)6174
29510 y(and)434 b(Communic)-62 b(ations)431 b(Se)-62
b(curity)p Ft(.)402 b(A)-34 b(CM,)405 b(No)-34 b(v)g(em)g(b)34
b(er)404 b(1993.)0 32012 y([BL96])2671 b(Dan)388 b(Boneh)f(and)g(Ric)
-34 b(hard)388 b(Lipton.)509 b(Algorithms)388 b(for)f(blac)-34
b(k-b)34 b(o)-34 b(x)387 b(\014elds)g(and)g(their)g(applications)6174
33517 y(to)342 b(cryptograph)-34 b(y)-101 b(.)436 b(In)342
b(M.)f(Wiener,)353 b(editor,)g Fr(A)-62 b(dvanc)g(es)374
b(in)j(Cryptolo)-62 b(gy|CR)-93 b(YPTO)375 b('96)p Ft(,)352
b(v)-34 b(olume)6174 35023 y(1109)514 b(of)f Fr(L)-62
b(e)g(ctur)g(e)533 b(Notes)f(in)i(Computer)f(Scienc)-62
b(e)p Ft(,)539 b(pages)514 b(283{297.)f(Springer-V)-101
b(erlag,)540 b(August)6174 36528 y(1996.)0 39030 y([CGH98])1553
b(Ran)352 b(Canetti,)363 b(Oded)352 b(Goldreic)-34 b(h,)361
b(and)352 b(Shai)g(Halevi.)450 b(The)352 b(random)g(oracle)e(metho)34
b(dology)-101 b(,)362 b(revis-)6174 40536 y(ited.)409
b(In)325 b Fr(Pr)-62 b(o)g(c)g(e)g(e)g(dings)359 b(of)h(the)g
(Thirtieth)f(A)-31 b(nnual)361 b(A)-31 b(CM)361 b(Symp)-62
b(osium)360 b(on)h(The)-62 b(ory)359 b(of)i(Computing)p
Ft(,)6174 42041 y(pages)405 b(209{218,)f(Dallas,)f(23{26)i(Ma)-34
b(y)404 b(1998.)0 44543 y([CMR98])1410 b(Ran)424 b(Canetti,)429
b(Daniele)423 b(Micciancio,)j(and)f(Omer)d(Reingold.)595
b(P)-34 b(erfectly)423 b(one-w)-34 b(a)g(y)424 b(probabilistic)6174
46048 y(hash)402 b(functions.)534 b(In)401 b Fr(Pr)-62
b(o)g(c)g(e)g(e)g(dings)428 b(of)i(the)f(Thirtieth)g(A)-31
b(nnual)431 b(A)-31 b(CM)430 b(Symp)-62 b(osium)430 b(on)g(The)-62
b(ory)430 b(of)6174 47554 y(Computing)p Ft(,)403 b(pages)h(131{140,)g
(Dallas,)g(23{26)g(Ma)-34 b(y)404 b(1998.)0 50055 y([CT00])2538
b(Christian)472 b(Collb)34 b(erg)471 b(and)h(Clark)f(Thom)-34
b(b)34 b(orson.)738 b(W)-101 b(atermarking,)487 b(tamp)34
b(er-pro)g(o\014ng,)489 b(and)471 b(ob-)6174 51561 y(fuscation)421
b({)e(to)34 b(ols)419 b(for)g(soft)-34 b(w)g(are)421
b(protection.)582 b(T)-101 b(ec)-34 b(hnical)419 b(Rep)34
b(ort)420 b(TR00-03,)j(The)d(Departmen)-34 b(t)6174 53066
y(of)405 b(Computer)g(Science,)e(Univ)-34 b(ersit)g(y)404
b(of)h(Arizona,)e(F)-101 b(ebruary)404 b(2000.)0 55568
y([DDPY98])702 b(Alfredo)422 b(De)e(San)-34 b(tis,)427
b(Gio)-34 b(v)-67 b(anni)421 b(Di)g(Crescenzo,)k(Giusepp)34
b(e)421 b(P)-34 b(ersiano,)425 b(and)d(Moti)f(Y)-101
b(ung.)589 b(Image)6174 57074 y(Densit)-34 b(y)500 b(is)f(complete)h
(for)f(non-in)-34 b(teractiv)g(e-SZK.)822 b(In)500 b
Fr(A)-31 b(utomata,)541 b(L)-62 b(anguages)519 b(and)i(Pr)-62
b(o)g(gr)g(am-)6174 58579 y(ming,)480 b(25th)469 b(International)g(Col)
62 b(lo)-62 b(quium)p Ft(,)451 b(Lecture)443 b(Notes)i(in)f(Computer)h
(Science,)453 b(pages)445 b(784{)6174 60084 y(795,)464
b(Aalb)34 b(org,)463 b(Denmark,)g(13{17)453 b(July)e(1998.)h
(Springer-V)-101 b(erlag.)679 b(See)451 b(also)h(preliminary)f(draft)
6174 61590 y(of)405 b(full)f(v)-34 b(ersion,)404 b(Ma)-34
b(y)404 b(1999.)0 64092 y([DH76])2453 b(Whit\014eld)377
b(Di\016e)f(and)h(Martin)f(E.)g(Hellman.)491 b(New)377
b(directions)f(in)g(cryptograph)-34 b(y)-101 b(.)492
b Fr(IEEE)409 b(T)-93 b(r)-62 b(ans-)6174 65597 y(actions)432
b(on)i(Information)f(The)-62 b(ory)p Ft(,)402 b(IT-22\(6\):644{654,)j
(1976.)0 68099 y([DDN00])1527 b(Dann)-34 b(y)538 b(Dolev,)569
b(Cyn)-34 b(thia)538 b(Dw)-34 b(ork,)570 b(and)537 b(Moni)g(Naor.)933
b(Nonmalleable)536 b(cryptograph)-34 b(y)-101 b(.)933
b Fr(SIAM)6174 69604 y(Journal)433 b(on)h(Computing)p
Ft(,)402 b(30\(2\):391{437)j(\(electronic\),)f(2000.)25394
74521 y(35)p eop
%%Page: 36 37
36 36 bop 0 1218 a Ft([ESY84])1881 b(Shimon)352 b(Ev)-34
b(en,)361 b(Alan)350 b(L.)g(Selman,)361 b(and)351 b(Y)-101
b(aco)-34 b(v)350 b(Y)-101 b(acobi.)449 b(The)351 b(complexit)-34
b(y)350 b(of)h(promise)f(problems)6174 2723 y(with)370
b(applications)g(to)f(public-k)-34 b(ey)368 b(cryptograph)-34
b(y)-101 b(.)481 b Fr(Information)400 b(and)h(Contr)-62
b(ol)p Ft(,)375 b(61\(2\):159{173,)6174 4229 y(1984.)0
6730 y([FM91])2386 b(Joan)277 b(F)-101 b(eigen)-34 b(baum)277
b(and)f(Mic)-34 b(hael)276 b(Merritt,)301 b(editors.)328
b Fr(Distribute)-62 b(d)313 b(c)-62 b(omputing)313 b(and)i(crypto)-62
b(gr)g(aphy)p Ft(,)6174 8236 y(Pro)-34 b(vidence,)404
b(RI,)f(1991.)h(American)g(Mathematical)g(So)34 b(ciet)-34
b(y)-101 b(.)0 10738 y([FS87])2824 b(Amos)370 b(Fiat)f(and)h(Adi)f
(Shamir.)480 b(Ho)-34 b(w)370 b(to)g(pro)-34 b(v)g(e)369
b(y)-34 b(ourself:)521 b(practical)368 b(solutions)i(to)g(iden)-34
b(ti\014cation)6174 12243 y(and)553 b(signature)e(problems.)977
b(In)551 b Fr(A)-62 b(dvanc)g(es)567 b(in)i(cryptolo)-62
b(gy|CR)-93 b(YPTO)567 b('86)h(\(Santa)h(Barb)-62 b(ar)g(a,)6174
13748 y(Calif.,)433 b(1986\))p Ft(,)401 b(pages)j(186{194.)h(Springer,)
e(Berlin,)g(1987.)0 16250 y([GT00])2462 b(Rosario)378
b(Gennaro)g(and)h(Luca)e(T)-101 b(revisan.)494 b(Lo)-34
b(w)g(er)378 b(b)34 b(ounds)379 b(on)f(the)h(e\016ciency)d(of)i
(generic)f(crypto-)6174 17756 y(graphic)301 b(constructions.)370
b(In)301 b Fr(41st)336 b(A)-31 b(nnual)339 b(Symp)-62
b(osium)337 b(on)h(F)-93 b(oundations)337 b(of)g(Computer)g(Scienc)-62
b(e)p Ft(,)6174 19261 y(Redondo)406 b(Beac)-34 b(h,)403
b(CA,)i(17{19)f(Octob)34 b(er)403 b(2000.)h(IEEE.)0 21763
y([GGM86])1275 b(Oded)441 b(Goldreic)-34 b(h,)449 b(Sha\014)442
b(Goldw)-34 b(asser,)450 b(and)441 b(Silvio)f(Micali.)644
b(Ho)-34 b(w)442 b(to)f(construct)g(random)g(func-)6174
23268 y(tions.)538 b Fr(Journal)433 b(of)g(the)f(Asso)-62
b(ciation)431 b(for)i(Computing)f(Machinery)p Ft(,)402
b(33\(4\):792{807,)j(1986.)0 25770 y([GO96])2394 b(Oded)377
b(Goldreic)-34 b(h)375 b(and)i(Rafail)f(Ostro)-34 b(vsky)-101
b(.)491 b(Soft)-34 b(w)g(are)378 b(protection)e(and)h(sim)-34
b(ulation)377 b(on)f(oblivious)6174 27276 y(RAMs.)538
b Fr(Journal)433 b(of)g(the)f(A)-31 b(CM)p Ft(,)404 b(43\(3\):431{473,)
h(1996.)0 29777 y([GSV99])1755 b(Oded)586 b(Goldreic)-34
b(h,)630 b(Amit)586 b(Sahai,)631 b(and)586 b(Salil)f(V)-101
b(adhan.)1079 b(Can)586 b(statistical)g(zero-kno)-34
b(wledge)585 b(b)34 b(e)6174 31283 y(made)561 b(non-in)-34
b(teractiv)g(e?,)601 b(or)561 b(On)g(the)g(relationship)h(of)f(SZK)f
(and)i(NISZK.)1004 b(In)561 b Fr(A)-62 b(dvanc)g(es)576
b(in)6174 32788 y(Cryptolo)-62 b(gy|CR)-93 b(YPTO)402
b('99)p Ft(,)376 b(Lecture)369 b(Notes)i(in)g(Computer)h(Science.)d
(Springer-V)-101 b(erlag,)377 b(1999,)6174 34294 y(15{19)405
b(August)g(1999.)538 b(T)-101 b(o)404 b(app)34 b(ear.)0
36795 y([GM84])2226 b(Sha\014)362 b(Goldw)-34 b(asser)361
b(and)g(Silvio)f(Micali.)464 b(Probabilistic)360 b(encryption.)466
b Fr(Journal)393 b(of)g(Computer)e(and)6174 38301 y(System)433
b(Scienc)-62 b(es)p Ft(,)403 b(28\(2\):270{299,)h(April)g(1984.)0
40803 y([Had00])2100 b(Satoshi)390 b(Hada.)512 b(Zero-kno)-34
b(wledge)389 b(and)g(co)34 b(de)388 b(obfuscation.)513
b(In)389 b(T.)g(Ok)-67 b(amoto,)391 b(editor,)g Fr(A)-62
b(dvanc)g(es)6174 42308 y(in)458 b(Cryptolo)-62 b(gy)455
b({)i(ASIACR)-93 b(YPT)460 b(')d(2000)p Ft(,)435 b(Lecture)429
b(Notes)i(in)f(Computer)h(Science,)436 b(pages)431 b(443{)6174
43813 y(457,)377 b(Ky)-34 b(oto,)378 b(Japan,)f(2000.)371
b(In)-34 b(ternational)371 b(Asso)34 b(ciation)371 b(for)f(Cryptologic)
h(Researc)-34 b(h,)377 b(Springer-)6174 45319 y(V)-101
b(erlag,)404 b(Berlin)e(German)-34 b(y)-101 b(.)0 47821
y([HILL99])1425 b(Johan)386 b(H)-152 b(\027)-758 b(astad,)390
b(Russell)385 b(Impagliazzo,)j(Leonid)c(A.)h(Levin,)i(and)f(Mic)-34
b(hael)384 b(Lub)-34 b(y)-101 b(.)506 b(A)385 b(pseudoran-)6174
49326 y(dom)429 b(generator)f(from)g(an)-34 b(y)429 b(one-w)-34
b(a)g(y)429 b(function.)610 b Fr(SIAM)457 b(Journal)e(on)h(Computing)p
Ft(,)432 b(28\(4\):1364{)6174 50832 y(1396)405 b(\(electronic\),)e
(1999.)0 53333 y([IL89])3092 b(Russell)398 b(Impagliazzo)e(and)i(Mic)
-34 b(hael)397 b(Lub)-34 b(y)-101 b(.)527 b(One-w)-34
b(a)g(y)398 b(functions)h(are)d(essen)-34 b(tial)398
b(for)f(complexit)-34 b(y)6174 54839 y(based)333 b(cryptograph)-34
b(y)332 b(\(extended)g(abstract\).)421 b(In)331 b Fr(30th)366
b(A)-31 b(nnual)366 b(Symp)-62 b(osium)366 b(on)h(F)-93
b(oundations)365 b(of)6174 56344 y(Computer)314 b(Scienc)-62
b(e)p Ft(,)300 b(pages)276 b(230{235,)302 b(Researc)-34
b(h)275 b(T)-101 b(riangle)276 b(P)-34 b(ark,)300 b(North)277
b(Carolina,)301 b(30)275 b(Octob)34 b(er{)6174 57850
y(1)404 b(No)-34 b(v)g(em)g(b)34 b(er)405 b(1989.)f(IEEE.)0
60351 y([KY00])2436 b(Jonathan)630 b(Katz)e(and)h(Moti)g(Y)-101
b(ung.)1206 b(Complete)629 b(c)-34 b(haracterization)628
b(of)g(securit)-34 b(y)628 b(notions)i(for)6174 61857
y(priv)-67 b(ate-k)-34 b(ey)343 b(encryption.)439 b(In)343
b Fr(Pr)-62 b(o)g(c)g(e)g(e)g(dings)376 b(of)h(the)g(32nd)g(A)-31
b(nnual)378 b(A)-31 b(CM)378 b(Symp)-62 b(osium)376 b(on)i(The)-62
b(ory)6174 63362 y(of)433 b(Computing)p Ft(,)403 b(pages)h(245{254,)g
(P)-34 b(ortland,)405 b(OR,)f(Ma)-34 b(y)404 b(2000.)g(A)-34
b(CM.)0 65864 y([LR88])2638 b(Mic)-34 b(hael)321 b(Lub)-34
b(y)321 b(and)h(Charles)f(Rac)-34 b(k)g(o\013.)402 b(Ho)-34
b(w)322 b(to)g(construct)g(pseudorandom)g(p)34 b(erm)-34
b(utations)323 b(from)6174 67370 y(pseudorandom)460 b(functions.)698
b Fr(SIAM)483 b(Journal)f(on)h(Computing)p Ft(,)469 b(17\(2\):373{386,)
j(1988.)697 b(Sp)34 b(ecial)6174 68875 y(issue)404 b(on)h(cryptograph)
-34 b(y)-101 b(.)25394 74521 y(36)p eop
%%Page: 37 38
37 37 bop 0 1218 a Ft([MMS)3232 778 y Fl(+)4020 1218
y Ft(98])605 b(Lesley)478 b(R.)h(Matheson,)498 b(Stephen)481
b(G.)e(Mitc)-34 b(hell,)497 b(T)-101 b(alal)479 b(G.)f(Shamo)34
b(on,)499 b(Rob)34 b(ert)479 b(E.)g(T)-101 b(arjan,)498
b(and)6174 2723 y(F)-101 b(rancis)351 b(Zane.)451 b(Robustness)353
b(and)e(securit)-34 b(y)351 b(of)g(digital)g(w)-34 b(atermarks.)451
b(In)351 b(H.)g(Imai)f(and)i(Y.)f(Zheng,)6174 4229 y(editors,)487
b Fr(Financial)494 b(Crypto)-62 b(gr)g(aphy|F)-31 b(C)492
b('98)p Ft(,)485 b(v)-34 b(olume)471 b(1465)f(of)h Fr(L)-62
b(e)g(ctur)g(e)493 b(Notes)h(in)h(Computer)6174 5734
y(Scienc)-62 b(e)p Ft(,)403 b(pages)i(227{240.)f(Springer,)g(F)-101
b(ebruary)404 b(1998.)0 8236 y([NSS99])2033 b(Da)-34
b(vid)474 b(Naccac)-34 b(he,)491 b(Adi)474 b(Shamir,)491
b(and)475 b(Julien)f(P)-101 b(.)473 b(Stern.)745 b(Ho)-34
b(w)475 b(to)f(cop)-34 b(yrigh)g(t)475 b(a)f(function?)1021
b(In)6174 9741 y(H.)506 b(Imai)f(and)i(Y.)e(Zheng,)531
b(editors,)g Fr(Public)526 b(Key)g(Crypto)-62 b(gr)g(aphy|PK)-31
b(C)524 b('99)p Ft(,)530 b(v)-34 b(olume)506 b(1560)g(of)6174
11247 y Fr(L)-62 b(e)g(ctur)g(e)432 b(Notes)h(in)g(Computer)g(Scienc)
-62 b(e)p Ft(,)402 b(pages)j(188{196.)f(Springer-V)-101
b(erlag,)403 b(Marc)-34 b(h)405 b(1999.)0 13748 y([NR97])2487
b(Moni)529 b(Naor)f(and)h(Omer)e(Reingold.)907 b(Num)-34
b(b)34 b(er-theoretic)529 b(constructions)g(of)g(e\016cien)-34
b(t)528 b(pseudo-)6174 15254 y(random)484 b(functions.)773
b(In)483 b Fr(38th)504 b(A)-31 b(nnual)506 b(Symp)-62
b(osium)504 b(on)j(F)-93 b(oundations)503 b(of)j(Computer)e(Scienc)-62
b(e)p Ft(,)6174 16759 y(pages)405 b(458{467,)f(Miami)g(Beac)-34
b(h,)403 b(Florida,)h(20{22)g(Octob)34 b(er)404 b(1997.)g(IEEE.)0
19261 y([P)-101 b(AK99])1712 b(F)-101 b(abien)472 b(A.)f(P)-101
b(.)470 b(P)-34 b(etitcolas,)487 b(Ross)472 b(J.)e(Anderson,)489
b(and)471 b(Markus)h(J.)e(Kuhn.)737 b(Information)472
b(hiding)6174 20767 y(|)405 b(a)f(surv)-34 b(ey)-101
b(.)537 b Fr(Pr)-62 b(o)g(c)g(e)g(e)g(dings)431 b(of)i(the)f(IEEE)p
Ft(,)405 b(87\(7\):1062{1078,)g(1999.)0 23268 y([RAD78])1561
b(Ronald)345 b(L.)e(Riv)-34 b(est,)355 b(Len)344 b(Adleman,)356
b(and)344 b(Mic)-34 b(hael)343 b(L.)g(Dertouzos.)439
b(On)344 b(data)g(banks)h(and)f(priv)-67 b(acy)6174 24774
y(homomorphisms.)328 b(In)275 b Fr(F)-93 b(oundations)313
b(of)h(se)-62 b(cur)g(e)313 b(c)-62 b(omputation)312
b(\(Workshop,)335 b(Ge)-62 b(or)g(gia)313 b(Inst.)h(T)-93
b(e)-62 b(ch.,)6174 26279 y(A)-31 b(tlanta,)432 b(Ga.,)h(1977\))p
Ft(,)401 b(pages)k(169{179.)f(Academic,)f(New)h(Y)-101
b(ork,)404 b(1978.)0 28781 y([SV97])2706 b(Amit)572 b(Sahai)g(and)h
(Salil)e(P)-101 b(.)571 b(V)-101 b(adhan.)1038 b(A)571
b(complete)h(promise)f(problem)h(for)g(statistical)g(zero-)6174
30286 y(kno)-34 b(wledge.)897 b(In)525 b Fr(38th)543
b(A)-31 b(nnual)544 b(Symp)-62 b(osium)543 b(on)i(F)-93
b(oundations)542 b(of)i(Computer)f(Scienc)-62 b(e)p Ft(,)554
b(pages)6174 31792 y(448{457,)405 b(Miami)e(Beac)-34
b(h,)404 b(Florida,)g(20{22)g(Octob)34 b(er)403 b(1997.)h(IEEE.)0
34294 y([SYY99])1797 b(Thomas)469 b(Sander,)483 b(Adam)467
b(Y)-101 b(oung,)484 b(and)468 b(Moti)f(Y)-101 b(ung.)725
b(Non-in)-34 b(teractiv)g(e)468 b(crypto)34 b(computing)468
b(for)6174 35799 y(NC)7959 35321 y Fl(1)8485 35799 y
Ft(.)k(In)364 b Fr(40th)396 b(A)-31 b(nnual)397 b(Symp)-62
b(osium)395 b(on)i(F)-93 b(oundations)395 b(of)i(Computer)e(Scienc)-62
b(e)p Ft(,)371 b(pages)365 b(554{566,)6174 37305 y(New)405
b(Y)-101 b(ork,)403 b(NY,)h(17{19)g(Octob)34 b(er)404
b(1999.)g(IEEE.)0 39806 y([vD98])2722 b(F)-101 b(rans)464
b(v)-67 b(an)463 b(Dorsselaer.)712 b(Obsolescen)-34 b(t)463
b(feature.)713 b(Winning)464 b(en)-34 b(try)463 b(for)h(the)f
Fr(1998)486 b(International)6174 41312 y(Obfusc)-62 b(ate)g(d)431
b(C)j(Co)-62 b(de)432 b(Contest)p Ft(,)403 b(1998.)537
b Fm(http://www.ioccc.org/)p Ft(.)0 45131 y Fs(A)1793
b(Generalizing)599 b(Rice's)f(Theorem)h(to)f(Promise)h(Problems.)0
47837 y Ft(W)-101 b(e)404 b(sa)-34 b(y)404 b(that)h(a)f(T)-101
b(uring)405 b(mac)-34 b(hine)404 b Fo(M)537 b Fr(de)-62
b(cides)500 b Ft(the)405 b(promise)f(problem)g(\005)337
b(=)f(\(\005)37673 48025 y Fj(Y)38483 47837 y Fo(;)202
b Ft(\005)39931 48025 y Fj(N)40828 47837 y Ft(\))404
b(if)20528 50863 y Fo(x)336 b Fp(2)h Ft(\005)23611 51051
y Fj(Y)24758 50863 y Fp(\))f Fo(M)132 b Ft(\()p Fo(x)p
Ft(\))338 b(=)e(1)20484 53586 y Fo(x)h Fp(2)g Ft(\005)23568
53774 y Fj(N)24801 53586 y Fp(\))g Fo(M)132 b Ft(\()p
Fo(x)p Ft(\))337 b(=)g(0)0 55811 y(In)433 b(suc)-34 b(h)434
b(a)f(case,)439 b(w)-34 b(e)434 b(sa)-34 b(y)433 b(that)h(\005)f(is)f
(is)h Fr(de)-62 b(cidable)p Ft(.)621 b(W)-101 b(e)433
b(sa)-34 b(y)433 b(that)h(\005)f(is)g(closed)f(under)i([)p
Fp(\001)p Ft(])e(if)h(for)g(all)f Fo(M)65 b(;)202 b(M)50206
55371 y FG(0)50516 55811 y Ft(,)440 b(if)0 57316 y([)p
Fo(M)132 b Ft(])336 b Fp(\021)h Ft([)p Fo(M)5243 56877
y FG(0)5553 57316 y Ft(])404 b(then)h(b)34 b(oth)405
b Fo(M)468 b Fp(2)337 b Ft(\005)15615 57504 y Fj(Y)17098
57316 y Fp(\()-202 b(\))674 b Fo(M)21302 56877 y FG(0)21949
57316 y Fp(2)337 b Ft(\005)24003 57504 y Fj(Y)25217 57316
y Ft(and)405 b Fo(M)469 b Fp(2)337 b Ft(\005)31273 57504
y Fj(N)32843 57316 y Fp(\()-202 b(\))673 b Fo(M)37046
56877 y FG(0)37693 57316 y Fp(2)337 b Ft(\005)39747 57504
y Fj(N)41048 57316 y Ft(hold.)1882 58822 y(The)404 b(straigh)-34
b(tforw)g(ard)407 b(w)-34 b(a)g(y)405 b(to)f(generalize)f(Rice's)g
(Theorem)h(to)h(promise)f(problems)g(is)g(the)g(follo)-34
b(wing:)0 61656 y Fz(Conjecture)465 b(A.1)f(\(Rice's)h(Theorem)f(|)h
(naiv)-39 b(e)464 b(generalization\))607 b Fr(L)-62 b(et)399
b Ft(\005)337 b(=)g(\(\005)41397 61844 y Fj(Y)42207 61656
y Fo(;)202 b Ft(\005)43655 61844 y Fj(N)44551 61656 y
Ft(\))401 b Fr(b)-62 b(e)399 b(a)h(pr)-62 b(omise)0 63161
y(pr)g(oblem)517 b(close)-62 b(d)517 b(under)g Ft([)p
Fp(\001)p Ft(])p Fr(.)812 b(If)519 b Ft(\005)g Fr(is)f(de)-62
b(cidable,)536 b(then)518 b Ft(\005)h Fr(is)f(trivial)f(in)i(the)e
(sense)h(that)f(either)g Ft(\005)47010 63349 y Fj(Y)48311
63161 y Ft(=)491 b Fp(;)518 b Fr(or)0 64667 y Ft(\005)909
64855 y Fj(N)2142 64667 y Ft(=)337 b Fp(;)p Fr(.)0 67500
y Ft(This)405 b(generalization)e(is)h(really)f(to)34
b(o)404 b(naiv)-34 b(e.)538 b(Consider)405 b(the)f(follo)-34
b(wing)405 b(promise)f(problem)g(\(\005)43713 67688 y
Fj(Y)44524 67500 y Fo(;)202 b Ft(\005)45972 67688 y Fj(N)46868
67500 y Ft(\))15714 70224 y(\005)16623 70412 y Fj(Y)17769
70224 y Ft(=)337 b Fp(f)p Fo(M)d Fp(j)202 b Fo(M)536
b Ft(alw)-34 b(a)g(ys)405 b(halts,)f Fo(M)132 b Ft(\(0\))338
b(=)e(1)p Fp(g)25394 74521 y Ft(37)p eop
%%Page: 38 39
38 38 bop 15670 1218 a Ft(\005)16579 1406 y Fj(N)17813
1218 y Ft(=)336 b Fp(f)p Fo(M)e Fp(j)202 b Fo(M)536 b
Ft(alw)-34 b(a)g(ys)405 b(halts,)f Fo(M)132 b Ft(\(0\))338
b(=)e(0)p Fp(g)0 3443 y Ft(It)404 b(is)g(ob)-34 b(viously)404
b(decidable,)g(non-trivial,)g(and)g(closed)g(under)h([)p
Fp(\001)p Ft(].)1882 4948 y(Our)363 b(next)g(attempt)h(at)g
(generalizing)e(Rice's)g(Theorem)h(to)h(promise)f(problems)g(is)g
(based)h(on)f(the)h(idea)e(of)i(a)0 6454 y(sim)-34 b(ulator,)369
b(whic)-34 b(h)360 b(w)-34 b(e)360 b(use)g(to)g(formalize)e(the)i(in)
-34 b(terpretation)361 b(of)f(Rice's)e(Theorem)i(as)g(\\the)g(only)f
(useful)h(thing)0 7959 y(y)-34 b(ou)370 b(can)g(do)f(with)i(a)e(mac)-34
b(hine)370 b(is)f(run)h(it.")527 b(Recall)369 b(that)i(for)e(a)h(T)-101
b(uring)370 b(mac)-34 b(hine)370 b Fo(M)132 b Ft(,)376
b(the)370 b(function)h Fp(h)p Fo(M)132 b Fp(i)p Ft(\(1)49901
7519 y Fj(t)50297 7959 y Fo(;)202 b(x)p Ft(\))0 9465
y(is)404 b(de\014ned)h(to)f(b)34 b(e)404 b Fo(y)448 b
Ft(if)404 b Fo(M)132 b Ft(\()p Fo(x)p Ft(\))405 b(halts)g(within)g
Fo(t)e Ft(steps)i(with)g(output)h Fo(y)43 b Ft(,)405
b(and)g Fp(?)e Ft(otherwise.)0 11966 y Fz(Theorem)464
b(A.2)h(\(Rice's)g(Theorem)f(|)g(second)h(generalization\))606
b Fr(L)-62 b(et)463 b Ft(\005)390 b(=)g(\(\005)41208
12154 y Fj(Y)42018 11966 y Fo(;)202 b Ft(\005)43466 12154
y Fj(N)44362 11966 y Ft(\))464 b Fr(b)-62 b(e)462 b(a)h(pr)-62
b(omise)0 13472 y(pr)g(oblem)389 b(close)-62 b(d)388
b(under)i Ft([)p Fp(\001)p Ft(])p Fr(.)542 b(Supp)-62
b(ose)389 b(that)g Ft(\005)i Fr(is)f(de)-62 b(cidable,)395
b(then)390 b Ft(\005)h Fr(is)f(trivial)f(in)i(the)e(sense)h(that)f
(ther)-62 b(e)389 b(exists)0 14977 y(a)433 b(T)-93 b(uring)434
b(machine)e Fo(S)504 b Fr(such)432 b(that)18830 16483
y Fo(M)469 b Fp(2)337 b Ft(\005)22529 16671 y Fj(Y)23675
16483 y Fp(\))g Fo(S)26037 15982 y FG(h)p Fj(M)94 b FG(i)27823
16483 y Ft(\(1)28900 15982 y FG(j)p Fj(M)g FG(j)30476
16483 y Ft(\))337 b(=)g(1)18787 18708 y Fo(M)469 b Fp(2)337
b Ft(\005)22486 18896 y Fj(N)23719 18708 y Fp(\))g Fo(S)26081
18207 y FG(h)p Fj(M)94 b FG(i)27866 18708 y Ft(\(1)28943
18207 y FG(j)p Fj(M)g FG(j)30519 18708 y Ft(\))337 b(=)g(0)0
21541 y Fz(Pro)39 b(of:)1260 b Ft(Supp)34 b(ose)332 b(that)f(\005)337
b(=)f(\(\005)16171 21729 y Fj(Y)16981 21541 y Fo(;)202
b Ft(\005)18429 21729 y Fj(N)19326 21541 y Ft(\))330
b(is)g(decided)g(b)-34 b(y)331 b(the)f(T)-101 b(uring)331
b(mac)-34 b(hine)330 b Fo(T)168 b Ft(.)515 b(W)-101 b(e)329
b(will)h(build)g(a)h(mac)-34 b(hine)0 23047 y Fo(S)474
b Ft(whic)-34 b(h)405 b(will)f(satisfy)g(the)h(conclusion)f(of)h(the)f
(theorem.)1882 24552 y(W)-101 b(e)433 b(sa)-34 b(y)433
b(that)i(a)e(mac)-34 b(hine)434 b Fo(N)565 b Ft(is)434
b Fo(n)p Fr(-c)-62 b(omp)g(atible)520 b Ft(with)434 b(a)g(mac)-34
b(hine)433 b Fo(M)566 b Ft(if)433 b Fp(h)p Fo(N)132 b
Fp(i)p Ft(\(1)39163 24113 y Fj(t)39560 24552 y Fo(;)202
b(x)p Ft(\))385 b(=)g Fp(h)p Fo(M)132 b Fp(i)p Ft(\(1)46303
24113 y Fj(t)46700 24552 y Fo(;)202 b(x)p Ft(\))433 b(for)h(all)0
26058 y Fp(j)p Fo(x)p Fp(j)p Fo(;)202 b(t)335 b Fp(\024)i
Fo(n)p Ft(.)538 b(Note)404 b(that:)1481 28560 y(1.)606
b Fo(n)p Ft(-compatibilit)-34 b(y)404 b(with)h Fo(M)536
b Ft(can)405 b(b)34 b(e)403 b(decided)i(using)f(oracle)f(access)h(to)g
Fp(h)p Fo(M)132 b Fp(i)p Ft(.)1481 31061 y(2.)606 b Fo(M)536
b Ft(is)404 b Fo(n)p Ft(-compatible)g(with)h(itself)f(for)g(all)g
Fo(n)p Ft(.)1481 33563 y(3.)606 b(If)440 b([)p Fo(M)132
b Ft(])395 b Fp(6\021)h Ft([)p Fo(N)132 b Ft(])439 b(then)i(there)e
(exists)g(a)h(n)-34 b(um)g(b)34 b(er)441 b Fo(n)25752
33123 y FG(0)26501 33563 y Ft(suc)-34 b(h)441 b(that)g
Fo(N)571 b Ft(is)440 b(not)g Fo(n)p Ft(-compatible)g(with)g
Fo(M)572 b Ft(for)440 b(all)3030 35069 y Fo(n)337 b(>)f(n)6102
34629 y FG(0)6412 35069 y Ft(.)1481 37570 y(4.)606 b(It)404
b(ma)-34 b(y)405 b(b)34 b(e)404 b(the)g(case)g(than)h([)p
Fo(M)132 b Ft(])336 b Fp(\021)h Ft([)p Fo(N)132 b Ft(])403
b(but)i Fo(N)536 b Ft(is)404 b(not)h Fo(n)p Ft(-compatible)f(with)h
Fo(M)536 b Ft(for)404 b(some)g Fo(n)p Ft(.)0 40072 y(With)g(oracle)f
Fp(h)p Fo(M)132 b Fp(i)405 b Ft(and)g(input)g(1)15486
39632 y FG(j)p Fj(M)94 b FG(j)17063 40072 y Ft(,)403
b Fo(S)475 b Ft(do)34 b(es)404 b(the)g(follo)-34 b(wing)405
b(for)f Fo(n)336 b Ft(=)h(0)p Fo(;)202 b Ft(1)p Fo(;)g
Ft(2)p Fo(;)g(:)g(:)g(:)n Ft(:)1481 42574 y(1.)606 b(Compute)359
b(the)g(set)f Fo(S)12867 42756 y Fj(n)13851 42574 y Ft(whic)-34
b(h)358 b(consists)h(of)f(all)f(the)h(mac)-34 b(hines)358
b(of)g(size)f Fp(j)p Fo(M)132 b Fp(j)357 b Ft(that)i(are)f
Fo(n)p Ft(-compatible)f(with)3030 44079 y Fo(M)536 b
Ft(\(this)405 b(can)g(b)34 b(e)404 b(done)g(in)g(\014nite)h(time)f(as)g
(there)g(are)g(only)g(\014nitely)g(man)-34 b(y)404 b(mac)-34
b(hines)405 b(of)f(size)g Fp(j)p Fo(M)132 b Fp(j)p Ft(\).)1481
46581 y(2.)606 b(Run)369 b Fo(T)535 b Ft(on)368 b(all)f(the)g(mac)-34
b(hines)368 b(in)f Fo(S)19533 46763 y Fj(n)20527 46581
y Ft(for)h Fo(n)e Ft(steps.)527 b(If)367 b Fo(T)536 b
Ft(halts)368 b(on)g(all)e(these)i(mac)-34 b(hines)368
b(and)g(returns)f(the)3030 48086 y(same)404 b(answ)-34
b(er)405 b Fo(\033)43 b Ft(,)404 b(then)h(halt)g(and)f(return)h
Fo(\033)43 b Ft(.)538 b(Otherwise,)404 b(con)-34 b(tin)g(ue.)1882
50588 y(It)545 b(is)h(clear)e(that)j(if)e Fo(S)616 b
Ft(halts)546 b(then)h(it)e(returns)h(the)g(same)g(answ)-34
b(er)546 b(as)g Fo(T)168 b Ft(\()p Fo(M)132 b Ft(\).)964
b(This)546 b(is)f(b)34 b(ecause)546 b Fo(M)677 b Ft(is)0
52094 y Fo(n)p Ft(-compatible)404 b(with)h(itself)f(for)g(all)g
Fo(n)f Ft(and)i(so)f Fo(M)469 b Fp(2)337 b Fo(S)25072
52276 y Fj(n)26102 52094 y Ft(for)405 b(all)e Fo(n)p
Ft(.)1882 53599 y(W)-101 b(e)422 b(claim)h(that)h Fo(S)493
b Ft(alw)-34 b(a)g(ys)424 b(halts.)595 b(F)-101 b(or)423
b(an)-34 b(y)424 b(mac)-34 b(hine)423 b Fo(N)555 b Ft(of)424
b(size)e Fp(j)p Fo(M)132 b Fp(j)422 b Ft(suc)-34 b(h)424
b(that)h([)p Fo(N)132 b Ft(])367 b Fp(6\021)h Ft([)p
Fo(M)132 b Ft(])423 b(,)k(there's)c(a)0 55104 y(n)-34
b(um)g(b)34 b(er)355 b Fo(n)5092 54665 y FG(0)5756 55104
y Ft(suc)-34 b(h)355 b(that)g Fo(n)e Ft(is)h(not)g(in)g
Fo(S)17477 55286 y Fj(n)18457 55104 y Ft(for)g(all)g
Fo(n)336 b(>)g(n)24967 54665 y FG(0)25278 55104 y Ft(.)521
b(Since)354 b(there)g(are)f(only)h(\014nitely)g(man)-34
b(y)354 b(suc)-34 b(h)355 b(mac)-34 b(hines,)0 56610
y(there's)372 b(a)h(n)-34 b(um)g(b)34 b(er)373 b Fo(n)9973
56170 y FG(00)10911 56610 y Ft(suc)-34 b(h)374 b(that)g(all)e(the)h
(mac)-34 b(hines)373 b Fo(N)468 b Fp(2)337 b Fo(S)28441
56792 y Fj(n)29440 56610 y Ft(for)373 b Fo(n)336 b(>)h(n)34336
56170 y FG(00)35273 56610 y Ft(satisfy)373 b([)p Fo(N)132
b Ft(])337 b Fp(\021)f Ft([)p Fo(M)132 b Ft(].)527 b(F)-101
b(or)373 b(an)-34 b(y)373 b(suc)-34 b(h)0 58115 y(mac)g(hine)446
b Fo(N)577 b Ft(with)447 b([)p Fo(N)132 b Ft(])405 b
Fp(\021)g Ft([)p Fo(M)132 b Ft(])445 b(,)456 b Fo(T)614
b Ft(halts)446 b(after)g(a)f(\014nite)h(n)-34 b(um)g(b)34
b(er)447 b(of)f(steps)g(and)g(outputs)i(the)e(same)f(answ)-34
b(er)0 59621 y(as)385 b Fo(T)168 b Ft(\()p Fo(M)132 b
Ft(\).)533 b(Again,)389 b(since)384 b(there)h(are)f(only)h(\014nitely)f
(man)-34 b(y)386 b(of)f(them)g(,)j(there's)d(a)f(n)-34
b(um)g(b)34 b(er)386 b Fo(n)336 b(>)h(n)44852 59181 y
FG(00)45802 59621 y Ft(suc)-34 b(h)385 b(that)h Fo(T)0
61126 y Ft(halts)405 b(on)f(all)g(the)g(mac)-34 b(hines)405
b(of)f Fo(S)15773 61308 y Fj(n)16804 61126 y Ft(in)g
Fo(n)g Ft(steps)g(and)h(returns)g(the)f(same)g(answ)-34
b(er)405 b(as)f Fo(T)168 b Ft(\()p Fo(M)132 b Ft(\).)p
51114 61126 886 886 v 1882 63960 a(Our)421 b(previous)h(pro)34
b(of)422 b(relied)e(hea)-34 b(vily)421 b(on)h(the)g(fact)g(that)h(the)f
(sim)-34 b(ulator)422 b(w)-34 b(as)422 b(giv)-34 b(en)422
b(an)g(upp)34 b(er)422 b(b)34 b(ound)423 b(on)0 65466
y(the)360 b(size)e(of)h(the)h(mac)-34 b(hine)359 b Fo(M)132
b Ft(.)524 b(While)358 b(in)i(the)f(con)-34 b(text)360
b(of)f(complexit)-34 b(y)359 b(w)-34 b(e)360 b(ga)-34
b(v)g(e)359 b(this)h(length)f(to)h(the)f(sim)-34 b(ulator)0
66971 y(to)373 b(allo)-34 b(w)373 b(it)g(enough)h(running)g(time,)379
b(one)373 b(ma)-34 b(y)373 b(w)-34 b(onder)374 b(whether)f(it)g(is)g
(justi\014able)h(to)f(giv)-34 b(e)373 b(this)g(b)34 b(ound)374
b(to)f(the)0 68477 y(sim)-34 b(ulator)405 b(in)f(the)g(computabilit)-34
b(y)405 b(con)-34 b(text.)540 b(That)405 b(is:)25394
74521 y(38)p eop
%%Page: 39 40
39 39 bop 0 1218 a Fz(Conjecture)465 b(A.3)f(\(Rice's)h(Theorem)f(|)h
(third)h(generalization\))606 b Fr(L)-62 b(et)430 b Ft(\005)337
b(=)f(\(\005)41304 1406 y Fj(Y)42115 1218 y Fo(;)202
b Ft(\005)43563 1406 y Fj(N)44459 1218 y Ft(\))431 b
Fr(b)-62 b(e)430 b(a)h(pr)-62 b(omise)0 2723 y(pr)g(oblem)463
b(close)-62 b(d)463 b(under)h Ft([)p Fp(\001)p Ft(])p
Fr(.)651 b(Supp)-62 b(ose)463 b(that)g Ft(\005)i Fr(is)g(de)-62
b(cidable.)647 b(Then)465 b Ft(\005)394 b(=)465 b Fr(is)f(trivial)f(in)
i(the)f(sense)g(that)f(ther)-62 b(e)0 4229 y(exists)432
b(a)h(T)-93 b(uring)434 b(machine)e Fo(S)504 b Fr(such)432
b(that)19921 6952 y Fo(M)469 b Fp(2)337 b Ft(\005)23620
7140 y Fj(Y)24767 6952 y Fp(\))f Fo(S)27128 6451 y FG(h)p
Fj(M)94 b FG(i)28914 6952 y Ft(\(\))337 b(=)g(1)19878
9675 y Fo(M)469 b Fp(2)337 b Ft(\005)23577 9863 y Fj(N)24810
9675 y Fp(\))g Fo(S)27172 9174 y FG(h)p Fj(M)94 b FG(i)28957
9675 y Ft(\(\))337 b(=)g(0)0 12177 y(It)404 b(turns)h(out)g(that)g
(this)g(small)f(c)-34 b(hange)404 b(mak)-34 b(es)405
b(a)f(di\013erence.)0 14678 y Fz(Theorem)464 b(A.4)606
b Fr(Conje)-62 b(ctur)g(e)432 b(A.3)h(is)g(false.)0 17512
y Fz(Pro)39 b(of:)1468 b Ft(Consider)404 b(the)h(follo)-34
b(wing)405 b(promise)f(problem)g(\005)337 b(=)f(\(\005)30651
17700 y Fj(Y)31461 17512 y Fo(;)202 b Ft(\005)32909 17700
y Fj(N)33806 17512 y Ft(\):)10205 20235 y(\005)11114
20423 y Fj(Y)12261 20235 y Ft(=)336 b Fp(f)p Fo(M)f Fp(j)202
b Fo(M)535 b Ft(alw)-34 b(a)g(ys)405 b(halts,)g Fp(9)p
Fo(x)336 b(<)h Ft(K)-34 b(C\([)p Fo(M)132 b Ft(]\))405
b(s.t.)538 b([)p Fo(M)132 b Ft(]\()p Fo(x)p Ft(\))337
b(=)f(1)q Fp(g)14776 22958 y Ft(\005)15685 23146 y Fj(N)16918
22958 y Ft(=)g Fp(f)p Fo(M)f Fp(j)202 b Fo(M)535 b Ft(alw)-34
b(a)g(ys)405 b(halts,)g Fp(8)p Fo(x)336 b(M)132 b Ft(\()p
Fo(x)p Ft(\))338 b(=)e(0)p Fp(g)0 25183 y Ft(where)516
b(for)g(a)g(partial)g(recursiv)-34 b(e)515 b(function)j
Fo(f)130 b Ft(,)544 b(K)-34 b(C)q(\()p Fo(f)130 b Ft(\))517
b(is)f(the)h(description)f(length)h(of)f(the)g(smallest)g(T)-101
b(uring)0 26689 y(mac)-34 b(hine)404 b(that)i(computes)f
Fo(f)130 b Ft(.)539 b(It)404 b(is)g(ob)-34 b(vious)405
b(that)g(\005)f(is)g(closed)g(under)h([)p Fp(\001)p Ft(].)1882
28194 y(W)-101 b(e)420 b(claim)g(that)i(\005)f(is)f(decidable.)588
b(Indeed,)425 b(consider)c(the)g(follo)-34 b(wing)421
b(T)-101 b(uring)422 b(mac)-34 b(hine)421 b Fo(T)168
b Ft(:)572 b(On)421 b(input)h Fo(M)132 b Ft(,)0 29700
y Fo(T)598 b Ft(in)-34 b(v)g(ok)g(es)429 b Fo(M)132 b
Ft(\()p Fo(x)p Ft(\))429 b(for)g(all)g Fo(x)378 b(<)f
Fp(j)p Fo(M)132 b Fp(j)429 b Ft(and)h(returns)f(1)g(i\013)g(it)g(gets)g
(a)g(non-zero)g(answ)-34 b(er.)613 b(Since)429 b(an)-34
b(y)430 b(mac)-34 b(hine)429 b(in)0 31205 y(\005)909
31393 y Fj(Y)2045 31205 y Fp([)326 b Ft(\005)4088 31393
y Fj(N)5473 31205 y Ft(alw)-34 b(a)g(ys)490 b(halts,)510
b Fo(T)657 b Ft(halts)490 b(in)f(\014nite)g(time.)793
b(If)488 b Fo(T)658 b Ft(returns)489 b(1)g(then)h(certainly)e
Fo(M)621 b Ft(is)488 b(not)i(in)f(\005)49166 31393 y
Fj(N)50062 31205 y Ft(.)793 b(If)0 32711 y Fo(M)469 b
Fp(2)337 b Ft(\005)3699 32899 y Fj(Y)4912 32711 y Ft(then)405
b Fo(M)132 b Ft(\()p Fo(x)p Ft(\))338 b(=)e(1)404 b(for)g(some)h
Fo(x)336 b(<)g Ft(K)-34 b(C)q(\([)p Fo(M)132 b Ft(]\))337
b Fp(\024)f(j)p Fo(M)132 b Fp(j)404 b Ft(and)h(so)f Fo(T)573
b Ft(returns)404 b(1.)1882 34216 y(W)-101 b(e)359 b(claim)g(that)i
(\005)f(is)g(not)g(trivial)f(in)h(the)g(sense)g(of)g(Conjecture)h(A.3.)
524 b(Indeed,)368 b(supp)34 b(ose)361 b(for)f(con)-34
b(tradiction)0 35722 y(that)405 b(there)f(exists)g(a)g(sim)-34
b(ulator)405 b Fo(S)474 b Ft(suc)-34 b(h)405 b(that)19921
38445 y Fo(M)469 b Fp(2)337 b Ft(\005)23620 38633 y Fj(Y)24767
38445 y Fp(\))f Fo(S)27128 37944 y FG(h)p Fj(M)94 b FG(i)28914
38445 y Ft(\(\))337 b(=)g(1)19878 41168 y Fo(M)469 b
Fp(2)337 b Ft(\005)23577 41356 y Fj(N)24810 41168 y Fp(\))g
Fo(S)27172 40667 y FG(h)p Fj(M)94 b FG(i)28957 41168
y Ft(\(\))337 b(=)g(0)1882 43393 y(Consider)404 b(the)h(mac)-34
b(hine)404 b Fo(Z)492 b Ft(whic)-34 b(h)405 b(reads)f(its)g(input)h
(and)g(then)g(returns)g(0.)538 b(W)-101 b(e)404 b(ha)-34
b(v)g(e)404 b(that)18530 46467 y Fp(h)p Fo(Z)87 b Fp(i)p
Ft(\(1)21463 45967 y Fj(t)21860 46467 y Fo(;)202 b(x)p
Ft(\))336 b(=)25179 45122 y Ff(n)26189 45866 y Fp(?)1212
b Fo(t)336 b(<)h Fp(j)p Fo(x)p Fp(j)26189 47371 y Ft(0)1549
b(otherwise)0 49804 y(As)497 b Fo(Z)578 b Fp(2)491 b
Ft(\005)5497 49992 y Fj(N)6394 49804 y Ft(,)519 b(w)-34
b(e)497 b(kno)-34 b(w)498 b(that)g Fo(S)15917 49364 y
FG(h)p Fj(Z)63 b FG(i)17404 49804 y Ft(\(\))497 b(will)g(halt)g(after)g
(a)f(\014nite)h(time)g(and)g(return)g(0.)817 b(Let)496
b Fo(n)g Ft(b)34 b(e)497 b(an)g(upp)34 b(er)0 51309 y(b)g(ound)405
b(on)g Fp(j)p Fo(x)p Fp(j)e Ft(and)i Fo(t)f Ft(o)-34
b(v)g(er)404 b(all)f(oracle)g(queries)g(\(1)23335 50869
y Fj(t)23731 51309 y Fo(;)202 b(x)p Ft(\))404 b(of)g
Fo(S)28031 50869 y FG(h)p Fj(Z)63 b FG(i)29519 51309
y Ft(\(\).)1882 52815 y(Let)481 b Fo(r)514 b Ft(b)34
b(e)481 b(a)g(string)g(of)h(Kolmogoro)-34 b(v)481 b(complexit)-34
b(y)480 b(2)p Fo(n)p Ft(.)769 b(Consider)482 b(the)f(mac)-34
b(hine)482 b Fo(N)41646 52997 y Fj(n;r)43465 52815 y
Ft(whic)-34 b(h)482 b(computes)0 54320 y(the)405 b(follo)-34
b(wing)404 b(function,)18889 56948 y Fo(N)19863 57130
y Fj(n;r)21202 56948 y Ft(\()p Fo(x)p Ft(\))337 b(=)24453
54875 y Ff(\()25632 55448 y Ft(0)1212 b Fo(x)336 b Fp(\024)h
Fo(n)25632 56953 y Ft(1)1212 b Fo(x)336 b Ft(=)h Fo(n)269
b Ft(+)g(1)25632 58459 y Fo(r)1271 b(x)336 b Fp(\025)h
Fo(n)269 b Ft(+)g(2)0 60363 y(and)405 b(runs)g(in)f(time)g
Fp(j)p Fo(x)p Fp(j)f Ft(on)h(inputs)i Fo(x)e Ft(suc)-34
b(h)405 b(that)g Fp(j)p Fo(x)p Fp(j)336 b(\024)g Fo(n)p
Ft(.)1882 61868 y(F)-101 b(or)418 b(an)-34 b(y)418 b
Fo(t;)202 b Fp(j)p Fo(x)p Fp(j)358 b(\024)i Fo(n)p Ft(,)420
b Fp(h)p Fo(Z)87 b Fp(i)p Ft(\(1)14797 61428 y Fj(t)15194
61868 y Fo(;)202 b(x)p Ft(\))359 b(=)h Fp(h)p Fo(N)20004
62050 y Fj(n;r)21342 61868 y Fp(i)p Ft(\(1)22890 61428
y Fj(t)23286 61868 y Fo(;)202 b(x)p Ft(\).)580 b(Therefore)418
b Fo(S)32228 61428 y FG(h)p Fj(N)33341 61539 y Fc(n;r)34566
61428 y FG(i)34987 61868 y Ft(\(\))361 b(=)e Fo(S)38405
61428 y FG(h)p Fj(Z)63 b FG(i)39893 61868 y Ft(\(\))360
b(=)f(0.)580 b(But)418 b Fo(N)47415 62050 y Fj(n;r)49113
61868 y Fp(2)360 b Ft(\005)51190 62056 y Fj(Y)0 63374
y Ft(since)490 b Fo(N)4030 63556 y Fj(n;r)5368 63374
y Ft(\()p Fo(n)326 b Ft(+)h(1\))480 b(=)f(1)490 b(and)h(K)-34
b(C)q(\([)p Fo(N)18248 63556 y Fj(n;r)19586 63374 y Ft(]\))480
b Fo(>)f(n)326 b Ft(+)h(1.)796 b(This)490 b(con)-34 b(tradicts)491
b(the)f(assumption)i(that)f Fo(S)560 b Ft(decides)0 64879
y(\005.)p 51114 64879 886 886 v 25394 74521 a(39)p eop
%%Page: 40 41
40 40 bop 0 1218 a Fs(B)1793 b(Pseudorandom)599 b(Oracles)0
3923 y Ft(In)387 b(this)g(section,)j(w)-34 b(e)388 b(sk)-34
b(etc)g(h)387 b(a)g(pro)34 b(of)387 b(of)h(the)f(follo)-34
b(wing)387 b(lemma,)j(whic)-34 b(h)388 b(states)g(that)g(a)e(random)i
(function)g(is)f(a)0 5429 y(pseudorandom)406 b(generator)e(relativ)-34
b(e)403 b(to)h(itself)h(with)f(high)h(probabilit)-34
b(y)-101 b(.)0 7928 y Fz(Lemma)464 b(4.17)606 b Fr(Ther)-62
b(e)475 b(is)g(a)g(c)-62 b(onstant)474 b Fo(\016)459
b(>)413 b Ft(0)475 b Fr(such)f(that)h(the)f(fol)62 b(lowing)474
b(holds)g(for)h(al)62 b(l)475 b(su\016ciently)e(lar)-62
b(ge)474 b Fo(K)0 9433 y Fr(and)525 b(any)h Fo(L)503
b Fp(\025)h Fo(K)8753 8994 y Fl(2)9279 9433 y Fr(.)834
b(L)-62 b(et)525 b Fo(D)559 b Fr(b)-62 b(e)525 b(an)h(algorithm)d(that)
i(makes)f(at)h(most)g Fo(K)35183 8994 y Fj(\016)36214
9433 y Fr(or)-62 b(acle)524 b(queries)f(and)j(let)f Fo(G)g
Fr(b)-62 b(e)525 b(a)0 10939 y(r)-62 b(andom)433 b(inje)-62
b(ctive)431 b(function)i Fo(G)336 b Ft(:)h([)p Fo(K)87
b Ft(])336 b Fp(!)h Ft([)p Fo(L)p Ft(])p Fr(.)557 b(Then)433
b(with)g(pr)-62 b(ob)g(ability)430 b(at)j(le)-62 b(ast)432
b Ft(1)269 b Fp(\000)g Ft(2)40589 10499 y FG(\000)p Fj(K)42177
10187 y Fc(\016)43121 10939 y Fr(over)432 b Fo(G)p Fr(,)12708
12236 y Ff(\014)12708 12963 y(\014)12708 13690 y(\014)12708
14417 y(\014)13730 13993 y Ft(Pr)13112 14859 y Fj(x)p
FG(2)p Fl([)p Fj(K)63 b Fl(])15850 13011 y Ff(\002)16355
13993 y Fo(D)17393 13493 y Fj(G)18181 13993 y Ft(\()p
Fo(G)p Ft(\()p Fo(x)p Ft(\)\))338 b(=)e(1)23934 13011
y Ff(\003)24709 13993 y Fp(\000)762 b Ft(Pr)25921 14859
y Fj(y)32 b FG(2)p Fl([)p Fj(L)p Fl(])28410 13011 y Ff(\002)28915
13993 y Fo(D)29953 13493 y Fj(G)30740 13993 y Ft(\()p
Fo(y)43 b Ft(\))339 b(=)d(1)34543 13011 y Ff(\003)35048
12236 y(\014)35048 12963 y(\014)35048 13690 y(\014)35048
14417 y(\014)35789 13993 y Fp(\024)37709 13173 y Ft(1)p
37201 13714 1622 49 v 37201 14825 a Fo(K)38317 14475
y Fj(\016)38955 13993 y Fo(:)11159 b Ft(\(4\))1882 17148
y(W)-101 b(e)529 b(pro)-34 b(v)g(e)531 b(the)f(lemma)g(via)g(a)g(coun)
-34 b(ting)532 b(argumen)-34 b(t)531 b(in)f(the)h(st)-34
b(yle)530 b(of)g(Gennaro)h(and)g(T)-101 b(revisan's)530
b(pro)34 b(of)0 18653 y(that)350 b(a)g(random)f(p)34
b(erm)-34 b(utation)351 b(is)e(one-w)-34 b(a)g(y)350
b(against)g(non)-34 b(uniform)351 b(adv)-34 b(ersaries)349
b([GT00].)520 b(Sp)34 b(eci\014cally)-101 b(,)359 b(w)-34
b(e)349 b(will)0 20159 y(sho)-34 b(w)449 b(that)g(\\most")f
Fo(G)f Ft(for)h(whic)-34 b(h)449 b(Inequalit)-34 b(y)447
b(\(4\))h(fails)g(ha)-34 b(v)g(e)448 b(a)g(\\short")g(description)g
(giv)-34 b(en)447 b Fo(D)34 b Ft(,)458 b(and)448 b(hence)0
21664 y(there)404 b(cannot)h(b)34 b(e)404 b(to)34 b(o)404
b(man)-34 b(y)405 b(of)f(them.)1882 23170 y(Let)489 b
Fp(G)561 b Ft(b)34 b(e)489 b(the)g(collection)g(of)g
Fo(G)p Ft('s)g(for)g(whic)-34 b(h)490 b(Inequalit)-34
b(y)489 b(\(4\))h(fails)f(\(for)h(a)f(su\016cien)-34
b(tly)490 b(small)f Fo(\016)46 b Ft(,)509 b(whose)0 24675
y(v)-67 b(alue)346 b(is)g(implicit)g(in)h(the)g(pro)34
b(of)346 b(b)34 b(elo)-34 b(w\).)520 b(W)-101 b(e)346
b(b)34 b(egin)347 b(b)-34 b(y)347 b(arguing)g(that,)358
b(for)347 b(ev)-34 b(ery)346 b Fo(G)336 b Fp(2)h(G)72
b Ft(,)358 b(there)346 b(is)g(a)h(large)f(set)0 26181
y Fo(S)743 26369 y Fj(G)1868 26181 y Fp(\032)337 b Ft([)p
Fo(K)87 b Ft(])331 b(of)g(inputs)i(on)e(whic)-34 b(h)332
b Fo(D)34 b Ft('s)330 b(b)34 b(eha)-34 b(vior)331 b(is)g(\\indep)34
b(enden)-34 b(t,")347 b(in)331 b(the)h(sense)f(that)h(for)g
Fo(x)k Fp(2)h Fo(S)70 b Ft(,)345 b(none)332 b(of)f(the)0
27686 y(oracle)407 b(queries)h(made)h(in)f(the)h(execution)f(of)h
Fo(D)22193 27246 y Fj(G)22981 27686 y Ft(\()p Fo(G)p
Ft(\()p Fo(x)p Ft(\)\))h(are)e(at)g(p)34 b(oin)-34 b(ts)410
b(in)e Fo(S)70 b Ft(,)410 b(y)-34 b(et)408 b Fo(D)442
b Ft(still)408 b(has)h(nonnegligible)0 29192 y(adv)-67
b(an)-34 b(tage)439 b(in)g(distinguishing)h Fo(G)p Ft(\()p
Fo(x)p Ft(\))f(from)g(random.)642 b(Actually)-101 b(,)447
b(w)-34 b(e)439 b(will)f(not)h(b)34 b(e)438 b(able)h(to)g(a\013ord)f
(sp)34 b(ecifying)0 30697 y Fo(S)743 30885 y Fj(G)2072
30697 y Ft(when)542 b(w)-34 b(e)541 b(\\describ)34 b(e")539
b Fo(G)p Ft(,)575 b(so)540 b(w)-34 b(e)541 b(actually)f(sho)-34
b(w)542 b(that)g(there)e(is)h(a)f(\014xed)h(set)g Fo(S)610
b Ft(\(indep)34 b(enden)-34 b(t)543 b(of)e Fo(G)p Ft(\))0
32203 y(suc)-34 b(h)422 b(that)h(for)f(most)g Fo(G)p
Ft(,)k(the)c(desired)f(set)h Fo(S)20864 32391 y Fj(G)22074
32203 y Ft(can)g(b)34 b(e)421 b(obtained)i(b)-34 b(y)422
b(just)h(thro)-34 b(wing)423 b(out)f(a)g(small)f(n)-34
b(um)g(b)34 b(er)423 b(of)0 33708 y(elemen)-34 b(ts)404
b(from)h Fo(S)70 b Ft(.)0 36207 y Fz(Claim)464 b(B.1)606
b Fr(Ther)-62 b(e)463 b(is)g(a)h(set)f Fo(S)e Fp(\032)392
b Ft([)p Fo(K)87 b Ft(])463 b Fr(with)h Fp(j)p Fo(S)70
b Fp(j)391 b Ft(=)g Fo(K)26401 35767 y Fl(1)p FG(\000)p
Fl(5)p Fj(\016)28579 36207 y Fr(,)471 b(and)464 b Fp(G)32600
35767 y FG(0)33302 36207 y Fp(\032)391 b(G)536 b Fr(with)464
b Fp(jG)39687 35767 y FG(0)39997 36207 y Fp(j)391 b Ft(=)h
Fp(jG)72 b(j)p Fo(=)p Ft(2)463 b Fr(such)f(that)h(for)0
37713 y(al)62 b(l)433 b Fo(G)336 b Fp(2)h(G)4962 37273
y FG(0)5273 37713 y Fr(,)433 b(ther)-62 b(e)432 b(is)h(a)g(set)g
Fo(S)14085 37901 y Fj(G)15210 37713 y Fp(\032)337 b Fo(S)504
b Fr(with)433 b(the)f(fol)62 b(lowing)432 b(pr)-62 b(op)g(erties:)1433
39935 y(1.)605 b Fp(j)p Fo(S)4110 40123 y Fj(G)4899 39935
y Fp(j)336 b Ft(=)h(\(1)269 b Fp(\000)h Fo(\015)67 b
Ft(\))p Fp(j)p Fo(S)j Fp(j)p Fr(,)433 b(wher)-62 b(e)433
b Fo(\015)403 b Ft(=)337 b Fo(K)19719 39495 y FG(\000)p
Fl(3)p Fj(\016)21426 39935 y Fr(.)1433 42326 y(2.)605
b(If)434 b Fo(x)336 b Fp(2)h Fo(S)7220 42514 y Fj(G)8009
42326 y Fr(,)433 b(then)g Fo(D)12547 41886 y Fj(G)13335
42326 y Ft(\()p Fo(G)p Ft(\()p Fo(x)p Ft(\)\))i Fr(never)e(queries)e
(its)i(or)-62 b(acle)432 b(at)h(an)g(element)g(of)g Fo(S)39835
42514 y Fj(G)40624 42326 y Fr(.)1433 44716 y(3.)13897
44826 y Ff(\014)13897 45554 y(\014)13897 46281 y(\014)13897
47008 y(\014)14860 46584 y Ft(Pr)14301 47391 y Fj(x)p
FG(2)p Fj(S)16030 47540 y Fc(G)16920 45602 y Ff(\002)17425
46584 y Fo(D)18463 46083 y Fj(G)19251 46584 y Ft(\()p
Fo(G)p Ft(\()p Fo(x)p Ft(\)\))338 b(=)e(1)25004 45602
y Ff(\003)25779 46584 y Fp(\000)845 b Ft(Pr)26991 47391
y Fj(y)32 b FG(2)p Fj(L)28755 47540 y Fc(G)29645 45602
y Ff(\002)30150 46584 y Fo(D)31188 46083 y Fj(G)31976
46584 y Ft(\()p Fo(y)43 b Ft(\))338 b(=)f(1)35779 45602
y Ff(\003)36284 44826 y(\014)36284 45554 y(\014)36284
46281 y(\014)36284 47008 y(\014)37025 46584 y Fo(>)39247
45764 y Ft(1)p 38437 46305 2228 49 v 38437 47416 a(2)p
Fo(K)40159 47066 y Fj(\016)40797 46584 y Fo(;)3030 49822
y Fr(wher)-62 b(e)412 b Fo(L)7257 50010 y Fj(G)8268 49134
y Fl(def)8446 49822 y Ft(=)401 b([)p Fo(L)p Ft(])222
b Fp(n)h Fo(G)p Ft(\([)p Fo(K)87 b Ft(])223 b Fp(n)h
Fo(S)17350 50010 y Fj(G)18138 49822 y Ft(\))p Fr(.)551
b(\(Note)411 b(that)g Fo(L)26203 50010 y Fj(G)27403 49822
y Fr(c)-62 b(ontains)411 b(mor)-62 b(e)412 b(than)g(a)g
Ft(1)223 b Fp(\000)g Fo(K)19 b(=L)413 b Fr(fr)-62 b(action)411
b(of)g Fo(L)p Fr(.\))0 52491 y Fz(Pro)39 b(of:)1722 b
Ft(First)463 b(consider)h(c)-34 b(ho)34 b(osing)463 b(b)34
b(oth)464 b(a)g(random)g Fo(G)28347 51804 y Fw(R)28034
52491 y Fp( )309 b(G)535 b Ft(and)465 b(a)e(random)h
Fo(S)533 b Ft(\(among)464 b(subsets)h(of)e([)p Fo(K)87
b Ft(])0 53997 y(of)487 b(size)f Fo(K)4958 53557 y Fl(1)p
FG(\000)p Fl(5)p Fj(\016)7136 53997 y Ft(\).)787 b(W)-101
b(e)487 b(will)f(sho)-34 b(w)488 b(that)g(with)g(probabilit)-34
b(y)487 b(at)h(least)e(1)p Fo(=)p Ft(2,)508 b(there)487
b(is)f(a)h(go)34 b(o)g(d)487 b(subset)h Fo(S)48506 54185
y Fj(G)49769 53997 y Fp(\032)475 b Fo(S)0 55502 y Ft(satisfying)458
b(Prop)34 b(erties)456 b(1{3.)698 b(By)456 b(a)-34 b(v)g(eraging,)471
b(this)457 b(implies)g(that)h(there)f(is)g(a)h(\014xed)f(set)g
Fo(S)527 b Ft(for)458 b(whic)-34 b(h)458 b(a)f(go)34
b(o)g(d)0 57008 y(subset)462 b(exists)g(for)f(at)h(least)f(half)h(the)g
Fo(G)432 b Fp(2)h(G)72 b Ft(,)475 b(as)461 b(desired.)711
b(Let's)460 b(b)34 b(egin)462 b(with)g(Prop)34 b(ert)-34
b(y)461 b(2.)710 b(F)-101 b(or)462 b(a)f(random)0 58513
y Fo(G)p Ft(,)d Fo(S)70 b Ft(,)458 b(and)448 b(a)f(random)h
Fo(x)408 b Fp(2)h Fo(S)70 b Ft(,)458 b(note)448 b(that)g
Fo(D)21670 58073 y Fj(G)22458 58513 y Ft(\()p Fo(G)p
Ft(\()p Fo(x)p Ft(\)\))h(initially)d(has)i(no)f(information)i(ab)34
b(out)448 b Fo(S)70 b Ft(,)458 b(whic)-34 b(h)448 b(is)f(a)0
60018 y(random)484 b(set)g(of)g(densit)-34 b(y)484 b
Fo(K)13336 59579 y FG(\000)p Fl(5)p Fj(\016)15043 60018
y Ft(.)776 b(Since)484 b Fo(D)516 b Ft(mak)-34 b(es)484
b(at)f(most)i Fo(K)30370 59579 y Fj(\016)31358 60018
y Ft(queries,)502 b(the)484 b(probabilit)-34 b(y)484
b(that)g(it)g(queries)0 61524 y(its)422 b(oracle)f(at)h(some)f(elemen)
-34 b(t)422 b(of)g Fo(S)492 b Ft(is)421 b(at)i(most)f
Fo(K)23752 61084 y Fj(\016)24538 61524 y Fp(\001)281
b Fo(K)26272 61084 y FG(\000)p Fl(5)p Fj(\016)28345 61524
y Ft(=)366 b Fo(K)30770 61084 y FG(\000)p Fl(4)p Fj(\016)32477
61524 y Ft(.)592 b(Th)-34 b(us,)427 b(with)c(probabilit)-34
b(y)422 b(at)g(least)g(3)p Fo(=)p Ft(4)0 63029 y(o)-34
b(v)g(er)368 b Fo(G)h Ft(and)g Fo(S)70 b Ft(,)375 b Fo(D)8766
62590 y Fj(G)9554 63029 y Ft(\()p Fo(G)p Ft(\()p Fo(x)p
Ft(\)\))370 b(queries)d(its)i(oracle)e(at)i(an)g(elemen)-34
b(t)369 b(of)f Fo(S)439 b Ft(for)368 b(at)h(most)g(a)g(4)p
Fo(=K)42191 62590 y FG(\000)p Fl(4)p Fj(\016)44235 63029
y Fo(<)336 b(\015)436 b Ft(fraction)369 b(of)0 64535
y Fo(x)336 b Fp(2)h Fo(S)70 b Ft(.)539 b(Thro)-34 b(wing)405
b(out)g(this)g Fo(\015)471 b Ft(fraction)404 b(of)h(elemen)-34
b(ts)404 b(of)g Fo(S)475 b Ft(giv)-34 b(es)403 b(a)h(set)h
Fo(S)35003 64723 y Fj(G)36196 64535 y Ft(satisfying)f(Prop)34
b(erties)403 b(1)i(and)f(2.)1882 66040 y(No)-34 b(w)270
b(let's)f(turn)h(to)g(Prop)34 b(ert)-34 b(y)269 b(3.)494
b(By)269 b(a)g(Cherno\013-lik)-34 b(e)269 b(b)34 b(ound,)298
b(with)270 b(probabilit)-34 b(y)270 b(at)g(least)f(1)p
Fp(\000)p Ft(exp\(\012\()p Fo(K)49979 65600 y Fl(1)p
FG(\000)p Fl(5)p Fj(\016)52158 66040 y Fp(\001)0 67546
y Ft(\()p Fo(K)1587 67106 y FG(\000)p Fj(\016)2824 67546
y Ft(\))3295 67106 y Fl(2)3821 67546 y Ft(\)\))338 b
Fo(>)e Ft(3)p Fo(=)p Ft(4)405 b(o)-34 b(v)g(er)404 b(the)g(c)-34
b(hoice)404 b(of)g Fo(S)70 b Ft(,)11684 68842 y Ff(\014)11684
69570 y(\014)11684 70297 y(\014)11684 71024 y(\014)12327
70600 y Ft(Pr)12088 71407 y Fj(x)p FG(2)p Fj(S)14069
69618 y Ff(\002)14574 70600 y Fo(D)15612 70100 y Fj(G)16399
70600 y Ft(\()p Fo(G)p Ft(\()p Fo(x)p Ft(\)\))338 b(=)f(1)22153
69618 y Ff(\003)22927 70600 y Fp(\000)887 b Ft(Pr)24139
71466 y Fj(x)p FG(2)p Fl([)p Fj(K)63 b Fl(])26877 69618
y Ff(\002)27382 70600 y Fo(D)28420 70100 y Fj(G)29208
70600 y Ft(\()p Fo(G)p Ft(\()p Fo(x)p Ft(\)\))338 b(=)e(1)34961
69618 y Ff(\003)35467 68842 y(\014)35467 69570 y(\014)35467
70297 y(\014)35467 71024 y(\014)36207 70600 y Fp(\024)38430
69780 y Ft(1)p 37620 70321 V 37620 71432 a(4)p Fo(K)39342
71082 y Fj(\016)39980 70600 y Fo(:)25394 74521 y Ft(40)p
eop
%%Page: 41 42
41 41 bop 0 1218 a Ft(Then)405 b(w)-34 b(e)404 b(ha)-34
b(v)g(e:)11453 2251 y Ff(\014)11453 2978 y(\014)11453
3706 y(\014)11453 4433 y(\014)12416 4009 y Ft(Pr)11857
4816 y Fj(x)p FG(2)p Fj(S)13586 4965 y Fc(G)14476 3027
y Ff(\002)14981 4009 y Fo(D)16019 3508 y Fj(G)16807 4009
y Ft(\()p Fo(G)p Ft(\()p Fo(x)p Ft(\)\))338 b(=)e(1)22560
3027 y Ff(\003)23335 4009 y Fp(\000)845 b Ft(Pr)24547
4816 y Fj(y)32 b FG(2)p Fj(L)26311 4965 y Fc(G)27201
3027 y Ff(\002)27706 4009 y Fo(D)28744 3508 y Fj(G)29532
4009 y Ft(\()p Fo(y)43 b Ft(\))338 b(=)f(1)33335 3027
y Ff(\003)33840 2251 y(\014)33840 2978 y(\014)33840 3706
y(\014)33840 4433 y(\014)12560 7360 y Fp(\025)14609 5603
y Ff(\014)14609 6330 y(\014)14609 7057 y(\014)14609 7785
y(\014)15631 7360 y Ft(Pr)15013 8227 y Fj(x)p FG(2)p
Fl([)p Fj(K)63 b Fl(])17751 6379 y Ff(\002)18256 7360
y Fo(D)19294 6860 y Fj(G)20082 7360 y Ft(\()p Fo(G)p
Ft(\()p Fo(x)p Ft(\)\))338 b(=)e(1)25835 6379 y Ff(\003)26610
7360 y Fp(\000)763 b Ft(Pr)27822 8227 y Fj(y)32 b FG(2)p
Fl([)p Fj(L)p Fl(])30311 6379 y Ff(\002)30816 7360 y
Fo(D)31854 6860 y Fj(G)32642 7360 y Ft(\()p Fo(y)43 b
Ft(\))338 b(=)e(1)36444 6379 y Ff(\003)36949 5603 y(\014)36949
6330 y(\014)36949 7057 y(\014)36949 7785 y(\014)14609
10770 y Fp(\000)15754 9012 y Ff(\014)15754 9739 y(\014)15754
10467 y(\014)15754 11194 y(\014)16717 10770 y Ft(Pr)16158
11577 y Fj(x)p FG(2)p Fj(S)17887 11726 y Fc(G)18778 9788
y Ff(\002)19283 10770 y Fo(D)20321 10269 y Fj(G)21108
10770 y Ft(\()p Fo(G)p Ft(\()p Fo(x)p Ft(\)\))i(=)f(1)26862
9788 y Ff(\003)27636 10770 y Fp(\000)770 b Ft(Pr)28848
11636 y Fj(x)p FG(2)p Fl([)p Fj(S)50 b Fl(])31352 9788
y Ff(\002)31857 10770 y Fo(D)32895 10269 y Fj(G)33683
10770 y Ft(\()p Fo(G)p Ft(\()p Fo(x)p Ft(\)\))338 b(=)e(1)39436
9788 y Ff(\003)39941 9012 y(\014)39941 9739 y(\014)39941
10467 y(\014)39941 11194 y(\014)14609 14179 y Fp(\000)15754
12421 y Ff(\014)15754 13148 y(\014)15754 13876 y(\014)15754
14603 y(\014)16398 14179 y Ft(Pr)16158 14986 y Fj(x)p
FG(2)p Fj(S)18139 13197 y Ff(\002)18644 14179 y Fo(D)19682
13678 y Fj(G)20470 14179 y Ft(\()p Fo(G)p Ft(\()p Fo(x)p
Ft(\)\))i(=)e(1)26223 13197 y Ff(\003)26998 14179 y Fp(\000)887
b Ft(Pr)28210 15045 y Fj(x)p FG(2)p Fl([)p Fj(K)63 b
Fl(])30947 13197 y Ff(\002)31453 14179 y Fo(D)32491 13678
y Fj(G)33278 14179 y Ft(\()p Fo(G)p Ft(\()p Fo(x)p Ft(\)\))338
b(=)f(1)39032 13197 y Ff(\003)39537 12421 y(\014)39537
13148 y(\014)39537 13876 y(\014)39537 14603 y(\014)14609
17588 y Fp(\000)15754 15830 y Ff(\014)15754 16558 y(\014)15754
17285 y(\014)15754 18012 y(\014)16652 17588 y Ft(Pr)16158
18454 y Fj(y)32 b FG(2)p Fl([)p Fj(L)p Fl(])18647 16606
y Ff(\002)19152 17588 y Fo(D)20190 17087 y Fj(G)20978
17588 y Ft(\()p Fo(y)43 b Ft(\))338 b(=)e(1)24780 16606
y Ff(\003)25555 17588 y Fp(\000)845 b Ft(Pr)26767 18395
y Fj(y)32 b FG(2)p Fj(L)28531 18544 y Fc(G)29421 16606
y Ff(\002)29926 17588 y Fo(D)30964 17087 y Fj(G)31752
17588 y Ft(\()p Fo(y)43 b Ft(\))338 b(=)f(1)35555 16606
y Ff(\003)36060 15830 y(\014)36060 16558 y(\014)36060
17285 y(\014)36060 18012 y(\014)12560 20355 y Fo(>)1106
b Ft(1)p Fo(=K)16937 19855 y Fj(\016)17712 20355 y Fp(\000)269
b Fo(\015)336 b Fp(\000)269 b Ft(1)p Fo(=)p Ft(4)p Fo(K)24034
19855 y Fj(\016)24809 20355 y Fp(\000)g Fo(K)19 b(=L)12560
22193 y(>)1106 b Ft(1)p Fo(=)p Ft(2)p Fo(K)17543 21692
y Fj(\016)p 51114 24621 886 886 v 1882 27159 a Ft(No)-34
b(w)405 b(w)-34 b(e)404 b(sho)-34 b(w)406 b(ho)-34 b(w)405
b(the)f(ab)34 b(o)-34 b(v)g(e)405 b(claim)e(implies)h(that)h(ev)-34
b(ery)403 b Fo(G)337 b Fp(2)f(G)33880 26720 y FG(0)34595
27159 y Ft(has)405 b(a)f(\\small")f(description.)0 29698
y Fz(Claim)464 b(B.2)606 b Fr(Every)441 b Fo(G)353 b
Fp(2)f(G)13505 29258 y FG(0)14258 29698 y Fr(c)-62 b(an)442
b(b)-62 b(e)442 b(uniquely)e(describ)-62 b(e)g(d)439
b(by)i Ft(\(log)219 b Fo(B)61 b Ft(\))277 b Fp(\000)e
Ft(\012\()p Fo(K)37204 29258 y Fl(1)p FG(\000)p Fl(7)p
Fj(\016)39383 29698 y Ft(\))442 b Fr(bits)f(given)h Fo(D)34
b Fr(,)443 b(wher)-62 b(e)442 b Fo(B)0 31204 y Fr(is)433
b(the)f(numb)-62 b(er)434 b(of)f(inje)-62 b(ctive)431
b(functions)h(fr)-62 b(om)433 b Ft([)p Fo(K)87 b Ft(])433
b Fr(to)g Ft([)p Fo(L)p Ft(])p Fr(.)0 33742 y Fz(Pro)39
b(of:)1527 b Ft(F)-101 b(or)418 b(starters,)k(the)c(description)g(of)h
Fo(G)e Ft(will)h(con)-34 b(tains)419 b(the)f(set)h Fo(S)35149
33930 y Fj(G)36355 33742 y Ft(and)g(the)f(v)-67 b(alues)418
b(of)g Fo(G)p Ft(\()p Fo(x)p Ft(\))g(for)h(all)0 35248
y Fo(x)471 b(=)-741 b Fp(2)337 b Fo(S)2917 35436 y Fj(G)3706
35248 y Ft(.)529 b(No)-34 b(w)375 b(w)-34 b(e'd)375 b(lik)-34
b(e)374 b(to)g(argue)h(that)g(this)h(information)f(is)f(enough)i(to)f
(determine)f Fo(D)41911 34808 y Fj(G)42699 35248 y Ft(\()p
Fo(y)43 b Ft(\))376 b(for)e(all)g Fo(y)43 b Ft(.)529
b(This)0 36753 y(w)-34 b(on't)463 b(exactly)d(b)34 b(e)461
b(the)h(case,)475 b(but)462 b(rather)g(w)-34 b(e'll)461
b(sho)-34 b(w)462 b(ho)-34 b(w)463 b(to)e(compute)h Fo(M)36743
36313 y Fj(G)37532 36753 y Ft(\()p Fo(y)43 b Ft(\))463
b(for)e(some)h Fo(M)593 b Ft(that)463 b(is)e(\\as)0 38259
y(go)34 b(o)g(d")404 b(as)g Fo(D)34 b Ft(.)538 b(F)-101
b(rom)404 b(Prop)34 b(ert)-34 b(y)404 b(3)g(in)g(Claim)g(B.1,)f(w)-34
b(e)405 b(ha)-34 b(v)g(e)13344 41129 y(Pr)12786 41936
y Fj(x)p FG(2)p Fj(S)14515 42085 y Fc(G)15405 40147 y
Ff(\002)15910 41129 y Fo(D)16948 40628 y Fj(G)17736 41129
y Ft(\()p Fo(G)p Ft(\()p Fo(x)p Ft(\)\))338 b(=)e(1)23489
40147 y Ff(\003)24264 41129 y Fp(\000)845 b Ft(Pr)25476
41936 y Fj(y)32 b FG(2)p Fj(L)27240 42085 y Fc(G)28130
40147 y Ff(\002)28635 41129 y Fo(D)29673 40628 y Fj(G)30461
41129 y Ft(\()p Fo(y)43 b Ft(\))338 b(=)f(1)34264 40147
y Ff(\003)35105 41129 y Fo(>)37328 40309 y Ft(1)p 36518
40850 2228 49 v 36518 41961 a(2)p Fo(K)38240 41611 y
Fj(\016)38878 41129 y Fo(:)0 44172 y Ft(\(W)-101 b(e'v)-34
b(e)323 b(dropp)34 b(ed)324 b(the)g(absolute)h(v)-67
b(alues.)510 b(The)325 b(other)e(case)g(is)h(handled)g(analogously)-101
b(,)340 b(and)324 b(the)g(only)g(cost)g(is)f(one)0 45677
y(bit)412 b(to)h(describ)34 b(e)411 b(whic)-34 b(h)412
b(case)g(holds.\))562 b(W)-101 b(e)412 b(will)f(describ)34
b(e)411 b(an)h(algorithm)g Fo(M)544 b Ft(for)412 b(whic)-34
b(h)413 b(the)f(same)g(inequalit)-34 b(y)0 47183 y(holds,)382
b(y)-34 b(et)375 b Fo(M)508 b Ft(will)375 b(only)g(use)h(the)g
(information)g(in)g(our)f(description)h(of)g Fo(G)g Ft(instead)g(of)g
(making)f(oracle)g(queries)0 48688 y(to)492 b Fo(G)p
Ft(.)800 b(Sp)34 b(eci\014cally)-101 b(,)512 b(on)492
b(input)h Fo(y)43 b Ft(,)513 b Fo(M)624 b Ft(sim)-34
b(ulates)492 b Fo(D)34 b Ft(\()p Fo(y)43 b Ft(\),)513
b(except)491 b(that)i(it)f(handles)g(eac)-34 b(h)491
b(oracle)g(query)f Fo(z)545 b Ft(as)0 50194 y(follo)-34
b(ws:)1481 52450 y(1.)606 b(If)364 b Fo(z)525 b(=)-741
b Fp(2)337 b Fo(S)7044 52638 y Fj(G)7833 52450 y Ft(,)371
b(then)365 b Fo(M)496 b Ft(resp)34 b(onds)365 b(with)g
Fo(G)p Ft(\()p Fo(z)53 b Ft(\))365 b(\(This)g(information)g(is)e
(included)i(in)f(our)g(description)g(of)h Fo(G)p Ft(\).)1481
54853 y(2.)606 b(If)420 b Fo(z)415 b Fp(2)363 b Fo(S)7151
55041 y Fj(G)7939 54853 y Ft(,)423 b(then)e Fo(M)551
b Ft(halts)420 b(and)g(outputs)i(0.)585 b(\(By)419 b(Prop)34
b(ert)-34 b(y)419 b(2)g(of)h(Claim)g(B.1,)i(this)e(cannot)g(happ)34
b(en)421 b(if)3030 56358 y Fo(y)381 b Fp(2)337 b Fo(G)p
Ft(\()p Fo(S)7317 56546 y Fj(G)8106 56358 y Ft(\),)404
b(hence)g(outputting)j(0)d(only)g(impro)-34 b(v)g(es)404
b Fo(M)132 b Ft('s)404 b(distinguishing)h(gap.\))0 58614
y(Th)-34 b(us,)405 b(giv)-34 b(en)404 b Fo(S)7315 58802
y Fj(G)8508 58614 y Ft(and)h Fo(G)p Fp(j)12155 58861
y Fl([)p Fj(K)63 b Fl(])p FG(n)p Fj(S)14575 59010 y Fc(G)15320
58614 y Ft(,)404 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)405
b(a)f(function)h Fo(M)537 b Ft(satisfying)14367 61717
y(Pr)13808 62524 y Fj(x)p FG(2)p Fj(S)15537 62673 y Fc(G)16428
61717 y Ft([)o Fo(M)132 b Ft(\()p Fo(G)p Ft(\()p Fo(x)p
Ft(\)\))339 b(=)d(1])269 b Fp(\000)846 b Ft(Pr)25644
62524 y Fj(y)32 b FG(2)p Fj(L)27408 62673 y Fc(G)28299
61717 y Ft([)o Fo(M)132 b Ft(\()p Fo(y)43 b Ft(\))339
b(=)d(1])g Fo(>)36642 60897 y Ft(1)p 35832 61438 V 35832
62549 a(2)p Fo(K)37554 62198 y Fj(\016)50451 61717 y
Ft(\(5\))0 64760 y(T)-101 b(o)422 b(complete)g(the)h(description)f(of)g
Fo(G)p Ft(,)k(w)-34 b(e)423 b(m)-34 b(ust)423 b(sp)34
b(ecify)422 b Fo(G)p Fp(j)28527 64948 y Fj(S)29099 65097
y Fc(G)29842 64760 y Ft(,)k(whic)-34 b(h)423 b(w)-34
b(e)423 b(can)f(think)g(of)h(as)f(\014rst)g(sp)34 b(ecifying)0
66265 y(the)495 b(image)e Fo(T)656 b Ft(=)486 b Fo(G)p
Ft(\()p Fo(S)10729 66453 y Fj(G)11518 66265 y Ft(\))h
Fp(\032)g Fo(L)14731 66453 y Fj(G)16013 66265 y Ft(and)495
b(then)g(the)g(bijection)f Fo(G)487 b Ft(:)f Fo(S)31534
66453 y Fj(G)32810 66265 y Fp(!)h Fo(T)168 b Ft(.)809
b(Ho)-34 b(w)g(ev)g(er,)517 b(w)-34 b(e)494 b(can)g(sa)-34
b(v)g(e)495 b(in)f(our)0 67771 y(description)404 b(b)34
b(ecause)404 b Fo(T)573 b Ft(is)404 b(constrained)h(b)-34
b(y)404 b(Inequalit)-34 b(y)404 b(\(5\),)h(whic)-34 b(h)405
b(can)f(b)34 b(e)404 b(rewritten)g(as:)15343 70708 y(Pr)15091
71515 y Fj(y)32 b FG(2)p Fj(T)17096 70708 y Ft([)o Fo(M)132
b Ft(\()p Fo(y)43 b Ft(\))339 b(=)d(1])269 b Fp(\000)846
b Ft(Pr)24361 71515 y Fj(y)32 b FG(2)p Fj(L)26125 71664
y Fc(G)27016 70708 y Ft([)o Fo(M)132 b Ft(\()p Fo(y)43
b Ft(\))339 b(=)d(1])h Fo(>)35359 69888 y Ft(1)p 34549
70429 V 34549 71540 a(2)p Fo(K)36271 71190 y Fj(\016)50451
70708 y Ft(\(6\))25394 74521 y(41)p eop
%%Page: 42 43
42 42 bop 0 1218 a Ft(Cherno\013)438 b(Bounds)h(sa)-34
b(y)437 b(that)i(most)f(large)f(subsets)i(are)e(go)34
b(o)g(d)437 b(appro)-34 b(ximators)439 b(of)f(the)g(a)-34
b(v)g(erage)437 b(of)h(a)f(b)34 b(o)g(olean)0 2723 y(function.)767
b(Sp)34 b(eci\014cally)-101 b(,)497 b(at)481 b(most)f(a)g(exp\()p
Fp(\000)p Ft(\012\(\(1)321 b Fp(\000)f Fo(\015)67 b Ft(\))p
Fo(K)27350 2283 y Fl(1)p FG(\000)p Fl(5)p Fj(\016)29848
2723 y Fp(\001)319 b Ft(\()p Fo(K)32091 2283 y FG(\000)p
Fj(\016)33329 2723 y Ft(\))33800 2283 y Fl(2)34326 2723
y Ft(\)\))463 b(=)g(exp\()p Fp(\000)p Ft(\012\()p Fo(K)42865
2283 y Fl(1)p FG(\000)p Fl(7)p Fj(\016)45044 2723 y Ft(\)\))480
b(fraction)h(of)0 4229 y(sets)404 b Fo(T)506 b Fp(\032)336
b Fo(L)5688 4417 y Fj(G)6881 4229 y Ft(of)404 b(size)f(\(1)270
b Fp(\000)f Fo(\015)67 b Ft(\))p Fo(K)15398 3789 y Fl(1)p
FG(\000)p Fl(5)p Fj(\016)17981 4229 y Ft(satisfy)404
b(Equation)h(6.)1882 5734 y(Th)-34 b(us,)380 b(using)374
b Fo(M)132 b Ft(,)379 b(w)-34 b(e)373 b(ha)-34 b(v)g(e)374
b(\\sa)-34 b(v)g(ed")374 b(\012\()p Fo(K)21862 5294 y
Fl(1)p FG(\000)p Fl(7)p Fj(\016)24040 5734 y Ft(\))f(bits)h(in)f
(describing)g Fo(G)p Ft(\()p Fo(S)36470 5922 y Fj(G)37259
5734 y Ft(\))g(\(o)-34 b(v)g(er)374 b(the)f(standard)i(\\truth-)0
7240 y(table")383 b(represen)-34 b(tation)383 b(of)g(a)f(function)i
Fo(G)p Ft(\).)532 b(Ho)-34 b(w)g(ev)g(er,)387 b(w)-34
b(e)383 b(had)g(to)g(describ)34 b(e)381 b(the)i(set)g
Fo(S)40996 7428 y Fj(G)42167 7240 y Ft(itself,)j(whic)-34
b(h)384 b(w)-34 b(ould)0 8745 y(ha)g(v)g(e)522 b(b)34
b(een)520 b(unnecessary)i(in)f(the)g(truth-table)i(represen)-34
b(tation.)890 b(F)-101 b(ortunately)g(,)551 b(w)-34 b(e)521
b(only)g(need)g(to)h(describ)34 b(e)0 10403 y Fo(S)743
10591 y Fj(G)1947 10403 y Ft(as)416 b(a)f(subset)i(of)f
Fo(S)70 b Ft(,)418 b(and)e(this)g(only)f(costs)h(log)23327
9421 y Ff(\000)25118 9865 y Fj(K)25974 9553 y Fw(1)p
Fy(\000)p Fw(5)p Fc(\016)23882 10821 y Fl(\(1)p FG(\000)p
Fj(\015)52 b Fl(\))p Fj(K)27209 10569 y Fw(1)p Fy(\000)p
Fw(5)p Fc(\016)29115 9421 y Ff(\001)30026 10403 y Ft(=)355
b Fo(O)34 b Ft(\()p Fo(H)33762 10585 y Fl(2)34287 10403
y Ft(\()p Fo(\015)67 b Ft(\))p Fo(K)37040 9963 y Fl(1)p
FG(\000)p Fl(5)p Fj(\016)39219 10403 y Ft(\))356 b Fo(<)f(O)34
b Ft(\()p Fo(K)43890 9963 y Fl(1)p FG(\000)p Fl(8)p Fj(\016)46270
10403 y Ft(log)218 b Fo(K)87 b Ft(\))416 b(bits)0 12062
y(\(where)443 b Fo(H)5023 12244 y Fl(2)5549 12062 y Ft(\()p
Fo(\015)67 b Ft(\))401 b(=)g Fo(O)34 b Ft(\()p Fo(\015)269
b Ft(log)17 b(\(1)p Fo(=\015)67 b Ft(\)\))444 b(denotes)f(the)g(binary)
g(en)-34 b(trop)g(y)444 b(function\).)656 b(So)444 b(w)-34
b(e)443 b(ha)-34 b(v)g(e)443 b(a)g(net)g(sa)-34 b(vings)444
b(of)0 13568 y(\012\()p Fo(K)2462 13128 y Fl(1)p FG(\000)p
Fl(7)p Fj(\016)4641 13568 y Ft(\))269 b Fp(\000)g Fo(O)34
b Ft(\()p Fo(K)9139 13128 y Fl(1)p FG(\000)p Fl(8)p Fj(\016)11519
13568 y Ft(log)219 b Fo(K)87 b Ft(\))337 b(=)f(\012\()p
Fo(K)18952 13128 y Fl(1)p FG(\000)p Fl(7)p Fj(\016)21131
13568 y Ft(\))404 b(bits.)p 51114 13568 886 886 v 1882
16401 a(F)-101 b(rom)457 b(Claim)h(B.2,)469 b Fp(G)12144
15962 y FG(0)12912 16401 y Ft(can)457 b(consist)h(of)g(at)g(most)g(an)f
(exp\()p Fp(\000)p Ft(\012\()p Fo(K)32683 15962 y Fl(1)p
FG(\000)p Fl(7)p Fj(\016)34862 16401 y Ft(\)\))426 b
Fo(<)f(K)38714 15962 y FG(\000)p Fj(\016)39951 16401
y Fo(=)p Ft(2)458 b(fraction)f(of)h(injectiv)-34 b(e)0
17907 y(functions)406 b([)p Fo(K)87 b Ft(])336 b Fp(!)h
Ft([)p Fo(L)p Ft(],)403 b(and)i(th)-34 b(us)405 b Fp(G)476
b Ft(has)405 b(densit)-34 b(y)405 b(smaller)e(than)i
Fo(K)31833 17467 y FG(\000)p Fj(\016)33070 17907 y Ft(,)f(as)g
(desired.)25394 74521 y(42)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF