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23 2 v 1436 2679 a Fu(2)1892 2648 y(\(1\))964 2795 y(7)p eop %%Page: 8 9 8 8 bop 0 42 a Fu(Let)583 144 y Fo(e)604 151 y Fm(i)618 144 y Fu(\()p Fo(A;)8 b(B)r Fu(\))785 119 y Fk(def)791 144 y Fu(=)47 b(2)896 125 y Fi(\000)p Fm(n)954 144 y Fn(\001)10 b(jf)p Fo(x)i Fn(2)5 b Fo(A)13 b Fu(:)f Fo(g)1180 151 y Fm(i)1194 144 y Fu(\()p Fo(x)p Fu(\))5 b Fn(2)g Fo(B)r Fn(gj)524 b Fu(\(2\))0 246 y(Th)o(us,)15 b(Eq.)f(\(1\))h(can)g (b)q(e)h(rewritten)f(as)594 305 y Fd(\014)594 330 y(\014)594 355 y(\014)594 380 y(\014)594 405 y(\014)614 348 y Fu(1)p 613 368 24 2 v 613 410 a Fo(d)652 379 y Fn(\001)696 326 y Fm(d)675 338 y Fd(X)678 427 y Fm(i)p Fk(=1)742 379 y Fo(e)763 386 y Fm(i)777 379 y Fu(\()p Fo(A;)8 b(B)r Fu(\))i Fn(\000)g Fo(\032)p Fu(\()p Fo(A)p Fu(\))p Fo(\032)p Fu(\()p Fo(B)r Fu(\))1149 305 y Fd(\014)1149 330 y(\014)1149 355 y(\014)1149 380 y(\014)1149 405 y(\014)1187 379 y Fn(\024)1255 348 y Fo(\017)p 1252 368 23 2 v 1252 410 a Fu(2)1892 379 y(\(3\))0 534 y Fv(Ov)o(erview.)44 b Fu(Eq.)16 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Fu(and)f(consider)i(ho)o(w)e(w)o(ell)1669 689 y Fd(P)1713 732 y Fm(\013)1744 721 y Fo(e)1765 728 y Fm(i)1777 732 y Fh(\013)1800 721 y Fu(\()p Fo(A)1852 728 y Fm(\013)1875 721 y Fo(;)8 b(B)r Fu(\))0 783 y(appro)o(ximates)283 751 y Fd(P)326 794 y Fm(\013)358 783 y Fo(\032)p Fu(\()p Fo(A)434 790 y Fm(\013)456 783 y Fu(\))p Fo(\032)p Fu(\()p Fo(B)r Fu(\))15 b(=)h Fo(\032)p Fu(\()p Fo(A)p Fu(\))p Fo(\032)p Fu(\()p Fo(B)r Fu(\).)24 b(W)l(e)17 b(sho)o(w)g(that)f(when)i (the)f Fo(i)1375 790 y Fm(\013)1398 783 y Fu('s)g(are)f(uniformly)i (distributed)0 845 y(in)e(a)g(pairwise)g(indep)q(enden)o(t)i(manner,)e (as)f(is)h(the)g(case)f(when)h(setting)g Fo(i)1267 852 y Fm(\013)1303 845 y Fu(=)e Fo(h)p Fu(\()p Fo(\013)p Fu(\))h(for)g(one)h(uniformly)h(c)o(hosen)0 907 y Fo(h)c Fn(2)g Fo(H)t Fu(,)h(the)h(appro)o(ximation)g(is)h(go)q(o)q(d)f(with)h (high)g(probabilit)o(y)l(.)71 992 y(Returning)h(to)f(the)g(formal)g (pro)q(of,)f(w)o(e)h(consider)h(a)f(partition)h(of)f Fo(A)g Fu(in)o(to)g Fo(L)1416 967 y Fk(def)1421 992 y Fu(=)k(2)1499 976 y Fm(l)1528 992 y Fu(subsets)c(so)g(that)g Fo(A)1877 999 y Fm(\013)1915 992 y Fu(=)0 1054 y Fn(f)p Fo(x)h Fn(2)g Fo(A)g Fu(:)f(lsb)q(\()p Fo(x)p Fu(\))9 b(=)g Fo(\013)p Fn(g)p Fu(,)18 b(for)f(ev)o(ery)h Fo(\013)f Fn(2)g(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)848 1038 y Fm(l)859 1054 y Fu(.)28 b(W)l(e)18 b(de\014ne)g Fo(L)g Fu(random)f(v)m (ariables,)j Fo(\020)1557 1061 y Fk(0)1574 1053 y Fh(l)1587 1054 y Fo(;)8 b(:::;)g(\020)1688 1061 y Fk(1)1705 1053 y Fh(l)1716 1054 y Fu(,)18 b(so)f(that)g Fo(\020)1926 1061 y Fm(\013)0 1116 y Fu(represen)o(ts)h(the)g(densit)o(y)g(of)g(the) g(set)g Fn(f)p Fo(x)e Fn(2)10 b Fo(A)806 1123 y Fm(\013)847 1116 y Fu(:)17 b Fo(g)899 1123 y Fm(h)p Fk(\()p Fm(\013)p Fk(\))968 1116 y Fu(\()p Fo(x)p Fu(\))f Fn(2)i Fo(B)r Fn(g)g Fu(\(in)g Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1357 1100 y Fm(n)1378 1116 y Fu(\).)28 b(Note)17 b(that)g(the)h Fo(\020)1751 1123 y Fm(\013)1775 1116 y Fu('s)f(are)h(all)0 1179 y(determined)e(b)o(y)f(the)g(c)o(hoice)g(of)f Fo(h)p Fu(,)h(and)g(th)o(us)g(the)f(probabilit)o(y)i(space)f(is)h(uniform)f(o) o(v)o(er)f(all)h(c)o(hoices)h(of)e Fo(h)f Fn(2)g Fo(H)t Fu(.)0 1241 y(Alternativ)o(ely)l(,)j(since)h(lsb\()p Fo(x)p Fu(\))12 b(=)h Fo(\013)j Fu(for)e(ev)o(ery)h Fo(x)e Fn(2)g Fo(A)923 1248 y Fm(\013)947 1241 y Fu(,)h(w)o(e)h(can)h(write) 597 1343 y Fo(\020)617 1350 y Fm(\013)653 1343 y Fu(=)d(2)724 1324 y Fi(\000)p Fm(n)782 1343 y Fn(\001)d(jf)p Fo(x)i Fn(2)5 b Fo(A)948 1350 y Fm(\013)984 1343 y Fu(:)13 b Fo(g)1032 1350 y Fm(h)p Fk(\(lsb)q(\()p Fm(x)p Fk(\)\))1164 1343 y Fu(\()p Fo(x)p Fu(\))f Fn(2)h Fo(B)r Fn(gj)0 1445 y Fu(Observ)o(e)k(that)f(the)h(set)g Fn(f)p Fo(x)e Fn(2)8 b Fo(A)15 b Fu(:)g Fo(g)630 1452 y Fm(h)p Fk(\(lsb)q(\()p Fm(x)p Fk(\)\))763 1445 y Fu(\()p Fo(x)p Fu(\))g Fn(2)g Fo(B)r Fn(g)j Fu(can)f(b)q(e)g(written)g(as)1339 1412 y Fm(:)1328 1445 y Fn([)1359 1452 y Fm(\013)1398 1445 y Fn(f)p Fo(x)e Fn(2)8 b Fo(A)1534 1452 y Fm(\013)1573 1445 y Fu(:)15 b Fo(g)1623 1452 y Fm(h)p Fk(\(lsb)q(\()p Fm(x)p Fk(\)\))1756 1445 y Fu(\()p Fo(x)p Fu(\))f Fn(2)i Fo(B)r Fn(g)p Fu(.)0 1507 y(Th)o(us,)131 1475 y Fd(P)175 1519 y Fm(\013)206 1507 y Fo(\020)226 1514 y Fm(\013)267 1507 y Fu(=)h(2)342 1491 y Fi(\000)p Fm(n)402 1507 y Fn(\001)12 b(jf)p Fo(x)k Fn(2)10 b Fo(A)17 b Fu(:)f Fo(g)647 1514 y Fm(h)p Fk(\(lsb)q(\()p Fm(x)p Fk(\)\))780 1507 y Fu(\()p Fo(x)p Fu(\))g Fn(2)i Fo(B)r Fn(gj)g Fu(and)g(the)f(mixing)i (prop)q(ert)o(y)f(\(to)f(b)q(e)h(pro)o(v)o(en\))f(can)h(b)q(e)0 1569 y(rephrased)e(as)e(asserting)509 1709 y(Prob)613 1624 y Fd(0)613 1699 y(@)650 1622 y(\014)650 1647 y(\014)650 1672 y(\014)650 1697 y(\014)650 1722 y(\014)650 1747 y(\014)700 1668 y(X)663 1760 y Fm(\013)p Fi(2f)p Fk(0)p Fm(;)p Fk(1)p Fi(g)785 1752 y Fh(l)804 1709 y Fo(\020)824 1716 y Fm(\013)858 1709 y Fn(\000)c Fo(\032)p Fu(\()p Fo(A)p Fu(\))f Fn(\001)h Fo(\032)p Fu(\()p Fo(B)r Fu(\))1125 1622 y Fd(\014)1125 1647 y(\014)1125 1672 y(\014)1125 1697 y(\014)1125 1722 y(\014)1125 1747 y(\014)1151 1709 y Fo(>)j(\017)1217 1624 y Fd(1)1217 1699 y(A)1279 1709 y Fn(\024)26 b Fo(\017)534 b Fu(\(4\))0 1850 y(where)16 b(the)g(probabilit)o(y)h(space)g(is)f(o)o(v)o(er)f(all)i(p)q(ossible)h (c)o(hoices)f(of)e Fo(h)f Fn(2)h Fo(H)k Fu(\(or,)c(equiv)m(alen)o(tly)j (of)e Fo(f)j Fn(2)14 b Fo(F)6 b Fu(\).)22 b(This)0 1912 y(claim)e(is)g(v)o(ery)f(app)q(ealing)i(since)g(eac)o(h)e Fo(\020)731 1919 y Fm(\013)774 1912 y Fu(is)h(exp)q(ected)g(to)f(appro) o(ximate)g Fo(\032)p Fu(\()p Fo(A)1420 1919 y Fm(\013)1443 1912 y Fu(\))12 b Fn(\001)h Fo(\032)p Fu(\()p Fo(B)r Fu(\).)32 b(Indeed,)21 b(Eq.)e(\(4\))0 1974 y(follo)o(ws)c(b)o(y)g(com) o(bining)i(the)e(t)o(w)o(o)f(items)h(of)g(the)g(next)h(lemma.)0 2080 y Fv(Lemma)h(3.2)23 b Fp(L)n(et)15 b(the)i Fo(\020)449 2087 y Fm(\013)472 2080 y Fp('s)f(b)n(e)g(as)g(ab)n(ove)g(and)g Fo(I)883 2055 y Fk(def)888 2080 y Fu(=)i Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1054 2063 y Fm(l)1065 2080 y Fp(.)21 b(Then)726 2136 y Fd(\014)726 2160 y(\014)726 2185 y(\014)726 2210 y(\014)726 2235 y(\014)740 2169 y(X)739 2258 y Fm(\013)p Fi(2)p Fm(I)808 2209 y Fu(Exp\()p Fo(\020)926 2216 y Fm(\013)949 2209 y Fu(\))10 b Fn(\000)g Fo(\032)p Fu(\()p Fo(A)p Fu(\))s Fn(\001)s Fo(\032)p Fu(\()p Fo(B)r Fu(\))1231 2136 y Fd(\014)1231 2160 y(\014)1231 2185 y(\014)1231 2210 y(\014)1231 2235 y(\014)1284 2209 y Fo(<)1368 2178 y(\017)p 1366 2199 23 2 v 1366 2240 a Fu(2)1892 2209 y(\(5\))556 2354 y(Prob)661 2282 y Fd( )694 2281 y(\014)694 2306 y(\014)694 2331 y(\014)694 2356 y(\014)694 2380 y(\014)708 2314 y(X)707 2403 y Fm(\013)p Fi(2)p Fm(I)776 2354 y Fo(\020)796 2361 y Fm(\013)830 2354 y Fn(\000)875 2314 y Fd(X)875 2403 y Fm(\013)p Fi(2)p Fm(I)943 2354 y Fu(Exp)q(\()p Fo(\020)1062 2361 y Fm(\013)1085 2354 y Fu(\))1103 2281 y Fd(\014)1103 2306 y(\014)1103 2331 y(\014)1103 2356 y(\014)1103 2380 y(\014)1129 2354 y Fo(>)1184 2324 y(\017)p 1182 2344 V 1182 2386 a Fu(2)1210 2282 y Fd(!)1284 2354 y Fo(<)42 b(\017)513 b Fu(\(6\))0 2491 y Fv(Pro)q(of:)22 b Fu(Using)16 b(the)h(fact)e(that)h Fo(h)p Fu(\()p Fo(\013)p Fu(\))g(is)g(uniformly)i(distributed)f(on)f([) p Fo(d)p Fu(])f(and)i(recalling)h(the)e(de\014nition)i(of)e Fo(\020)1914 2498 y Fm(\013)1937 2491 y Fu(,)0 2553 y(w)o(e)f(ha)o(v)o (e)570 2634 y(Exp\()p Fo(\020)688 2641 y Fm(\013)711 2634 y Fu(\))25 b(=)820 2604 y(1)p 820 2624 24 2 v 820 2666 a Fo(d)858 2634 y Fn(\001)902 2582 y Fm(d)881 2594 y Fd(X)884 2682 y Fm(i)p Fk(=1)954 2604 y Fn(jf)p Fo(x)11 b Fn(2)i Fo(A)1104 2611 y Fm(\013)1141 2604 y Fu(:)f Fo(g)1188 2611 y Fm(i)1202 2604 y Fu(\()p Fo(x)p Fu(\))5 b Fn(2)g Fo(B)r Fn(gj)p 954 2624 422 2 v 1142 2666 a Fu(2)1165 2653 y Fm(n)964 2795 y Fu(8)p eop %%Page: 9 10 9 9 bop 0 42 a Fu(Using)16 b Fo(A)c Fu(=)220 8 y Fm(:)209 42 y Fn([)240 49 y Fm(\013)276 42 y Fo(A)310 49 y Fm(\013)349 42 y Fu(and)k(Eq.)e(\(2\),)g(w)o(e)h(ha)o(v)o(e)462 117 y Fd(X)462 206 y Fm(\013)p Fi(2)p Fm(I)530 158 y Fu(Exp)q(\()p Fo(\020)649 165 y Fm(\013)672 158 y Fu(\))41 b(=)813 127 y(1)p 813 147 24 2 v 813 189 a Fo(d)852 158 y Fn(\001)896 105 y Fm(d)874 117 y Fd(X)877 206 y Fm(i)p Fk(=1)947 95 y Fd(P)991 138 y Fm(\013)p Fi(2)p Fm(I)1061 127 y Fn(jf)p Fo(x)12 b Fn(2)h Fo(A)1212 134 y Fm(\013)1248 127 y Fu(:)g Fo(g)1296 134 y Fm(i)1309 127 y Fu(\()p Fo(x)p Fu(\))5 b Fn(2)g Fo(B)r Fn(gj)p 947 147 537 2 v 1192 189 a Fu(2)1215 176 y Fm(n)731 306 y Fu(=)813 276 y(1)p 813 296 24 2 v 813 338 a Fo(d)852 306 y Fn(\001)896 254 y Fm(d)874 266 y Fd(X)877 354 y Fm(i)p Fk(=1)942 306 y Fo(e)963 313 y Fm(i)977 306 y Fu(\()p Fo(A;)j(B)r Fu(\))0 420 y(Using)16 b(Eq.)e(\(3\),)g(w)o(e)h(establish)i(Eq.)d (\(5\).)71 482 y(Next)20 b(w)o(e)f(use)i(Cheb)o(yshev)g(Inequalit)o(y)g (to)f(pro)o(v)o(e)f(Eq.)h(\(6\):)29 b(here)20 b(w)o(e)g(use)h(the)f (fact)f(that)h(the)g Fo(\020)1815 489 y Fm(\013)1839 482 y Fu('s)f(are)0 545 y(pairwise)d(indep)q(enden)o(t)i(\(since)e(the) f Fo(h)p Fu(\()p Fo(\013)p Fu(\)'s)g(are)f(pairwise)j(indep)q(enden)o (t\).)423 657 y(Prob)527 585 y Fd( )560 583 y(\014)560 608 y(\014)560 633 y(\014)560 658 y(\014)560 683 y(\014)574 617 y(X)574 706 y Fm(\013)p Fi(2)p Fm(I)642 657 y Fo(\020)662 664 y Fm(\013)696 657 y Fn(\000)742 617 y Fd(X)742 706 y Fm(\013)p Fi(2)p Fm(I)810 657 y Fu(Exp\()p Fo(\020)928 664 y Fm(\013)951 657 y Fu(\))969 583 y Fd(\014)969 608 y(\014)969 633 y(\014)969 658 y(\014)969 683 y(\014)996 657 y Fo(>)1051 626 y(\017)p 1048 647 23 2 v 1048 688 a Fu(2)1076 585 y Fd(!)1151 657 y Fo(<)1232 626 y Fu(V)l(ar\()1321 594 y Fd(P)1365 638 y Fm(\013)1396 626 y Fo(\020)1416 633 y Fm(\013)1440 626 y Fu(\))p 1232 647 225 2 v 1286 688 a(\()p Fo(\017=)p Fu(2\))1386 675 y Fk(2)1151 795 y Fn(\024)1232 764 y Fu(4)10 b Fn(\001)1288 732 y Fd(P)1332 775 y Fm(\013)1363 764 y Fu(Exp\()p Fo(\020)1484 747 y Fk(2)1481 775 y Fm(\013)1504 764 y Fu(\))p 1232 784 290 2 v 1359 826 a Fo(\017)1377 813 y Fk(2)0 890 y Fu(Using)129 858 y Fd(P)172 902 y Fm(\013)204 890 y Fu(Exp\()p Fo(\020)325 874 y Fk(2)322 901 y Fm(\013)345 890 y Fu(\))k Fn(\024)426 858 y Fd(P)470 902 y Fm(\013)501 890 y Fo(\032)p Fu(\()p Fo(A)577 897 y Fm(\013)600 890 y Fu(\))618 874 y Fk(2)650 890 y Fn(\024)h Fo(L)10 b Fn(\001)h Fu(\()790 872 y Fk(1)p 787 879 23 2 v 787 906 a Fm(L)815 890 y Fu(\))833 874 y Fk(2)851 890 y Fu(,)16 b(and)g(the)h(de\014nition)g(of)f Fo(L)g Fu(\(=)e(8)p Fo(=\017)1482 874 y Fk(3)1501 890 y Fu(\),)h(w)o(e)h(upp)q(er)h(b)q(ound)g(the)0 952 y(ab)q(o)o(v)o(e)e (b)o(y)204 934 y Fk(4)p 197 941 31 2 v 197 968 a Fm(\017)211 960 y Fc(2)242 952 y Fn(\001)273 934 y Fk(1)p 270 941 23 2 v 270 968 a Fm(L)310 952 y Fu(=)e Fo(\017=)p Fu(2,)i(and)g(so)g (Eq.)g(\(6\))f(follo)o(ws.)p 966 958 34 40 v 0 1086 a Fl(3.4)56 b(Lo)n(w)n(er)19 b(Bound)0 1178 y Fv(Theorem)e(3.3)22 b Fp(A)17 b(family)f(with)h(mixing)e(pr)n(op)n(erty)i(of)g(ac)n(cur)n (acy)f Fo(\017)p Fp(,)h(must)f(have)h(size)e(at)i(le)n(ast)1676 1130 y Fd(q)p 1717 1130 27 2 v 1722 1160 a Fk(4)p 1722 1167 17 2 v 1723 1193 a Fm(\017)1744 1178 y Fp(.)0 1264 y Fv(Pro)q(of:)22 b Fu(Otherwise,)17 b(let)g Fo(F)22 b Fu(=)15 b Fn(f)p 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Fm(z)958 1539 y Fu(\))p Fo(;)8 b(\021)r Fu(]\))i Fo(<)1123 1514 y Fc(3)1112 1498 y Fn(p)p 1150 1498 46 2 v 41 x Fo(\016)r(d)i Fu(=)1261 1508 y Fo(\017)1279 1491 y Fk(2)p 1261 1528 38 2 v 1268 1570 a Fu(2)0 1647 y(where)j Fo(\021)g Fu(is)h(uniformly)f(distributed) h(o)o(v)o(er)e Fn(f)p Fu(1)p Fo(;)8 b(:::;)g(d)o Fn(g)p Fu(.)17 b(Recalling)g(the)d(de\014nition)j(of)d Fo(A)1514 1654 y Fm(h)1535 1647 y Fu(,)h(this)g(means)f(that)g(the)0 1709 y Fp(exp)n(e)n(cte)n(d)19 b Fu(di\013erence)c(b)q(et)o(w)o(een)g (corresp)q(onding)h(en)o(tries)f(in)g(the)g(matrices)g Fo(A)f Fu(and)h Fo(A)1475 1716 y Fm(h)1511 1709 y Fu(is)g(at)f(most)g Fo(\017)1740 1692 y Fk(2)1759 1709 y Fo(=)p Fu(2.)19 b(Th)o(us,)0 1771 y(for)c(ev)o(ery)g(probabilit)o(y)h(v)o(ector)f Fo(p)g Fu(\(and)g(in)h(particular)g(for)e Fo(p)h Fu(corresp)q(onding)h (to)f Fo(Z)1455 1754 y Fi(0)1467 1771 y Fu(\),)717 1891 y(Exp)798 1902 y Fm(h)p Fi(2)p Fm(H)871 1891 y Fu(\(\001[)p Fo(A)974 1898 y Fm(h)995 1891 y Fo(p;)8 b(Ap)p Fu(]\))j 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Fu(\()p Fo(f)r(;)h(U)903 2477 y Fm(m)931 2470 y Fu(\)]\))41 b Fo(<)h Fu(2)10 b Fn(\001)1158 2440 y Fu(2)1181 2423 y Fi(\000)p Fk(\()p Fm(n)p Fi(\000)p Fm(m)p Fi(\000)p Fm(k)q Fk(\))p 1158 2460 197 2 v 1246 2501 a Fo(\016)1370 2470 y Fu(+)1421 2440 y Fo(\017)1439 2423 y Fk(2)p 1421 2460 38 2 v 1428 2501 a Fu(2)1473 2470 y(+)1523 2440 y Fo(\017)1541 2423 y Fk(2)p 1523 2460 V 1531 2501 a Fu(4)1869 2470 y(\(14\))0 2589 y(Recalling)17 b(that)e Fo(d)d Fu(=)h(\()403 2571 y Fk(2)420 2559 y Fh(k)p 403 2578 35 2 v 413 2605 a Fm(\017)442 2589 y Fu(\))460 2573 y Fm(c)492 2589 y Fu(and)i Fo(\016)g Fu(=)670 2571 y Fm(\017)684 2559 y Fc(6)p 668 2578 34 2 v 668 2605 a Fk(8)p Fm(d)719 2589 y Fu(=)782 2571 y Fm(\017)796 2559 y Fc(6+)p Fh(c)p 772 2578 84 2 v 772 2605 a Fk(2)789 2597 y Fc(3+)p Fh(ck)861 2589 y Fu(,)f(and)i(using)g Fo(n)10 b Fn(\000)g Fo(m)g Fn(\000)g Fo(k)k Fu(=)f(6)c(+)h Fo(ck)h Fu(+)f(\(8)f(+)i Fo(c)p Fu(\))e Fn(\001)g Fu(log)q(\(1)p Fo(=\017)p Fu(\),)14 b(the)0 2651 y(\014rst)h(term)g(in)h(Eq.)e(\(14\)) g(is)i(b)q(ounded)h(b)o(y)e Fo(\017)747 2635 y Fk(2)766 2651 y Fo(=)p Fu(4,)f(and)h(the)h(lemma)f(follo)o(ws.)p 1369 2657 34 40 v 952 2795 a(13)p eop %%Page: 14 15 14 14 bop 0 42 a Fl(4.4)56 b(Lo)n(w)n(er)19 b(Bound)0 133 y Fu(W)l(e)14 b(conclude)h(b)o(y)e(observing)h(that)f(a)g(lo)o(w)o (er)g(b)q(ound)h(of)f([29)o(])g(\(i.e.,)g([29)o(,)h(Thm)f(3]\))g (implies)i(that,)e(for)g Fo(\017)g Fu(=)g(2)1842 116 y Fi(\000)p Fk(\012\()p Fm(k)q Fk(\))1937 133 y Fu(,)0 195 y(our)i(construction)h(is)g(optimal.)22 b(This)16 b(holds)h(ev)o(en)f(when)g(trying)g(to)f(extract)f(just)i(one)g(bit.)21 b(Sp)q(eci\014call)q(y)l(,)d Fp(any)0 257 y(family)e(of)g(functions)f (fr)n(om)h Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)608 241 y Fm(n)645 257 y Fp(to)16 b Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)p Fp(,)14 b(with)i(extr)n(action)g(pr)n(op)n(erty)h(of)f(ac)n(cur) n(acy)g Fo(\017)d(<)g Fu(0)p Fo(:)p Fu(5)i Fp(with)h(r)n(esp)n(e)n(ct)0 319 y(to)h(r)n(andom)f(variables)g(of)g(min-entr)n(opy)g Fo(n)11 b Fn(\000)f Fo(k)k Fn(\024)f Fo(n)d Fn(\000)h Fu(1)p Fp(,)16 b(must)g(have)h(size)e(at)i(le)n(ast)e Fu(max)p Fn(f)p Fo(k)c Fu(+)f(1)p Fo(;)e Fu(\(1)p Fo(=\017)p Fu(\))g Fn(\000)j Fu(1)p Fn(g)p Fu(.)0 467 y Fq(5)67 b(Tin)n(y)24 b(F)-6 b(amilies)24 b(Extracting)g(Lo)n(w)e(Min-En)n(trop) n(y)0 574 y Fu(Here)16 b(w)o(e)f(treat)f(the)i(case)f(of)g(random)g(v)m (ariables)i(with)f(min-en)o(trop)o(y)f Fo(k)q Fu(,)g(for)g Fo(k)f Fn(\034)g Fo(n)p Fu(.)21 b(W)l(e)15 b(construct)g(a)g(family)0 636 y(of)i(p)q(oly)q(\(2)181 619 y Fm(k)201 636 y Fo(n=\017)p Fu(\))h(functions)g(mapping)g Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)805 619 y Fm(n)843 636 y Fu(to)16 b Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1013 619 y Fm(m)1043 636 y Fu(,)18 b(where)f Fo(m)f Fu(=)h(\012\()p Fo(k)q Fu(\).)25 b(\(Again,)18 b Fo(\017)g Fu(is)f(the)h(accuracy)0 698 y(parameter.\))j(Lo)q(osely)16 b(sp)q(eaking,)h(this)f(means)g(that)f 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Fn(k)g(\025)g Fu(2)p Fo(\017=)1297 1769 y Fn(p)p 1335 1769 V 29 x Fo(p)i Fu(and)h(this)f Fo(j)j Fu(cannot)d(b)q(e)h Fo(p)f Fu(\(since)0 1828 y Fd(P)44 1871 y Fm(i)65 1860 y Fo(v)87 1867 y Fm(i)101 1860 y Fo(!)131 1843 y Fm(pi)178 1860 y Fu(=)229 1828 y Fd(P)272 1871 y Fm(i)294 1860 y Fo(v)316 1867 y Fm(i)345 1860 y Fu(=)h(0\).)25 b(Applying)19 b(this)f(argumen)o(t)e(to)g(the)h (v)o(ector)g(represen)o(ting)g(the)g(di\013erence)i(b)q(et)o(w)o(een)0 1922 y(the)14 b(probabilit)o(y)h(function)g(of)535 1890 y Fd(P)579 1900 y Fm(t)579 1933 y(i)p Fk(=1)642 1922 y Fo(X)680 1929 y Fm(i)694 1922 y Fo(s)715 1929 y Fm(i)743 1922 y Fu(and)f(the)g(uniform)g(distributional)i(function,)e(w)o(e)g (conclude)h(that)f(for)0 1984 y(ev)o(ery)j(\026)-25 b Fo(s)154 1959 y Fk(def)159 1984 y Fu(=)18 b(\()p Fo(s)251 1991 y Fk(1)270 1984 y Fo(;)8 b(:::;)g(s)372 1991 y Fm(t)384 1984 y Fu(\))k Fn(2)h Fo(S)488 1967 y Fi(0)515 1984 y Fu(there)i(exists)g(some)g Fo(j)g Fn(2)e(f)p Fu(1)p Fo(;)8 b(:::;)g(p)r Fn(\000)s 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b(that)f(for)h(ev)o(ery)i(\026)-25 b Fo(s)378 2515 y Fk(def)383 2540 y Fu(=)18 b(\()p Fo(s)475 2547 y Fk(1)494 2540 y Fo(;)8 b(:::;)g(s)596 2547 y Fm(t)608 2540 y Fu(\))k Fn(2)h Fo(S)712 2523 y Fi(00)711 2591 y Fd(\015)711 2616 y(\015)711 2641 y(\015)734 2640 y Fu(Exp)822 2593 y Fd(\020)847 2640 y Fo(!)877 2620 y Fm(j)898 2592 y Fd(P)942 2602 y Fh(t)942 2635 y(i)p Fc(=1)996 2620 y Fm(X)1023 2624 y Fh(i)1036 2620 y Fm(s)1052 2624 y Fh(i)1067 2593 y Fd(\021)1092 2591 y(\015)1092 2616 y(\015)1092 2641 y(\015)1128 2640 y Fn(\025)1190 2609 y Fu(2)p Fo(\017)p 1181 2629 61 2 v 1181 2643 a Fn(p)p 1219 2643 23 2 v 28 x Fo(p)1869 2640 y Fu(\(16\))952 2795 y(15)p eop %%Page: 16 17 16 16 bop 0 42 a Fu(Assume,)18 b(without)g(loss)g(of)f(generalit)o(y)i (that)e Fo(j)i Fu(=)e(1.)28 b(By)18 b(partitioning)h(these)f(sequences) g(according)h(to)e(the)0 104 y(appro)o(ximate)g(direction)i(of)e(the)g (exp)q(onen)o(tial)i(sum)e(and)h(applying)h(a)e(pigeon-hole)i(argumen)o (t,)e(w)o(e)g(obtain)g(a)0 166 y(set)e Fo(B)g Fn(\022)e Fo(S)199 149 y Fi(00)235 166 y Fu(of)i(cardinalit)o(y)h(\012\()p Fo(\017)p Fn(j)p Fo(S)s Fn(j)p Fo(=p)p Fu(\))d(so)i(that)532 215 y Fd(\015)532 240 y(\015)532 265 y(\015)532 289 y(\015)532 314 y(\015)532 339 y(\015)580 270 y Fu(1)p 560 290 63 2 v 560 332 a Fn(j)p Fo(B)r Fn(j)697 260 y Fd(X)635 351 y Fk(\()p Fm(s)664 355 y Fc(1)679 351 y Fm(;:::;s)745 355 y Fh(t)758 351 y Fk(\))p Fi(2)p Fm(B)828 301 y Fu(Exp)915 254 y Fd(\020)940 301 y Fo(!)970 252 y Fd(P)1014 263 y Fh(t)1014 296 y(i)p Fc(=1)1068 280 y Fm(X)1095 284 y Fh(i)1109 280 y Fm(s)1125 284 y Fh(i)1139 254 y Fd(\021)1164 215 y(\015)1164 240 y(\015)1164 265 y(\015)1164 289 y(\015)1164 314 y(\015)1164 339 y(\015)1200 301 y Fu(=)e(\012\()p Fo(\017=)1340 270 y Fn(p)p 1377 270 23 2 v 1377 301 a Fo(p)p Fu(\))0 433 y(Sp)q(eci\014cally)l(,)k(w)o(e)12 b(ma)o(y)h(partition)g(the)g(v)o(ectors)g(according)g(quarters)g(of)f (the)i(plain)g(and)f(consider)h(the)g(direction)0 495 y(whic)o(h)i(resides)g(in)g(the)f(middle)i(of)e(the)h(quarter)e(with)i (the)f(largest)g(n)o(um)o(b)q(er)g(of)g(v)o(ectors.)k(This)d(yields,)g Fo(B)g Fn(\022)d Fo(S)1939 479 y Fi(0)0 558 y Fu(so)i(that)969 653 y Fn(j)p Fo(B)r Fn(j)42 b(\025)1155 622 y Fu(1)p 1155 643 V 1155 684 a(4)1193 653 y Fn(\001)9 b(j)p Fo(S)1259 634 y Fi(00)1280 653 y Fn(j)25 b(\025)1397 622 y Fo(\017)p 1383 643 46 2 v 1383 684 a Fu(4)p Fo(p)1444 653 y Fn(\001)9 b(j)p Fo(S)s Fn(j)346 b Fu(\(17\))376 710 y Fd(\015)376 735 y(\015)376 760 y(\015)376 784 y(\015)376 809 y(\015)376 834 y(\015)424 765 y Fu(1)p 404 785 63 2 v 404 827 a Fn(j)p Fo(B)r Fn(j)541 755 y Fd(X)479 846 y Fk(\()p Fm(s)508 850 y Fc(1)524 846 y Fm(;:::;s)590 850 y Fh(t)602 846 y Fk(\))p Fi(2)p Fm(B)672 796 y Fu(Exp)759 749 y Fd(\020)784 796 y Fo(!)814 747 y Fd(P)858 758 y Fh(t)858 791 y(i)p Fc(=1)912 775 y Fm(X)939 779 y Fh(i)953 775 y Fm(s)969 779 y Fh(i)984 749 y Fd(\021)1008 710 y(\015)1008 735 y(\015)1008 760 y(\015)1008 784 y(\015)1008 809 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(ows)h(have)0 1215 y(inner-pr)n(o)n(duct)d(b)n(ounde)n(d)f Fu(\(in)g(magnitude\))h Fp(by)f Fo(\017)853 1198 y Fi(0)865 1215 y Fo(M)5 b Fp(.)25 b(L)n(et)17 b Fo(u)g Fp(b)n(e)g(an)h Fo(N)5 b Fp(-dimensional)16 b(pr)n(ob)n(ability)h(ve)n(ctor)h(with)0 1277 y(e)n(ach)23 b(c)n(omp)n(onents)e(b)n(ounde)n(d)h(ab)n(ove)h(by)f Fo(\016)r Fp(,)i(and)f Fo(v)h Fp(b)n(e)e(an)g Fo(M)5 b Fp(-dimensional)22 b(pr)n(ob)n(ability)h(ve)n(ctor)f(with)h(e)n(ach)0 1339 y(c)n(omp)n(onents)15 b(b)n(eing)g(either)502 1321 y Fk(1)p 496 1328 30 2 v 496 1355 a Fm(K)547 1339 y Fp(or)i(zer)n(o.)j (Then,)728 1476 y Fn(k)p Fo(uAv)835 1457 y Fi(>)862 1476 y Fn(k)13 b(\024)946 1396 y Fd(s)p 987 1396 235 2 v 987 1476 a Fu(\()p Fo(\017)1023 1463 y Fi(0)1045 1476 y Fu(+)e Fo(\016)r Fu(\))e Fn(\001)1168 1445 y Fo(M)p 1168 1465 50 2 v 1172 1507 a(K)0 1669 y Fu(Lindsey's)21 b(Lemma)e(is)h(obtained)g (from)f(the)g(ab)q(o)o(v)o(e)g(b)o(y)g(requiring)i(the)e(ro)o(ws)g(of)g Fo(A)g Fu(to)g(b)q(e)h(orthogonal)f(\(i.e.,)0 1731 y Fo(\017)18 1715 y Fi(0)43 1731 y Fu(=)13 b(0\))h(and)i(considering)g (only)g(\\\015at")e(distributions)j(\(i.e.,)e(eac)o(h)g Fo(u)1196 1738 y Fm(i)1225 1731 y Fu(b)q(eing)h(either)g Fo(\016)h Fu(or)e(0\).)1623 1715 y Fk(6)0 1835 y Fv(Pro)q(of:)20 b Fu(Denote,)14 b(\001)381 1810 y Fk(def)386 1835 y Fu(=)k Fn(k)p Fo(uAv)546 1818 y Fi(>)574 1835 y Fn(k)p Fu(.)h(Then,)c(using)h (Cauc)o(h)o(y)f(Sc)o(h)o(w)o(artz)g(Inequalit)o(y)l(,)h(w)o(e)f(get)606 1930 y(\001)644 1912 y Fk(2)705 1930 y Fn(\024)41 b Fu(\()p Fo(v)12 b Fn(\001)e Fo(v)880 1912 y Fi(>)907 1930 y Fu(\))g Fn(\001)g Fu(\(\()p Fo(uA)p Fu(\))f Fn(\001)h Fu(\()p Fo(uA)p Fu(\))1200 1912 y Fi(>)1227 1930 y Fu(\))705 2018 y(=)796 1988 y(1)p 786 2008 42 2 v 786 2050 a Fo(K)843 2018 y Fn(\001)g Fu(\(\()902 1978 y Fd(X)925 2066 y Fm(i)969 2018 y Fo(u)995 2025 y Fm(i)1009 2018 y Fo(A)1043 2025 y Fm(i)1057 2018 y Fu(\))f Fn(\001)h Fu(\()1125 1978 y Fd(X)1149 2066 y Fm(i)1192 2018 y Fo(u)1218 2025 y Fm(i)1232 2018 y Fo(A)1266 2025 y 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Fo(;)8 b Fu(1],)19 b(there)h(exists)0 2185 y(a)f Fo(\026)12 b Fu(:)f Fn(f)p Fu(0)p Fo(;)d Fu(1)p Fn(g)218 2168 y Fm(n)250 2185 y Fn(7!)k(f)p Fu(\()p Fo(i)f Fn(\000)428 2167 y Fk(1)p 428 2174 V 428 2200 a(2)450 2185 y Fu(\))h Fn(\001)512 2167 y Fm(\017)p 510 2174 V 510 2200 a Fk(2)551 2185 y Fu(:)18 b Fo(i)12 b Fu(=)f(1)p Fo(;)d(::;)g Fu(2)p Fo(\017)788 2168 y Fi(\000)p Fk(1)830 2185 y Fn(g)19 b Fu(so)g(that)f Fn(j)p Fo(\027)s Fu(\()p Fo(x)p Fu(\))12 b Fn(\000)h Fo(\026)p Fu(\()p Fo(x)p Fu(\))p Fn(j)18 b(\024)1375 2167 y Fm(\017)p 1374 2174 V 1374 2200 a Fk(4)1395 2185 y Fu(,)i Fn(8)p Fo(x)f Fn(2)g(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1660 2168 y Fm(n)1681 2185 y Fu(.)31 b(Th)o(us,)20 b(b)q(oth)0 2247 y Fn(j)s Fu(\026)-26 b Fo(\027)s Fu(\()p Fo(x)p Fu(\))9 b Fn(\000)14 b Fu(\026)-26 b Fo(\026)p Fu(\()p Fo(x)p Fu(\))p Fn(j)12 b(\024)323 2229 y Fm(\017)p 322 2236 V 322 2263 a Fk(4)358 2247 y Fu(and)k Fn(j)s Fu(~)-26 b Fo(\027)s Fu(\()p Fo(x)p Fu(\))9 b Fn(\000)14 b Fu(~)-26 b Fo(\026)p Fu(\()p 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Fu(0)p Fo(;)8 b Fu(1)p Fn(g)193 2651 y Fm(n)214 2668 y Fu(,)248 2650 y Fk(1)p 247 2657 18 2 v 247 2684 a Fm(d)280 2668 y Fn(\001)303 2636 y Fd(P)347 2679 y Fm(u)p Fi(2N)t Fk(\()p Fm(v)q Fk(\))474 2668 y Fo(\026)p Fu(\()p Fo(u)p Fu(\))15 b(is)h(within)h Fo(\017=)p Fu(4)e(of)k(\026)-27 b Fo(\026)q Fu(,)15 b(where)h Fn(N)7 b Fu(\()p Fo(v)r Fu(\))14 b(denotes)i(the)g(set)f(of)g(neigh)o (b)q(ors)h(of)f(v)o(ertex)952 2795 y(21)p eop %%Page: 22 23 22 22 bop 0 42 a Fo(v)r Fu(.)20 b(W)l(e)c(conclude)h(b)o(y)f(recalling) h(that)e(with)h(probabilit)o(y)g(at)f(least)h(1)10 b Fn(\000)1251 24 y Fm(\016)p 1251 31 17 2 v 1251 57 a Fk(2)1288 42 y Fu(a)15 b(pairwise)h(indep)q(enden)o(t)i(sample)e(of)0 104 y Fo(m)f Fu(v)o(ertices)h(in)g Fn(N)7 b Fu(\()p Fo(v)r Fu(\))14 b(appro)o(ximates)g(the)i(ab)q(o)o(v)o(e)f(up-to)g Fo(\017=)p Fu(4.)71 166 y(Supp)q(ose,)g(for)g(simplicit)o(y)l(,)i(that) e Fo(\017)g Fu(is)h(suc)o(h)g(that)e Fo(t)955 141 y Fk(def)960 166 y Fu(=)k(2)p Fo(\017)1054 149 y Fi(\000)p Fk(1)1114 166 y Fu(is)e(an)f(in)o(teger.)20 b(F)l(or)15 b(ev)o(ery)g Fo(i)5 b Fu(=)g(1)p Fo(;)j(::;)g(t)p Fu(,)13 b(let)554 257 y Fo(A)588 264 y Fm(i)643 232 y Fk(def)649 257 y Fu(=)47 b Fn(f)p Fo(v)14 b Fu(:)e Fo(\026)p Fu(\()p Fo(v)r Fu(\))g(=)h(\()p Fo(i)c Fn(\000)i Fu(0)p Fo(:)p Fu(5\))e Fn(\001)g Fo(\017=)p Fu(2)p Fn(g)623 b Fu(\(21\))554 351 y Fo(B)588 358 y Fm(i)643 326 y Fk(def)649 351 y Fu(=)731 291 y Fd(\032)762 351 y Fo(v)14 b Fu(:)823 289 y Fd(\014)823 314 y(\014)823 339 y(\014)823 364 y(\014)842 320 y Fn(jN)7 b Fu(\()p Fo(v)r Fu(\))i Fn(\\)h Fo(A)1042 327 y Fm(i)1056 320 y Fn(j)p 842 340 227 2 v 944 382 a Fo(d)1084 351 y Fn(\000)g Fo(\032)p Fu(\()p Fo(A)1205 358 y Fm(i)1218 351 y Fu(\))1236 289 y Fd(\014)1236 314 y(\014)1236 339 y(\014)1236 364 y(\014)1263 351 y Fo(>)1316 320 y(\017)1334 303 y Fk(2)p 1316 340 38 2 v 1323 382 a Fu(4)1358 291 y Fd(\033)1869 351 y Fu(\(22\))0 456 y(Observ)o(e)16 b(that)e(if)i Fo(v)e Fn(62)f([)424 440 y Fm(t)424 468 y(i)p Fk(=1)480 456 y Fo(B)514 463 y Fm(i)544 456 y Fu(then)473 545 y(1)p 473 565 24 2 v 473 607 a Fo(d)512 575 y Fn(\001)563 535 y Fd(X)534 626 y Fm(u)p Fi(2N)t Fk(\()p Fm(v)q Fk(\))659 575 y Fo(\026)p Fu(\()p Fo(u)p Fu(\))41 b(=)872 545 y(1)p 871 565 V 871 607 a Fo(d)910 575 y Fn(\001)956 523 y Fm(t)933 535 y Fd(X)936 623 y Fm(i)p Fk(=1)993 575 y Fu(\()p Fo(i)9 b Fn(\000)h Fu(0)p Fo(:)p Fu(5\))f Fn(\001)1197 545 y Fo(\017)p 1195 565 23 2 v 1195 607 a Fu(2)1233 575 y Fn(\001)h(jN)d Fu(\()p Fo(v)r Fu(\))h Fn(\\)j Fo(A)1456 582 y Fm(i)1470 575 y Fn(j)789 729 y Fu(=)890 676 y Fm(t)866 689 y Fd(X)869 777 y Fm(i)p Fk(=1)926 729 y Fu(\()p Fo(i)f Fn(\000)g Fu(0)p Fo(:)p Fu(5\))f Fn(\001)1131 698 y Fo(\017)p 1129 718 V 1129 760 a Fu(2)1166 729 y Fn(\001)1189 669 y Fd(\022)1220 729 y Fo(\032)p Fu(\()p Fo(A)1296 736 y Fm(i)1309 729 y Fu(\))g Fn(\006)1387 698 y Fo(\017)1405 682 y Fk(2)p 1387 718 38 2 v 1394 760 a Fu(4)1429 669 y Fd(\023)789 880 y Fu(=)46 b(\026)-27 b Fo(\026)11 b Fn(\006)955 824 y Fk(2)p Fm(=\017)949 840 y Fd(X)952 928 y Fm(i)p Fk(=1)1009 880 y Fu(\()p Fo(i)e Fn(\000)i Fu(0)p Fo(:)p Fu(5\))e Fn(\001)1211 849 y Fo(\017)1229 833 y Fk(3)p 1211 870 V 1219 911 a Fu(8)789 999 y(=)46 b(\026)-27 b Fo(\026)11 b Fn(\006)956 968 y Fo(\017)p 954 989 23 2 v 954 1030 a Fu(4)0 1098 y(Th)o(us,)k(the)g(a)o(v)o(erage)f(v)m(alue)i(of)f(the)g (neigh)o(b)q(ors)h(of)f(eac)o(h)g Fo(v)f Fn(62)f([)1084 1081 y Fm(t)1084 1109 y(i)p Fk(=1)1141 1098 y Fo(B)1175 1105 y Fm(i)1204 1098 y Fu(is)j(within)1397 1080 y Fm(\017)p 1396 1087 17 2 v 1396 1113 a Fk(4)1433 1098 y Fu(of)e(the)i(the)f (correct)g(v)m(alue)20 b(\026)-27 b Fo(\026)p Fu(.)71 1160 y(Next,)13 b(w)o(e)g(use)h(the)f(Expander)h(Mixing)h(Lemma)e (\(Lemma)g(2.2\))f(as)h(in)i(the)e(pro)q(of)g(of)g(Lemma)h(6.6)e(to)h (deriv)o(e)0 1222 y(a)18 b(b)q(ound)g(on)g(the)g(densit)o(y)g(of)g(eac) o(h)g Fo(B)683 1229 y Fm(i)697 1222 y Fu(.)28 b(Analogously)l(,)19 b(w)o(e)e(partition)h Fo(B)1303 1229 y Fm(i)1336 1222 y Fu(in)o(to)f Fo(B)1466 1203 y Fk(+)1464 1232 y Fm(i)1512 1197 y Fk(def)1517 1222 y Fu(=)22 b Fn(f)p Fo(v)d Fu(:)d Fn(jN)7 b Fu(\()p Fo(v)r Fu(\))k Fn(\\)h Fo(A)1871 1229 y Fm(i)1885 1222 y Fn(j)17 b Fo(>)0 1284 y Fu(\()p Fo(\032)p Fu(\()p Fo(A)94 1291 y Fm(i)107 1284 y Fu(\))10 b(+)g Fo(\017)198 1267 y Fk(2)217 1284 y Fo(=)p Fu(4\))f Fn(\001)h Fo(d)p Fn(g)15 b Fu(and)g Fo(B)499 1265 y Fi(\000)497 1294 y Fm(i)540 1259 y Fk(def)546 1284 y Fu(=)j Fo(B)633 1291 y Fm(i)657 1284 y Fn(n)10 b Fo(B)726 1265 y Fk(+)724 1294 y Fm(i)754 1284 y Fu(,)15 b(and)g(get)429 1374 y Fo(\017)447 1358 y Fk(2)p 429 1394 38 2 v 436 1436 a Fu(4)481 1405 y Fn(\001)10 b Fo(\032)p Fu(\()p Fo(B)582 1386 y Fk(+)580 1416 y Fm(i)610 1405 y Fu(\))41 b(=)746 1331 y Fd(\014)746 1356 y(\014)746 1381 y(\014)746 1406 y(\014)746 1431 y(\014)764 1372 y Fn(j)p Fo(B)813 1353 y Fk(+)811 1382 y Fm(i)841 1372 y Fn(j)10 b(\001)g Fu(\()p Fo(\032)p Fu(\()p Fo(A)981 1379 y Fm(i)994 1372 y Fu(\))f(+)1072 1354 y Fm(\017)1086 1342 y Fc(2)p 1072 1361 31 2 v 1079 1388 a Fk(4)1107 1372 y Fu(\))h Fn(\001)f Fo(d)p 764 1394 417 2 v 940 1436 a(dN)1196 1405 y Fn(\000)i Fo(\032)p Fu(\()p Fo(B)1320 1386 y Fk(+)1318 1416 y Fm(i)1347 1405 y Fu(\))f Fn(\001)g Fo(\032)p Fu(\()p Fo(A)1474 1412 y Fm(i)1487 1405 y Fu(\))1505 1331 y Fd(\014)1505 1356 y(\014)1505 1381 y(\014)1505 1406 y(\014)1505 1431 y(\014)669 1546 y Fn(\024)746 1472 y Fd(\014)746 1497 y(\014)746 1522 y(\014)746 1547 y(\014)746 1572 y(\014)764 1515 y Fn(j)p Fu(\()p Fo(B)831 1496 y Fk(+)829 1526 y Fm(i)869 1515 y Fn(\002)h Fo(A)949 1522 y Fm(i)963 1515 y Fu(\))e Fn(\\)i Fo(E)s Fn(j)p 764 1536 316 2 v 891 1577 a(j)p Fo(E)s Fn(j)1095 1546 y(\000)1145 1515 y(j)p Fo(A)1192 1522 y Fm(i)1206 1515 y Fn(j)p 1145 1536 74 2 v 1151 1577 a(j)p Fo(V)e Fn(j)1233 1546 y(\001)1261 1515 y(j)p Fo(B)1310 1496 y Fk(+)1308 1526 y Fm(i)1338 1515 y Fn(j)p 1261 1536 90 2 v 1275 1577 a(j)p Fo(V)g Fn(j)1356 1472 y Fd(\014)1356 1497 y(\014)1356 1522 y(\014)1356 1547 y(\014)1356 1572 y(\014)669 1707 y Fn(\024)751 1676 y Fo(\025)p 751 1697 27 2 v 752 1738 a(d)792 1707 y Fn(\001)820 1617 y Fd(q)p 861 1617 196 2 v 861 1667 a Fn(j)p Fo(A)908 1674 y Fm(i)922 1667 y Fn(j)h(\001)f(j)p Fo(B)1016 1648 y Fk(+)1014 1678 y Fm(i)1044 1667 y Fn(j)p 820 1697 238 2 v 918 1738 a Fo(N)669 1839 y Fn(\024)746 1758 y Fd(s)p 787 1758 69 2 v 792 1808 a Fo(\017)810 1795 y Fk(5)829 1808 y Fo(\016)p 792 1828 59 2 v 799 1870 a Fu(64)866 1839 y Fn(\001)889 1786 y Fd(q)p 930 1786 249 2 v 930 1839 a Fo(\032)p Fu(\()p Fo(A)p Fu(\))h Fn(\001)f Fo(\032)p Fu(\()p Fo(B)1134 1820 y Fk(+)1132 1849 y Fm(i)1162 1839 y Fu(\))0 1952 y(It)j(follo)o(ws)g(that)f Fo(\032)p Fu(\()p Fo(B)365 1933 y Fk(+)363 1962 y Fm(i)392 1952 y Fu(\))h Fn(\024)475 1934 y Fm(\017\016)p 475 1941 30 2 v 482 1968 a Fk(4)513 1952 y Fn(\001)s Fo(\032)p Fu(\()p Fo(A)605 1959 y Fm(i)618 1952 y Fu(\))g(and)f(similarly)j Fo(\032)p Fu(\()p Fo(B)993 1933 y Fi(\000)991 1962 y Fm(i)1021 1952 y Fu(\))e Fn(\024)1104 1934 y Fm(\017\016)p 1104 1941 V 1111 1968 a Fk(4)1142 1952 y Fn(\001)s Fu(\(1)s Fn(\000)s Fo(\032)p Fu(\()p Fo(A)1316 1959 y Fm(i)1329 1952 y Fu(\)\).)18 b(Th)o(us,)12 b Fo(\032)p Fu(\()p Fn([)1593 1936 y Fm(t)1593 1963 y(i)p Fk(=1)1648 1952 y Fo(B)1682 1959 y Fm(i)1697 1952 y Fu(\))g Fn(\024)h Fo(t)s Fn(\001)1815 1934 y Fm(\017\016)p 1815 1941 V 1822 1968 a Fk(4)1863 1952 y Fu(=)1916 1934 y Fm(\016)p 1916 1941 17 2 v 1916 1968 a Fk(2)1937 1952 y Fu(.)0 2014 y(Com)o(bined)j(with)f(the)h(ab)q(o)o(v)o(e)f(w)o(e)f(ha)o(v)o(e) 163 2133 y(Prob)260 2140 y Fm(v)q Fi(2f)p Fk(0)p Fm(;)p Fk(1)p Fi(g)378 2132 y Fh(n)407 2049 y Fd(0)407 2124 y(@)443 2047 y(\014)443 2072 y(\014)443 2097 y(\014)443 2122 y(\014)443 2147 y(\014)443 2172 y(\014)463 2103 y Fu(1)p 462 2123 24 2 v 462 2165 a Fo(d)501 2133 y Fn(\001)552 2093 y Fd(X)524 2184 y Fm(u)p Fi(2N)t Fk(\()p Fm(v)q Fk(\))648 2133 y Fo(\026)p Fu(\()p Fo(u)p Fu(\))c Fn(\000)k Fu(\026)-26 b Fo(\026)820 2047 y Fd(\014)820 2072 y(\014)820 2097 y(\014)820 2122 y(\014)820 2147 y(\014)820 2172 y(\014)847 2133 y Fo(>)902 2103 y(\017)p 900 2123 23 2 v 900 2165 a Fu(4)927 2049 y Fd(1)927 2124 y(A)1001 2133 y Fn(\024)39 b Fu(Prob)1171 2140 y Fm(v)q Fi(2f)p Fk(0)p Fm(;)p Fk(1)p Fi(g)1289 2132 y Fh(n)1311 2133 y Fu(\()p Fo(v)14 b Fn(2)f([)1438 2115 y Fm(t)1438 2145 y(i)p Fk(=1)1494 2133 y Fo(B)1528 2140 y Fm(i)1542 2133 y Fu(\))38 b Fn(\024)1676 2103 y Fo(\016)p 1676 2123 V 1676 2165 a Fu(2)1869 2133 y(\(23\))0 2264 y(where)15 b Fo(v)i Fu(is)f(uniformly)g(selected)h(in)f Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)759 2247 y Fm(n)779 2264 y Fu(.)71 2326 y(Finally)l(,)15 b(w)o(e)g(note)f(that,)f(for)h(ev)o(ery)h Fo(v)f Fn(2)f(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)891 2309 y Fm(n)912 2326 y Fu(,)14 b(if)h Fo(u)1006 2333 y Fm(i)1020 2326 y Fo(;)8 b(:::;)g(u)1127 2333 y Fm(m)1170 2326 y Fu(are)14 b(uniformly)h(distributed)h(in)g(a)e(pairwise-)0 2388 y(indep)q(enden)o(t)k(manner)d(in)h Fn(N)7 b Fu(\()p Fo(v)r Fu(\))14 b(then)214 2518 y(Prob)311 2525 y Fm(u)331 2529 y Fc(1)347 2525 y Fm(;:::;u)417 2529 y Fh(m)453 2434 y Fd(0)453 2509 y(@)489 2432 y(\014)489 2457 y(\014)489 2482 y(\014)489 2507 y(\014)489 2532 y(\014)489 2557 y(\014)517 2488 y Fu(1)p 508 2508 40 2 v 508 2550 a Fo(m)563 2518 y Fn(\001)601 2465 y Fm(m)586 2478 y Fd(X)589 2566 y Fm(i)p Fk(=1)653 2518 y Fo(\026)p Fu(\()p Fo(u)724 2525 y Fm(i)738 2518 y Fu(\))c Fn(\000)817 2488 y Fu(1)p 816 2508 24 2 v 816 2550 a Fo(d)855 2518 y Fn(\001)906 2478 y Fd(X)878 2569 y Fm(u)p Fi(2N)t Fk(\()p Fm(v)q Fk(\))1003 2518 y Fo(\026)p Fu(\()p Fo(u)p Fu(\))1092 2432 y Fd(\014)1092 2457 y(\014)1092 2482 y(\014)1092 2507 y(\014)1092 2532 y(\014)1092 2557 y(\014)1118 2518 y Fo(>)1173 2488 y(\017)p 1171 2508 23 2 v 1171 2550 a Fu(4)1198 2434 y Fd(1)1198 2509 y(A)1273 2518 y Fn(\024)1378 2486 y Fo(m)g Fn(\001)1456 2468 y Fk(1)p 1456 2475 17 2 v 1456 2501 a(4)p 1351 2508 154 2 v 1351 2550 a Fu(\()p Fo(m)g Fn(\001)1448 2532 y Fm(\017)p 1446 2539 17 2 v 1446 2565 a Fk(4)1468 2550 y Fu(\))1486 2536 y Fk(2)1547 2518 y Fu(=)1626 2488 y Fo(\016)p 1625 2508 23 2 v 1625 2550 a Fu(2)1869 2518 y(\(24\))0 2651 y(where)15 b(the)h(equalit)o(y)g (is)f(due)h(to)f Fo(m)d Fu(=)690 2633 y Fk(8)p 675 2640 47 2 v 675 2667 a Fm(\017)689 2659 y Fc(2)705 2667 y Fm(\016)726 2651 y Fu(.)20 b(Com)o(bining)c(Eq.)f(\(23\))f(and)h(Eq.)g (\(24\),)f(the)h(lemma)g(follo)o(ws.)p 1890 2657 34 40 v 952 2795 a(22)p eop %%Page: 23 24 23 23 bop 0 42 a Fq(Ac)n(kno)n(wledgmen)n(ts)0 149 y Fu(W)l(e)14 b(are)g(grateful)g(to)f(the)h(anon)o(ymous)g(referees)g (for)g(their)g(useful)h(commen)o(ts,)f(and)g(to)f(Noga)h(Alon)g(for)g (helpful)0 211 y(discussions.)0 361 y Fq(References)72 469 y Fu([1])22 b(M.)e(Ajtai,)h(J.)g(Komlos,)h(E.)e(Szemer)o(\023)-21 b(edi,)22 b(\\Deterministic)g(Sim)o(ulation)g(in)g(LogSpace",)f Fp(Pr)n(o)n(c.)g(19th)143 531 y(STOC)p Fu(,)13 b(1987,)h(pp.)h (132{140.)72 630 y([2])22 b(N.)f(Alon,)j(\\Eigen)o(v)m(alues,)g (Geometric)d(Expanders,)j(Sorting)d(in)i(Rounds)f(and)g(Ramsey)g (Theory",)143 692 y Fp(Combinatoric)n(a)p Fu(,)14 b(6)h(\(1986\),)e (pp.)i(231{243.)72 792 y([3])22 b(N.)15 b(Alon,)h(J.)f(Bruc)o(k,)h(J.)f (Naor,)g(M.)f(Naor)h(and)h(R.)f(Roth,)h(\\Construction)f(of)g (Asymptotically)h(Go)q(o)q(d,)143 854 y(Lo)o(w-Rate)j(Error-Correcting) g(Co)q(des)g(through)g(Pseudo-Random)h(Graphs",)g Fp(IEEE)f(T)m(r)n (ansactions)143 916 y(on)d(Information)g(The)n(ory)f Fv(38)g Fu(\(1992\),)e(pp.)i(509{516.)72 1015 y([4])22 b(N.)g(Alon,)i(O.)e(Goldreic)o(h,)j(J.)d(Hastad,)i(R.)e(P)o(eralta,)h (\\Simple)h(Constructions)e(of)g(Almost)g Fo(k)q Fu(-wise)143 1077 y(Indep)q(enden)o(t)15 b(Random)e(V)l(ariables",)g Fp(Journal)h(of)g(R)n(andom)h(structur)n(es)f(and)g(A)o(lgorithms)p Fu(,)e(V)l(ol.)h(3,)g(No.)143 1140 y(3,)h(\(1992\),)f(pp.)j(289{304.)72 1239 y([5])22 b(N.)10 b(Alon)i(and)f(V.D.)f(Milman,)i Fo(\025)699 1246 y Fk(1)717 1239 y Fu(,)f(Isop)q(erimetric)h (Inequalities)i(for)c(Graphs)g(and)h(Sup)q(erconcen)o(trators,)143 1301 y Fp(J.)16 b(Combinatorial)g(The)n(ory,)g(Ser.)g(B)f Fu(38)f(\(1985\),)f(pp.)j(73{88.)72 1401 y([6])22 b(N.)15 b(Alon)g(and)h(J.H.)f(Sp)q(encer,)h Fp(The)g(Pr)n(ob)n(abilistic)f (Metho)n(d)p Fu(,)g(John)g(Wiley)i(&)e(Sons,)g(Inc.,)g(1992.)72 1500 y([7])22 b(M.)15 b(Bellare,)i(O.)e(Goldreic)o(h,)i(and)f(S.)g (Goldw)o(asser)f(\\Randomness)h(in)h(In)o(teractiv)o(e)f(Pro)q(ofs",)e Fp(Compu-)143 1562 y(tational)i(Complexity)p Fu(,)f(V)l(ol.)g(4,)g(No.) f(4)h(\(1993\),)e(pp.)i(319{354.)72 1662 y([8])22 b(R.)17 b(Canetti,)h(G.)f(Ev)o(en)g(and)h(O.)f(Goldreic)o(h,)i(\\Lo)o(w)o(er)d (Bounds)i(for)f(Sampling)i(Algorithms)f(for)f(Esti-)143 1724 y(mating)e(the)g(Av)o(erage",)f Fp(IPL)p Fu(,)g(V)l(ol.)h(53,)f (pp.)i(17{25,)d(1995.)72 1823 y([9])22 b(L.)12 b(Carter)f(and)i(M.)e(W) l(egman,)i(\\Univ)o(ersal)g(Classes)f(of)g(Hash)g(F)l(unctions",)h Fp(J.)h(Computer)g(and)g(System)143 1885 y(Scienc)n(es)t Fu(,)e(V)l(ol.)k(18,)e(pp.)h(143{154)e(\(1979\).)49 1985 y([10])22 b(B.)17 b(Chor)h(and)g(O.)f(Goldreic)o(h,)j(\\Un)o(biased)e (Bits)g(from)g(Sources)g(of)f(W)l(eak)h(Randomness)g(and)g(Prob-)143 2047 y(abilistic)i(Comm)o(unication)f(Complexit)o(y",)g Fp(SIAM)f(J.)h(Comput.)p Fu(,)h(V)l(ol.)e(17,)h(No.)f(2,)g(April)i (1988,)e(pp.)143 2109 y(230{261.)49 2208 y([11])k(B.)16 b(Chor)g(and)g(O.)g(Goldreic)o(h,)h(\\On)g(the)f(P)o(o)o(w)o(er)f(of)h (Tw)o(o{P)o(oin)o(t)f(Based)h(Sampling,")i Fp(Jour.)f(of)h(Com-)143 2270 y(plexity)p Fu(,)c(V)l(ol)i(5,)f(1989,)e(pp.)j(96{106.)49 2370 y([12])22 b(A.)14 b(Cohen)h(and)f(A.)g(Wigderson,)h(\\Disp)q (ensers,)g(Deterministic)h(Ampli\014cation,)g(and)e(W)l(eak)h(Random) 143 2432 y(Sources",)g Fp(30th)i(F)o(OCS)p Fu(,)c(1989,)h(pp.)h(14{19.) 49 2531 y([13])22 b(P)l(.)15 b(Erd\177)-23 b(os)14 b(and)i(J.)f(Sp)q (encer,)h Fp(Pr)n(ob)n(abilistic)f(Metho)n(ds)h(in)g(Combinatorics)p Fu(,)e(Academic)i(Press,)f(1974.)952 2795 y(23)p eop %%Page: 24 25 24 24 bop 49 42 a Fu([14])22 b(G.)17 b(Ev)o(en,)h(\\Construction)g(of)g (Small)h(Probabilit)o(y)g(Spaces)f(for)g(Deterministic)h(Sim)o (ulation",)h(M.Sc.)143 104 y(thesis,)d(Computer)f(Science)j(Departmen)o (t,)d(T)l(ec)o(hnion,)i(Haifa,)f(Israel,)g(1991.)f(\(In)h(Hebrew,)g (abstract)143 166 y(in)f(English\))49 265 y([15])22 b(G.)14 b(Ev)o(en,)i(O.)f(Goldreic)o(h,)h(M.)f(Lub)o(y)l(,)h(N.)f(Nisan,)g(and) h(B.)f(V)l(eli)o(\024)-21 b(ck)o(o)o(vi)o(\023)g(c,)15 b(\\Appro)o(ximations)h(of)f(General)143 327 y(Indep)q(enden)o(t)i (Distributions",)f Fp(24th)h(STOC)p Fu(,)c(pp.)i(10{16,)f(1992.)49 427 y([16])22 b(O.)16 b(Gab)q(er)g(and)h(Z.)e(Galil,)j(\\Explicit)g (Constructions)e(of)g(Linear)h(Size)h(Sup)q(erconcen)o(trators",)e Fp(JCSS)p Fu(,)143 489 y Fv(22)f Fu(\(1981\),)e(pp.)i(407-420.)49 588 y([17])22 b(O.)13 b(Goldreic)o(h,)h(H.)f(Kra)o(w)o(cyzk)f(and)h(M.) g(Lub)o(y)l(,)h(\\On)f(the)g(Existence)h(of)f(Pseudorandom)g (Generators",)143 650 y Fp(SIAM)i(J.)h(on)g(Computing)p Fu(,)f(V)l(ol.)g(22-6)f(\(Dec.)h(1993\),)e(pp.)j(1163{1175.)49 750 y([18])22 b(S.)15 b(Goldw)o(asser)h(and)g(M.)f(Sipser,)i(\\Priv)m (ate)f(Coins)g(v)o(ersus)g(Public)h(Coins)g(in)f(In)o(teractiv)o(e)g (Pro)q(of)g(Sys-)143 812 y(tems",)d Fp(A)n(dvanc)n(es)g(in)h(Computing) h(R)n(ese)n(ar)n(ch:)k(a)c(r)n(ese)n(ar)n(ch)f(annual)p Fu(,)f(V)l(ol.)h(5)f(\(Randomness)h(and)f(Com-)143 874 y(putation,)i(S.)g(Micali,)h(ed.\),)e(pp.)i(73{90,)d(1989.)49 974 y([19])22 b(R.)g(Impagliazzo)i(and)e(M.)g(Lub)o(y)l(,)j(\\One-W)l (a)o(y)d(F)l(unctions)i(are)e(Essen)o(tial)h(for)f(Complexit)o(y)h (Based)143 1036 y(Cryptograph)o(y",)13 b Fp(30th)k(F)o(OCS)p Fu(,)c(pp.)j(230{235,)c(1989.)49 1135 y([20])22 b(R.)e(Impagliazzo)i (and)f(L.A.)f(Levin,)k(\\No)c(Better)g(W)l(a)o(ys)g(to)g(Generate)h (Hard)f(NP)h(Instances)g(than)143 1197 y(Pic)o(king)16 b(Uniformly)g(at)f(Random)g(",)f Fp(31st)j(F)o(OCS)p Fu(,)c(pp.)i(812-821,)f(1990.)49 1297 y([21])22 b(R.)12 b(Impagliazzo,)h(L.A.)f(Levin,)h(and)g(M.G.)d(Lub)o(y)l(,)j (\\Pseudorandom)f(Generators)f(from)g(an)o(y)h(One-W)l(a)o(y)143 1359 y(F)l(unctions",)j Fp(21st)h(STOC)p Fu(,)e(pp.)h(12{24,)e(1989.)49 1458 y([22])22 b(R.)15 b(Impagliazzo)i(and)e(D.)g(Zuc)o(k)o(erman,)g (\\Ho)o(w)g(to)f(Recycle)k(Random)e(Bits",)f Fp(30th)i(F)o(OCS)p Fu(,)d(1989,)g(pp.)143 1520 y(248-253.)49 1620 y([23])22 b(R.M.)12 b(Karp,)i(N.)f(Pippinger)i(and)e(M.)g(Sipser,)h(\\A)f (Time-Randomness)i(T)l(radeo\013)t(",)c Fp(AMS)j(Confer)n(enc)n(e)143 1682 y(on)i(Pr)n(ob)n(abilistic)f(Computational)h(Complexity)t Fu(,)f(Durham,)g(New)g(Hampshire)h(\(1982\).)49 1781 y([24])22 b(A.)12 b(Lub)q(otzky)l(,)i(R.)f(Phillips,)j(P)l(.)c(Sarnak,) h(\\Explicit)i(Expanders)e(and)g(the)g(Raman)o(ujan)g(Conjectures",)143 1843 y Fp(Pr)n(o)n(c.)i(18th)i(STOC)p Fu(,)d(1986,)f(pp.)j(240-246.)49 1943 y([25])22 b(G.A.)c(Margulis,)i(\\Explicit)h(Construction)e(of)g (Concen)o(trators",)f Fp(Pr)n(ob.)i(Per.)f(Infor.)g Fu(9)g(\(4\))f (\(1973\),)143 2005 y(71{80.)13 b(\(In)j(Russian,)g(English)g (translation)f(in)h Fp(Pr)n(oblems)g(of)g(Infor.)g(T)m(r)n(ans.)d Fu(\(1975\),)g(325{332.\))49 2104 y([26])22 b(J.)14 b(Naor)f(and)i(M.)e (Naor,)g(\\Small-bias)j(Probabilit)o(y)f(Spaces:)20 b(E\016cien)o(t)15 b(Constructions)f(and)g(Applica-)143 2167 y(tions",)g Fp(SIAM)h(J.)h(on)g(Computing)p Fu(,)f(V)l(ol)h(22,)e(1993,)g(pp.)h (838{856.)49 2266 y([27])22 b(N.)13 b(Nisan,)h(\\Pseudorandom)f (Generators)g(for)g(Space)h(Bounded)h(Computation",)e Fp(Combinatoric)n(a)h Fu(12)143 2328 y(\(4\),)g(1992,)f(pp.)j(449{461.) 49 2428 y([28])22 b(N.)15 b(Nisan,)g(\\)p Fn(RL)d(\022)h(S)s(C)s Fu(",)h Fp(Journal)j(of)f(Computational)h(Complexity)e Fu(4,)f(1994,)g(pp.)h(1{11.)49 2527 y([29])22 b(N.)14 b(Nisan)g(and)h(D.)f(Zuc)o(k)o(erman,)f(\\Randomness)i(is)f(Linear)i (in)f(Space",)f(to)g(app)q(ear)g(in)h Fp(JCSS)p Fu(.)e(Prelim-)143 2589 y(inary)i(v)o(ersion)h(in)g Fp(25th)h(STOC)p Fu(,)c(pp.)i (235{244,)e(1993.)952 2795 y(24)p eop %%Page: 25 26 25 25 bop 49 42 a Fu([30])22 b(M.)e(Sipser,)k(\\A)d(Complexit)o(y)h (Theoretic)g(Approac)o(h)f(to)g(Randomness",)i Fp(15th)g(STOC)p Fu(,)c(1983,)j(pp.)143 104 y(330{335.)49 203 y([31])g(M.)c(Sipser,)i (\\Expanders,)f(Randomness)h(or)e(Time)h(vs)g(Space",)h Fp(Structur)n(e)g(in)f(Complexity)g(The)n(ory)143 265 y Fu(\(pro)q(ceedings\),)c(1986.)49 365 y([32])22 b(L.)10 b(Sto)q(c)o(kmey)o(er,)h(\\The)g(Complexit)o(y)g(of)f(Appro)o(ximate)g (Coun)o(ting",)h Fp(15th)i(STOC)p Fu(,)c(1983,)h(pp.)g(118{126.)49 464 y([33])22 b(A.)e(Sriniv)m(asan)j(and)e(D.)f(Zuc)o(k)o(erman,)i (\\Computing)f(with)g(V)l(ery)g(W)l(eak)g(Random)g(Sources",)h Fp(35th)143 526 y(F)o(OCS)p Fu(,)13 b(pp.)i(264{275,)e(1994.)49 626 y([34])22 b(L.)16 b(V)l(alian)o(t)i(and)f(V.V.)f(V)l(azirani,)h (\\NP)g(is)g(as)f(Easy)h(as)f(Detecting)h(Unique)h(Solutions",)f Fp(The)n(or)n(etic)n(al)143 688 y(Computer)g(Scienc)n(e)p Fu(,)c(V)l(ol.)i(47,)f(1986,)g(pp.)h(85{93.)49 787 y([35])22 b(A.)14 b(Wigderson,)h(D.)f(Zuc)o(k)o(erman,)g(\\)h(Expanders)g(that)g (Beat)f(the)h(Eigen)o(v)m(alue)i(Bound,)e(Explicit)i(Con-)143 849 y(struction)k(and)h(Applications",)j Fp(Pr)n(o)n(c.)d(of)g(the)g (25th)h(STOC)p Fu(,)d(pp.)i(245{251,)f(1993.)f(T)l(o)h(app)q(ear)h(in) 143 911 y Fp(Combinatoric)n(a)p Fu(.)49 1011 y([36])g(D.)d(Zuc)o(k)o (erman,)h(\\Sim)o(ulating)h(BPP)e(Using)i(a)e(General)h(W)l(eak)g (Random)g(Source,")h Fp(A)o(lgorithmic)n(a)p Fu(,)143 1073 y(V)l(ol.)15 b(16,)f(pp.)i(367{391,)c(1996.)49 1172 y([37])22 b(D.)11 b(Zuc)o(k)o(erman,)h(\\Randomness-Optimal)i (Sampling,)g(Extractors,)d(and)h(Constructiv)o(e)g(Leader)h(Elec-)143 1235 y(tion",)h Fp(28th)k(STOC)p Fu(,)13 b(1996,)g(pp.)j(286{295.)952 2795 y(25)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF