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%%Creator: dvips 5.55 Copyright 1986, 1994 Radical Eye Software
%%Title: primjocfin2.dvi
%%CreationDate: Mon Nov 14 08:42:43 1994
%%Pages: 35
%%PageOrder: Ascend
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%%EndComments
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y Fy(>)1033 1680 y Fu(p)p 1070 1680 101 2 v 1070 1714
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y FB(1)221 1773 y FE(of)g Fy(p)8 b Fu(\000)g FE(1)14
b(and)h(one)f(prime)h(factor)e Fy(q)832 1756 y Fr(0)830
1784 y FB(1)864 1773 y FE(of)h Fy(q)c Fu(\000)e FE(1)14
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1840 y(factored)f(parts)f Fy(F)354 1850 y Ft(p)372 1838
y Fo(0)372 1861 y Fx(1)406 1840 y FE(of)g Fy(p)479 1824
y Fr(0)479 1852 y FB(1)513 1840 y FE(and)h Fy(F)629 1850
y Ft(q)646 1838 y Fo(0)645 1861 y Fx(1)678 1840 y FE(of)g
Fy(q)751 1824 y Fr(0)749 1852 y FB(1)783 1840 y FE(are)f(greater)h
(than)1115 1792 y Fp(q)p 1157 1792 121 2 v 48 x Fy(p)1180
1825 y Fr(0)1180 1853 y FB(1)1210 1840 y Fu(\000)c FE(1)k(and)1379
1792 y Fp(q)p 1420 1792 119 2 v 1420 1840 a Fy(q)1442
1825 y Fr(0)1440 1853 y FB(1)1471 1840 y Fu(\000)c FE(1,)k(resp)q
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Fy(l)q(cm)p FE(\()p Fy(F)504 1907 y Ft(p)522 1898 y Fo(0)522
1915 y Fs(1)541 1900 y Fy(;)8 b(:)g(:)g(:)d(;)j(F)672
1907 y Ft(p)690 1898 y Fo(0)690 1915 y Fs(r)709 1900
y Fy(;)g(F)759 1907 y Ft(q)776 1898 y Fo(0)775 1915 y
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1898 y Fo(0)941 1915 y Fs(s)958 1900 y FE(\))18 b(=)h
Fy(l)q(cm)p FE(\()p Fy(F)1170 1907 y Ft(p)1188 1898 y
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y Ft(q)1273 1898 y Fo(0)1272 1915 y Fs(1)1292 1900 y
FE(\))18 b(of)g(Theorem)g(6)g(is)h(greater)35 1959 y(than)c
Fy(N)183 1942 y FB(1)p Ft(=)p FB(4)253 1959 y FE(when)h(\()p
Fy(F)419 1968 y Ft(p)437 1957 y Fo(0)437 1980 y Fx(1)456
1959 y Fy(;)8 b(F)506 1968 y Ft(q)523 1957 y Fo(0)522
1980 y Fx(1)540 1959 y FE(\))13 b(=)g(1.)106 2030 y(Theorem)h(7)h
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24 23 bop 35 140 a FE(It)14 b(follo)o(ws)g(from)f(\(13\))g(that)g(for)g
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199 y(o)o(v)o(er)i(the)h(in)o(terv)m(al)g([)p Fy(\015)s(;)8
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259 y(sizes)e(in)h(the)e(range)g(of)f(1)p Fy(=)p FE(2.)106
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508 y([)p Fy(N)85 515 y FB(1)104 508 y Fy(;)c(N)162 515
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638 y(result)14 b(in)f(an)g(en)o(tirely)g(di\013eren)o(t)g
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1751 y(of)j(the)g(pap)q(er.)35 1905 y Fn(App)r(endix)j(1:)25
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25 24 bop 35 140 a FE(where)16 b(the)f(functions)h Fy(F)470
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Ft(N)330 866 y(N)362 856 y Fs(\015)474 798 y Fy(d\031)r
FE(\()p Fy(x)p FE(\))p 424 818 215 2 v 424 860 a Fy(x)8
b FE(log\()p Fy(N)q(=x)p FE(\))655 829 y(=)703 771 y
Fp(Z)745 756 y Fr(p)p 774 756 32 2 v 28 x Ft(N)726 866
y(N)758 856 y Fs(\015)952 798 y Fy(dx)p 820 818 314 2
v 820 860 a(x)g FE(log)g Fy(x)g FE(log\()p Fy(N)q(=x)p
FE(\))1149 829 y(+)1195 771 y Fp(Z)1236 756 y Fr(p)p
1266 756 32 2 v 1266 784 a Ft(N)1218 866 y(N)1250 856
y Fs(\015)1367 798 y Fy(d\017)p FE(\()p Fy(x)p FE(\))p
1312 818 215 2 v 1312 860 a Fy(x)g FE(log\()p Fy(N)q(=x)p
FE(\))1531 829 y Fy(:)182 b FE(\(18\))35 955 y(The)15
b(\014rst)f(in)o(tegral)g(can)h(b)q(e)g(computed)f(b)o(y)g(using)h(the)
g(v)m(ariable)g(substitution)g Fy(y)g FE(=)e(\(log)8
b Fy(x)p FE(\))p Fy(=)p FE(\(log)f Fy(N)e FE(\),)35 1014
y(with)16 b Fy(dy)e FE(=)f Fy(dx=)p FE(\()p Fy(x)8 b
FE(log)f Fy(N)e FE(\),)14 b(whic)o(h)i(giv)o(es)262 1093
y Fp(Z)303 1078 y Fr(p)p 333 1078 32 2 v 333 1107 a Ft(N)285
1188 y(N)317 1178 y Fs(\015)511 1120 y Fy(dx)p 379 1140
314 2 v 379 1182 a(x)8 b FE(log)g Fy(x)g FE(log\()p Fy(N)q(=x)p
FE(\))710 1151 y(=)806 1120 y(1)p 763 1140 108 2 v 763
1182 a(log)g Fy(N)883 1093 y Fp(Z)925 1107 y FB(1)p Ft(=)p
FB(2)907 1188 y Ft(\015)1049 1120 y Fy(dy)p 993 1140
162 2 v 993 1182 a(y)r FE(\(1)h Fu(\000)h Fy(y)r FE(\))1172
1151 y(=)1225 1120 y(log\(1)g Fu(\000)g Fy(\015)s FE(\))f
Fu(\000)i FE(log)d Fy(\015)p 1225 1140 347 2 v 1344 1182
a FE(log)g Fy(N)1576 1151 y(:)137 b FE(\(19\))35 1275
y(Using)16 b(the)f(rule)i(for)d(in)o(tegration)h(b)o(y)g(parts,)528
1334 y Fp(Z)577 1391 y Fy(g)r FE(\()p Fy(x)p FE(\))p
Fy(d)-8 b(f)5 b FE(\()p Fy(x)p FE(\))11 b(=)i Fy(f)5
b FE(\()p Fy(x)p FE(\))p Fy(g)r FE(\()p Fy(x)p FE(\))j
Fu(\000)1056 1334 y Fp(Z)1105 1391 y Fy(f)d FE(\()p Fy(x)p
FE(\))p Fy(dg)r FE(\()p Fy(x)p FE(\))p Fy(;)35 1510 y
FE(for)16 b Fy(f)5 b FE(\()p Fy(x)p FE(\))13 b(=)i Fy(\017)p
FE(\()p Fy(x)p FE(\))h(and)g Fy(g)r FE(\()p Fy(x)p FE(\))d(=)h(1)p
Fy(=)p FE(\()p Fy(x)8 b FE(log)o(\()p Fy(N)q(=x)p FE(\)\),)15
b(allo)o(ws)h(us)g(to)f(transform)g(the)h(second)h(in)o(tegral)f(in)35
1569 y(\(18\):)259 1610 y Fp(Z)301 1595 y Fr(p)p 330
1595 32 2 v 28 x Ft(N)283 1705 y(N)315 1695 y Fs(\015)432
1637 y Fy(d\017)p FE(\()p Fy(x)p FE(\))p 377 1657 215
2 v 377 1699 a Fy(x)8 b FE(log\()p Fy(N)q(=x)p FE(\))608
1668 y(=)656 1608 y Fp(\024)750 1637 y Fy(\017)p FE(\()p
Fy(x)p FE(\))p 683 1657 V 683 1699 a Fy(x)g FE(log\()p
Fy(N)q(=x)p FE(\))902 1608 y Fp(\025)924 1588 y Fr(p)p
953 1588 32 2 v 28 x Ft(N)924 1712 y(N)956 1703 y Fs(\015)997
1668 y FE(+)1043 1610 y Fp(Z)1084 1595 y Fr(p)p 1113
1595 V 28 x Ft(N)1066 1705 y(N)1098 1695 y Fs(\015)1155
1668 y Fy(\017)p FE(\()p Fy(x)p FE(\))1252 1637 y(1)i(+)g(log)q(\()p
Fy(N)q(=x)p FE(\))p 1252 1657 259 2 v 1255 1702 a Fy(x)1281
1689 y FB(2)1308 1702 y FE(log)1367 1684 y FB(2)1386
1702 y FE(\()p Fy(N)q(=x)p FE(\))1528 1668 y Fy(dx:)135
b FE(\(20\))35 1784 y(W)l(e)14 b(no)o(w)e(mak)o(e)h(use)h(of)e(the)i(b)
q(ound)g Fy(\017)p FE(\()p Fy(x)p FE(\))e(=)h Fy(O)q
FE(\()p Fy(x=)8 b FE(log)971 1766 y FB(2)998 1784 y Fy(x)p
FE(\))13 b(whic)o(h)h(sho)o(ws)f(that)f(the)h(\014rst)g(term)g(on)g
(the)35 1843 y(righ)o(t)j(side)g(of)f(\(20\))g(is)h Fy(O)q
FE(\(1)p Fy(=)8 b FE(log)597 1825 y FB(3)624 1843 y Fy(N)d
FE(\).)21 b(In)16 b(order)f(to)g(sho)o(w)g(that)g(the)g(second)i(term)e
(is)h Fy(O)q FE(\(1)p Fy(=)8 b FE(log)1708 1825 y FB(2)1735
1843 y Fy(N)d FE(\),)35 1903 y(note)17 b(that)e(the)h(function)h(to)f
(b)q(e)h(in)o(tegrated)f(is)h Fy(O)q FE(\(1)p Fy(=)p
FE(\()p Fy(x)8 b FE(log)1088 1884 y FB(2)1115 1903 y
Fy(x)g FE(log)g Fy(N)d FE(\)\))15 b(whic)o(h)i(for)f
Fy(N)1552 1886 y Ft(\015)1588 1903 y Fu(\024)f Fy(x)f
Fu(\024)1728 1864 y(p)p 1766 1864 42 2 v 39 x Fy(N)35
1971 y FE(is)23 b(also)e Fy(O)q FE(\(1)p Fy(=)p FE(\()p
Fy(x)8 b FE(log)394 1952 y FB(3)421 1971 y Fy(N)d FE(\),)22
b(and)g(that)716 1936 y Fp(R)744 1921 y Fr(p)p 773 1921
32 2 v 29 x Ft(N)736 1985 y(N)768 1975 y Fs(\015)814
1971 y Fy(dx=x)h FE(=)g Fy(O)q FE(\(log)8 b Fy(N)d FE(\).)38
b(The)22 b(pro)q(of)f(of)h(the)f(theorem)h(is)35 2030
y(completed)17 b(b)o(y)e(com)o(bining)h(equations)g(\(17\))e(to)g
(\(20\).)49 b Fq(2)35 2271 y Fv(References)62 2381 y
Fi([1])20 b(L.M.)e(Adleman)g(and)h(M.A.)f(Huang,)h(Primalit)o(y)e
(testing)i(and)g(ab)q(elian)g(v)n(arieties)g(o)o(v)o(er)g(\014nite)g
(\014elds,)127 2433 y Fe(L)n(e)n(ctur)n(e)14 b(Notes)h(in)g
(Mathematics)p Fi(,)f(V)m(ol.)e(1512,)g(Springer)j(V)m(erlag,)d(1992.)
62 2523 y([2])20 b(L.M.)12 b(Adleman,)e(C.)j(P)o(omerance)f(and)g(R.S.)
g(Rumely)m(,)e(On)i(distinguishing)g(prime)f(n)o(um)o(b)q(ers)i(from)d
(com-)127 2575 y(p)q(osite)k(n)o(um)o(b)q(ers,)f Fe(A)o(nnals)i(of)g
(Mathematics)p Fi(,)f(V)m(ol.)e(117,)h(pp.)g(173-206,)f(1983.)899
2699 y FE(29)p eop
%%Page: 30 30
30 29 bop 62 140 a Fi([3])20 b(A.V.)11 b(Aho,)g(J.E.)g(Hop)q(croft)h
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(Computer)h(A)o(lgorithms)p Fi(,)127 193 y(Reading,)f(MA:)i(Addison-W)m
(esley)m(,)f(1974.)62 282 y([4])20 b(E.)14 b(Bac)o(h,)h(Ho)o(w)g(to)f
(generate)i(factored)g(random)d(n)o(um)o(b)q(ers,)h Fe(SIAM)i(Journal)g
(on)g(Computing)p Fi(,)f(V)m(ol.)e(17,)127 334 y(No.)g(4,)g(pp.)h
(173-193,)e(1988.)62 424 y([5])20 b(E.)14 b(Bac)o(h,)f(p)q(ersonal)i
(comm)o(unicati)o(on,)c(April)i(1992.)62 514 y([6])20
b(E.)10 b(Bac)o(h,)h(Exact)g(analysis)f(of)g(a)g(priorit)o(y)f(queue)j
(algorithm)c(for)i(random)f(v)n(ariate)g(generation,)i(to)g(app)q(ear)
127 566 y(in)i Fe(Pr)n(o)n(c.)i(5th)g(A)o(CM/SIAM)g(Symp.)g(on)h(Discr)
n(ete)e(A)o(lgorithms)g(\(SOD)o(A\))p Fi(,)f(1994.)62
656 y([7])20 b(E.)12 b(Bac)o(h)h(and)f(J.)g(Shallit,)f(F)m(actoring)g
(with)h(cyclotomic)f(p)q(olynomials,)e Fe(Mathematics)14
b(of)f(Computation)p Fi(,)127 708 y(V)m(ol.)f(52,)h(pp.)h(201-219,)d
(1989.)62 798 y([8])20 b(E.)f(Bac)o(h)i(and)e(J.)h(Shallit,)f
Fe(A)o(lgorithmic)g(numb)n(er)i(the)n(ory,)g(V)m(olume)f(I:)g
(E\016cient)g(A)o(lgorithms)p Fi(,)g(MIT)127 850 y(Press,)15
b(to)f(app)q(ear.)62 940 y([9])20 b(E.)d(Bac)o(h)i(and)e(J.)h
(Sorensen,)i(Siev)o(e)e(algorithms)d(for)i(p)q(erfect)j(p)q(o)o(w)o(er)
e(testing,)h Fe(A)o(lgorithmic)n(a)p Fi(,)e(V)m(ol.)f(9,)127
992 y(pp.)d(313-328,)f(1993.)42 1082 y([10])19 b(A.)14
b(Balog,)f Fg(p)c Fi(+)h Fg(a)k Fi(without)f(large)h(prime)f(factors,)i
(Seminaire)d(de)j(theorie)g(des)g(nom)o(bres)e(de)i(Bordeaux,)127
1134 y(No.)e(31,)g(1983.)42 1224 y([11])19 b(P)m(.)c(Beauc)o(hemin,)g
(G.)f(Brassard,)j(C.)d(Cr)o(\023)-20 b(ep)q(eau,)17 b(C.)e(Goutier)g
(and)g(C.)g(P)o(omerance,)g(The)h(generation)g(of)127
1276 y(random)f(n)o(um)o(b)q(ers)i(that)f(are)i(probably)e(prime,)g
Fe(Journal)h(of)h(Cryptolo)n(gy)p Fi(,)e(V)m(ol.)g(1,)g(No.)h(2,)f(pp.)
h(53-64,)127 1328 y(1988.)42 1418 y([12])i(B.)g(Blak)o(ely)f(and)g
(G.B.)g(Blak)o(ely)m(,)h(Securit)o(y)g(of)g(n)o(um)o(b)q(er)f
(theoretic)i(cryptosystems)f(against)g(random)127 1470
y(attac)o(ks,)14 b(I,)f Fe(Cryptolo)n(gia)p Fi(,)g(V)m(ol.)f(2,)h(No.)g
(4,)h(pp.)f(305-320,)f(Oct.)i(1978.)42 1560 y([13])19
b(D.)c(Bleic)o(hen)o(bac)o(her,)j(On)e(the)g(p)q(o)o(w)o(er)h(of)e
(pseudo-primalit)o(y)f(tests,)j(T)m(ec)o(h.)f(Rep.,)g(Dept.)f(of)h
(Computer)127 1612 y(Science,)f(ETH)f(Zuric)o(h,)g(Sept.)g(1993.)42
1702 y([14])19 b(D.)13 b(Bleic)o(hen)o(bac)o(her)i(and)f(U.M.)f
(Maurer,)h(Finding)f(All)g(Strong)h(Pseudoprimes)g Fd(\024)e
Fg(x)p Fi(,)h(preprin)o(t,)h(1993.)42 1792 y([15])19
b(M.)10 b(Blum)e(and)i(S.)g(Micali,)f(Ho)o(w)h(to)g(generate)i
(cryptographically)d(strong)i(sequences)h(of)e(pseudo-random)127
1844 y(bits,)j Fe(SIAM)j(Journal)e(on)i(Computing)p Fi(,)e(V)m(ol.)e
(13,)h(No.)g(4,)g(pp.)h(850-864,)d(1984.)42 1933 y([16])19
b(D.M.)13 b(Bressoud,)i Fe(F)m(actorization)f(and)i(Primality)e(T)m
(esting)p Fi(,)e(Berlin:)19 b(Springer-V)m(erlag,)13
b(1989.)42 2023 y([17])19 b(J.)g(Brillhart,)h(D.H.)e(Lehmer)h(and)g
(J.L.)f(Selfridge,)i(New)g(primalit)o(y)c(criteria)k(and)f
(factorizations)g(of)127 2075 y(2)148 2060 y Fc(m)188
2075 y Fd(\006)10 b Fi(1,)j Fe(Mathematics)i(of)g(Computation)p
Fi(,)f(V)m(ol.)e(29,)h(pp.)g(620-647,)f(1975.)42 2165
y([18])19 b(R.D.)e(Carmic)o(hael,)g(On)i(comp)q(osite)e(n)o(um)o(b)q
(ers)i Fg(P)k Fi(whic)o(h)c(satisfy)f(the)h(F)m(ermat)e(congruence)k
Fg(a)1686 2150 y Fc(P)t Fb(\000)p Fa(1)1775 2165 y Fd(\021)127
2217 y Fi(1)13 b(\(mo)q(d)g Fg(P)6 b Fi(\),)13 b Fe(A)o(meric)n(an)h
(Math.)h(Monthly)p Fi(,)f(V)m(ol.)f(19,)g(pp.)g(22-27,)f(1912.)42
2307 y([19])19 b(A.)12 b(Cobham,)e(The)j(recognition)f(problem)f(for)h
(the)h(set)g(of)f(p)q(erfect)i(squares,)f Fe(Pr)n(o)n(c.)g(7th)h(A)o
(nnual)g(Syp.)g(on)127 2359 y(Switching)h(and)g(A)o(utomata)g(The)n
(ory)p Fi(,)f(pp)f(78-87,)g(1966.)42 2449 y([20])19 b(H.)c(Cohen)g(and)
g(A.K.)g(Lenstra,)h(Implemen)o(tation)c(of)j(a)g(new)g(primalit)o(y)e
(test,)j Fe(Mathematics)g(of)g(Com-)127 2501 y(putation)p
Fi(,)e(V)m(ol.)e(48,)h(No.)g(177,)g(pp.)g(103-121,)f(1987.)899
2699 y FE(30)p eop
%%Page: 31 31
31 30 bop 42 140 a Fi([21])19 b(D.)13 b(Copp)q(ersmith,)g(A.M.)g
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(in)i Fg(GF)6 b Fi(\()p Fg(p)p Fi(\),)12 b Fe(A)o(lgorith-)127
193 y(mic)n(a)p Fi(,)h(V)m(ol.)f(1,)i(pp.)f(1-15,)g(1986.)42
282 y([22])19 b(C.)12 b(Couvreur)h(and)g(J.J.)f(Quisquater,)i(An)e(in)o
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(b)q(ers,)127 334 y Fe(Philips)i(Journal)h(of)g(R)n(ese)n(ar)n(ch)p
Fi(,)e(V)m(ol.)g(37,)g(pp.)g(231-264,)f(1982,)g(\(errata:)19
b(id,)13 b(V)m(ol.)f(38.,)h(p.)g(77,)g(1983\).)42 424
y([23])19 b(I.)d(Damg)-5 b(\027)-26 b(ard,)14 b(P)m(.)i(Landro)q(c)o(k)
g(and)h(C.)f(P)o(omerance,)g(Av)o(erage)h(Case)g(Error)g(Estimates)f
(for)g(the)h(Strong)127 476 y(Probable)d(Prime)f(T)m(est,)h
Fe(Mathematics)h(of)g(Computation)p Fi(,)e(V)m(ol.)g(61,)g(pp.)g
(177-194,)f(1993.)42 566 y([24])19 b(J.)13 b(v)n(an)g(de)h(Lune)g(and)g
(E.)f(W)m(attel,)f(On)i(the)g(n)o(umerical)e(solution)h(of)g(a)g
(di\013eren)o(tial-di\013erence)i(equation)127 618 y(arising)e(in)g
(analytic)g(n)o(um)o(b)q(er)h(theory)m(,)f Fe(Mathematics)i(of)g
(Computation)p Fi(,)f(V)m(ol.)e(23,)h(pp.)h(417-421,)d(1969.)42
708 y([25])19 b(R.)c(De)g(Moliner,)g(E\016zien)o(te)i(Konstruktion)f
(zuf\177)-21 b(alliger)14 b(grosser)j(Primzahlen,)d(Diploma)f(Thesis,)j
(Inst.)127 760 y(for)d(Signal)g(and)h(Information)d(Pro)q(cessing,)k
(Swiss)f(F)m(ederal)g(Institute)h(of)e(T)m(ec)o(hnology)m(,)f(Zuric)o
(h,)i(1989.)42 850 y([26])19 b(H.G.)12 b(Diamond,)f(Elemen)o(tary)i
(metho)q(ds)g(in)h(the)g(study)g(of)g(the)g(distribution)f(of)h(prime)e
(n)o(um)o(b)q(ers,)h Fe(Bul-)127 902 y(letin)h(A)o(m.)h(Math.)g(So)n
(c.)f Fi(\(New)g(Series\),)h(V)m(ol.)d(7,)h(No.)h(3,)f(1982.)42
992 y([27])19 b(K.)h(Dic)o(kman,)f(On)h(the)h(frequency)h(of)d(n)o(um)o
(b)q(ers)h(con)o(taining)f(prime)g(factors)i(of)e(a)h(certain)h
(relativ)o(e)127 1044 y(magnitude,)12 b Fe(A)o(rkiv)i(for)g(Matematik,)
h(Astr)n(onomi)f(o)n(ch)h(Fysik)p Fi(,)f(V)m(ol.)e(22A,)h(No.)g(10,)g
(pp.)h(1-14,)e(1930.)42 1134 y([28])19 b(W.)c(Di\016e)f(and)i(M.E.)f
(Hellman,)e(New)j(directions)g(in)f(cryptograph)o(y)m(,)h
Fe(IEEE)g(T)m(r)n(ansactions)h(on)f(Infor-)127 1186 y(mation)f(The)n
(ory)p Fi(,)e(V)m(ol.)f(22,)h(No.)h(6,)f(pp.)g(644-654,)f(1976.)42
1276 y([29])19 b(B.)14 b(Dixon)g(and)g(A.K.)g(Lenstra,)h(Massiv)o(ely)f
(parallel)g(elliptic)f(curv)o(e)j(factoring,)d Fe(A)n(dvanc)n(es)k(in)e
(Cryptol-)127 1328 y(o)n(gy)h(-)g(EUR)o(OCR)m(YPT)g('92)p
Fi(,)f(Lecture)i(Notes)f(in)e(Computer)h(Science,)h(V)m(ol.)e(658,)g
(pp.)h(183-193,)e(Berlin:)127 1381 y(Springer-V)m(erlag,)g(1993.)42
1470 y([30])19 b(T.)14 b(El-Gamal)o(,)d(A)j(public)g(k)o(ey)h
(cryptosystem)f(and)g(a)g(signature)h(sc)o(heme)f(based)h(on)f(the)h
(discrete)g(loga-)127 1523 y(rithm,)d Fe(IEEE)j(T)m(r)n(ansactions)g
(on)g(Information)g(The)n(ory)p Fi(,)e(V)m(ol.)g(31,)f(No.)i(4,)f(pp.)g
(469-472,)f(1985.)42 1612 y([31])19 b(P)m(.)9 b(Erd\177)-21
b(os,)10 b(On)g(the)g(normal)e(n)o(um)o(b)q(er)h(of)g(prime)f(factors)i
(of)f Fg(p)q Fd(\000)q Fi(1)f(and)i(some)f(related)h(problems)e
(concerning)127 1664 y(Euler's)14 b Fg(')p Fi(-function,)f
Fe(Quarterly)h(Journal)h(of)g(Mathematics)p Fi(,)e(Oxford,)h(V)m(ol.)e
(6,)h(pp.)h(205-213,)d(1935.)42 1754 y([32])19 b(A.)9
b(Fiat)g(and)g(A.)g(Shamir,)f(Ho)o(w)h(to)g(pro)o(v)o(e)g(y)o(ourself:)
16 b(practical)9 b(solution)g(to)g(iden)o(ti\014cation)g(and)g
(signature)127 1806 y(problems,)k Fe(A)n(dvanc)n(es)j(in)g(Cryptolo)n
(gy)f(-)g(CR)m(YPTO)g('86)p Fi(,)f(Lecture)j(Notes)e(in)f(Computer)g
(Science,)i(V)m(ol.)127 1859 y(263,)d(pp.)g(186-194,)f(Berlin:)18
b(Springer-V)m(erlag,)13 b(1987.)42 1948 y([33])19 b(J.B.)c(F)m
(riedlander,)g(Shifted)g(primes)f(without)h(large)f(prime)g(factors,)h
(in)g Fe(Numb)n(er)h(the)n(ory)f(and)i(applic)n(a-)127
2001 y(tions)p Fi(,)c(R.A.)g(Mollin)f(\(ed.\),)i(Klu)o(w)o(er)g
(Academic)f(Publishers,)i(pp.)e(393-401,)f(1989.)42 2090
y([34])19 b(M.)10 b(Goldfeld,)f(On)i(the)h(n)o(um)o(b)q(er)d(of)h
(primes)g Fg(p)h Fi(for)f(whic)o(h)g Fg(p)s Fi(+)s Fg(a)g
Fi(has)h(a)f(large)h(prime)e(factor,)i Fe(Mathematika)p
Fi(,)127 2143 y(V)m(ol.)h(16,)h(pp.)h(23-27,)e(1969.)42
2232 y([35])19 b(S.)e(Goldw)o(asser)h(and)g(J.)g(Kilian,)f(Almost)g
(all)f(primes)i(can)g(b)q(e)g(quic)o(kly)f(certi\014ed,)j
Fe(Pr)n(o)n(c.)e(of)h(the)g(18th)127 2285 y(A)o(nnual)c(A)o(CM)g(Symp)n
(osium)g(on)h(the)f(The)n(ory)f(of)h(Computing)p Fi(,)f(pp.)f(316-329,)
f(1986.)42 2374 y([36])19 b(S.)14 b(Goldw)o(asser)h(and)f(S.)g(Micali,)
f(Probabilistic)h(encryption,)h Fe(J.)g(of)h(Computer)f(and)h(System)g
(Scienc)n(es)p Fi(,)127 2427 y(V)m(ol.)c(28,)h(pp.)h(270-299,)d(1984.)
42 2516 y([37])19 b(J.)14 b(Gordon,)e(Strong)i(RSA)g(Keys,)g
Fe(Ele)n(ctr)n(onics)g(L)n(etters)p Fi(,)f(V)m(ol.)f(20,)h(No.)g(12,)g
(1984.)899 2699 y FE(31)p eop
%%Page: 32 32
32 31 bop 42 140 a Fi([38])19 b(A.)f(Gran)o(ville,)g(Primalit)o(y)e(T)m
(esting)i(and)h(Carmic)o(hael)e(Num)o(b)q(ers,)i Fe(Notic)n(es)g(of)g
(the)g(A)o(meric)n(an)g(Math.)127 193 y(So)n(ciety)p
Fi(,)14 b(pp.)f(696-700,)f(1992.)42 282 y([39])19 b(L.C.)g(Guillou)f
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Fe(A)n(dvanc)n(es)15 b(in)f(Cryptolo)n(gy)f(-)h(EUR)o(O-)127
387 y(CR)m(YPT)k('88)p Fi(,)h(Lecture)g(Notes)g(in)e(Computer)g
(Science,)j(V)m(ol.)d(330,)g(pp.)h(123-128,)f(Berlin:)26
b(Springer-)127 439 y(V)m(erlag,)13 b(1988.)42 529 y([40])19
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(1-70,)f(1922.)42 671 y([41])19 b(C.)13 b(Ho)q(oley)m(,)g(On)h(the)h
(largest)f(prime)e(factor)i(of)g Fg(p)9 b Fi(+)g Fg(a)p
Fi(,)k Fe(Mathematika)p Fi(,)h(V)m(ol.)f(20,)g(pp.)g(135-143,)f(1973.)
42 760 y([42])19 b(G.)12 b(Jaesc)o(hk)o(e,)h(On)g(strong)g
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(Computation)p Fi(,)g(V)m(ol.)e(61,)127 813 y(pp.)i(915-926,)f(1993.)42
902 y([43])19 b(S.H.)d(Kim)f(and)i(C.)f(P)o(omerance,)h(The)h
(probabilit)o(y)d(that)i(a)g(random)e(probable)i(prime)f(is)g(comp)q
(osite,)127 955 y Fe(Mathematics)f(of)g(Computation)p
Fi(,)f(V)m(ol.)e(53,)h(pp.)g(721-741,)f(1989.)42 1044
y([44])19 b(D.E.)d(Kn)o(uth)i(and)g(L.)f(T)m(rabb)g(P)o(ardo,)h
(Analysis)f(of)g(a)g(simple)f(factorization)h(algorithm,)e
Fe(The)n(or)n(etic)n(al)127 1097 y(Computer)f(Scienc)n(e)p
Fi(,)h(V)m(ol.)d(3,)h(pp.)g(321-348,)f(1976.)42 1186
y([45])19 b(N.)13 b(Koblitz,)h Fe(A)g(Course)h(in)g(Numb)n(er)f(The)n
(ory)h(and)h(Crypto)n(gr)n(aphy)p Fi(,)d(Berlin:)18 b(Springer-V)m
(erlag,)13 b(1987.)42 1276 y([46])19 b(N.)c(Koblitz,)f(Primalit)o(y)f
(of)h(the)i(n)o(um)o(b)q(er)e(of)h(p)q(oin)o(ts)g(on)g(an)g(elliptic)f
(curv)o(e)i(o)o(v)o(er)f(a)g(\014nite)g(\014eld,)g Fe(Paci\014c)127
1328 y(Journal)g(of)f(Mathematics)p Fi(,)g(V)m(ol.)e(131,)h(No.)g(1,)h
(pp.)f(157-165,)f(1988.)42 1418 y([47])19 b(K.)f(Ko)o(y)o(ama,)f(U.M.)g
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1470 y Fi(,)h Fe(A)n(dvanc)n(es)i(in)e(Cryptolo)n(gy)h(-)f(CR)m(YPTO)g
('91)p Fi(,)g(Lecture)127 1523 y(Notes)e(in)e(Computer)g(Science,)i(V)m
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b(1992.)42 1612 y([48])19 b(E.)12 b(Kranakis,)h Fe(Primality)g(and)h
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(ork:)k(John)c(Wiley)f(&)127 1664 y(Sons,)i(1986.)42
1754 y([49])19 b(A.K.)14 b(Lenstra,)h(Primalit)o(y)d(testing,)j(in)f
Fe(Cryptolo)n(gy)i(and)g(c)n(omputational)g(numb)n(er)g(the)n(ory)p
Fi(,)e(C.)g(P)o(omer-)127 1806 y(ance)19 b(\(ed.\),)h(Pro)q(c.)f(of)f
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(American)g(Mathematical)127 1859 y(So)q(ciet)o(y)m(,)13
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(-)f(Dec.)h(1,)g(1994.)42 2090 y([51])19 b(A.K.)14 b(Lenstra)i(and)e
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y(V)m(erlag,)c(1991.)42 2285 y([52])19 b(A.K.)d(Lenstra)h(and)f(H.W.)g
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Fi(,)e(pp.)h(564-572,)e(1990.)899 2699 y FE(32)p eop
%%Page: 33 33
33 32 bop 42 140 a Fi([54])19 b(H.W.)14 b(Lenstra,)j(Jr.,)e(F)m
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Fe(A)o(nnals)f(of)h(Mathematics)p Fi(,)e(V)m(ol.)f(126,)127
193 y(pp.)f(649-673,)f(1987.)42 282 y([55])19 b(U.M.)10
b(Maurer,)i(F)m(ast)f(generation)g(of)g(secure)i(RSA-mo)q(duli)8
b(with)j(almost)e(maxima)o(l)f(div)o(ersit)o(y)m(,)i
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Fe(A)n(dvanc)n(es)j(in)g(Cryptolo)n(gy)g(-)g(CR)m(YPTO)f('94)p
Fi(,)g(Y.)f(Desmedt\(Ed.\),)h(Lecture)h(Notes)127 775
y(in)g(Computer)g(Science,)i(Berlin:)j(Springer-V)m(erlag,)c(V)m(ol.)e
(839,)h(pp.)g(271-281,)f(1994.)42 865 y([58])19 b(U.M.)14
b(Maurer)h(and)f(Y.)g(Y)m(acobi,)f(Non-in)o(teractiv)o(e)i(public-k)o
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917 y(o)n(gy)h(-)g(EUR)o(OCR)m(YPT)g('91)p Fi(,)f(Lecture)i(Notes)f(in)
e(Computer)h(Science,)h(V)m(ol.)e(547,)g(pp.)h(498-507,)e(Berlin:)127
970 y(Springer-V)m(erlag,)g(1991.)42 1059 y([59])19 b(K.)14
b(McCurley)m(,)f(The)i(discrete)g(logarithm)d(problem,)g(in)h
Fe(Cryptolo)n(gy)i(and)h(c)n(omputational)f(numb)n(er)g(the-)127
1112 y(ory)p Fi(,)k(C.)f(P)o(omerance)h(\(ed.\),)h(Pro)q(c.)f(of)f
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(American)127 1164 y(Mathematical)12 b(So)q(ciet)o(y)m(,)h(1990.)42
1254 y([60])19 b(A.)13 b(Menezes,)j Fe(El)r(liptic)e(curve)h(public)g
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(1993.)42 1343 y([61])19 b(P)m(.)11 b(Mihailescu,)h(F)m(ast)f
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(282-293,)d(1994.)42 1537 y([62])19 b(G.L.)13 b(Miller,)h(Riemann's)f
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(algo-)127 1732 y(rithms,)12 b Fe(The)n(or)n(etic)n(al)i(Computer)h
(Scienc)n(e)p Fi(,)f(V)m(ol.)e(12,)h(pp.)g(97-108,)f(1980.)42
1821 y([64])19 b(F.)f(Morain,)h(Distributed)h(primalit)o(y)c(pro)o
(ving)i(and)g(the)i(primalit)o(y)c(of)i(\(2)1349 1806
y Fa(3539)1430 1821 y Fi(+)13 b(1\))p Fg(=)p Fi(3,)19
b Fe(A)n(dvanc)n(es)h(in)127 1874 y(Cryptolo)n(gy)c(-)g(EUR)o(OCR)m
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(LIX/92/RR/11,)d(Lab)q(oratoire)i(d'Informatique)e(de)j(l'Ecole)f(P)o
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b(D.A.)10 b(Plaisted,)h(F)m(ast)h(v)o(eri\014cation,)f(testing,)g(and)g
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(1979,)h(\(errata:)18 b(id.,)13 b(V)m(ol)g(14.,)f(p.)i(345,)e(1981\).)
899 2699 y FE(33)p eop
%%Page: 34 34
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245 y(1914-1916.)42 332 y([70])h(S.C.)11 b(P)o(ohlig)g(and)h(M.E.)g
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(logarithms)g(o)o(v)o(er)j Fg(GF)6 b Fi(\()p Fg(p)p Fi(\))127
384 y(and)12 b(its)g(cryptographic)g(signi\014cance,)h
Fe(IEEE)g(T)m(r)n(ansactions)g(on)h(Information)f(The)n(ory)p
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523 y([71])19 b(J.M.)13 b(P)o(ollard,)f(Theorems)h(on)g(factorization)g
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(the)g(Cambridge)127 575 y(Philosophic)n(al)h(So)n(ciety)p
Fi(,)e(V)m(ol.)g(76,)g(pp.)g(521-528,)f(1974.)42 662
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(Euler's)g(function,)f Fe(Mathematika)p Fi(,)h(V)m(ol.)e(27,)h(pp.)h
(84-89,)e(1980.)42 749 y([73])19 b(C.)g(P)o(omerance,)i(F)m(actoring,)f
(in)g Fe(Cryptolo)n(gy)g(and)h(c)n(omputational)g(numb)n(er)f(the)n
(ory)p Fi(,)h(C.)e(P)o(omerance)127 801 y(\(ed.\),)14
b(Pro)q(c.)h(of)f(Symp.)f(in)h(Applied)g(Math.,)g(V)m(ol.)f(42,)g(pp.)h
(27-47,)f(American)h(Mathematical)f(So)q(ciet)o(y)m(,)127
853 y(1990.)42 940 y([74])19 b(K.)13 b(Prac)o(har,)357
930 y(\177)352 940 y(Ub)q(er)h(die)g(Anzahl)f(der)i(T)m(eiler)e(einer)h
(nat)q(\177)-22 b(urlic)o(hen)14 b(Zahl,)e(w)o(elc)o(he)i(die)g(F)m
(orm)e Fg(p)c Fd(\000)h Fi(1)k(hab)q(en,)127 993 y Fe(Monatshefte)i(f)q
(\177)-22 b(ur)15 b(Mathematik)p Fi(,)e(V)m(ol.)g(59,)g(pp.)g(91-97,)f
(1955.)42 1079 y([75])19 b(V.R.)9 b(Pratt,)i(Ev)o(ery)g(prime)f(has)g
(a)g(succinct)i(certi\014cate,)h Fe(SIAM)f(Journal)f(on)i(Computing)p
Fi(,)e(V)m(ol.)e(4,)h(No.)g(3,)127 1132 y(pp.)j(214-220,)f(1975.)42
1218 y([76])19 b(M.O.)11 b(Rabin,)f(Probabilistic)g(algorithm)f(for)i
(testing)g(primalit)o(y)m(,)d Fe(Journal)13 b(of)f(Numb)n(er)g(The)n
(ory)p Fi(,)f(V)m(ol.)f(12,)127 1271 y(pp.)j(128-138,)f(1980.)42
1358 y([77])19 b(H.)14 b(Riesel,)g Fe(Prime)g(numb)n(ers)i(and)g(c)n
(omputer)f(metho)n(ds)h(for)f(factorization)p Fi(,)e(Boston,)i(Basel,)f
(Stuttgart:)127 1410 y(Birkh\177)-21 b(auser,)14 b(1985.)42
1497 y([78])19 b(R.L.)11 b(Riv)o(est,)h(Remarks)f(on)h(a)g(prop)q(osed)
h(cryptanalytic)f(attac)o(k)h(on)f(the)h(M.I.T.)e(public)h(k)o(ey)g
(cryptosys-)127 1549 y(tem,)g Fe(Cryptolo)n(gia)p Fi(,)h(V)m(ol.)f(2,)i
(No.)f(1,)g(pp.)g(62-65,)g(Jan.)g(1978.)42 1636 y([79])19
b(R.L.)f(Riv)o(est,)j(A.)e(Shamir,)g(and)g(L.)g(Adleman,)h(A)f(metho)q
(d)g(for)h(obtaining)e(digital)g(signatures)i(and)127
1688 y(public-k)o(ey)13 b(cryptosystems,)i Fe(Communic)n(ations)g(of)g
(the)g(A)o(CM)p Fi(,)e(V)m(ol.)f(21,)h(No.)g(2,)g(pp.)h(120-126,)d
(1978.)42 1775 y([80])19 b(C.P)m(.)12 b(Sc)o(hnorr,)i(E\016cien)o(t)f
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(Notes)f(in)f(Computer)f(Science,)i(V)m(ol.)e(435,)g(pp.)h(239-252,)e
(Berlin:)18 b(Springer-)127 1879 y(V)m(erlag,)13 b(1990.)42
1966 y([81])19 b(A.)11 b(Sc)o(h\177)-21 b(onhage)12 b(and)g(V.)f
(Strassen,)i(Sc)o(hnelle)f(Multiplik)n(ation)d(grosser)k(Zahlen,)f
Fe(Computing)p Fi(,)g(V)m(ol.)e(7,)h(pp.)127 2018 y(281-292,)h(1971.)42
2105 y([82])19 b(A.)14 b(Shamir,)e(E\016cien)o(t)j(signature)g(sc)o
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(ear)h(in)f(Pro)q(c.)127 2158 y(of)f(CR)m(YPTO)h('93.)42
2244 y([83])19 b(J.)13 b(Sha)o(w)o(e-T)m(a)o(ylor,)f(Generating)i
(strong)g(primes,)e Fe(Ele)n(ctr)n(onics)i(L)n(etters)p
Fi(,)e(V)m(ol.)g(22,)h(No.)g(16,)g(pp.)g(875-877,)127
2297 y(1986.)42 2383 y([84])19 b(G.)d(Simmons)e(and)j(M.)g(Norris,)g
(Preliminary)e(commen)o(ts)h(on)g(the)i(M.I.T)e(public)h(k)o(ey)g
(cryptosystem,)127 2436 y Fe(Cryptolo)n(gia)p Fi(,)c(V)m(ol.)f(1,)h
(No.)g(4,)g(pp.)h(406-414,)e(Oct.)i(1977.)42 2523 y([85])19
b(R.)c(Solo)o(v)n(a)o(y)f(and)h(V.)h(Strassen,)h(A)f(fast)g(Mon)o
(te-Carlo)f(test)i(for)e(primalit)o(y)m(,)e Fe(SIAM)k(Journal)f(on)h
(Com-)127 2575 y(puting)p Fi(,)d(V)m(ol.)e(6,)h(No.)g(1,)h(pp.)f
(84-85,)f(1977)h(\(errata:)19 b(ibid.,)12 b(V)m(ol.)g(7,)h(p.)h(118,)e
(1978\).)899 2699 y FE(34)p eop
%%Page: 35 35
35 34 bop 42 140 a Fi([86])19 b(G.)13 b(T)m(ren)o(ta,)i(W)m(erkzeuge)g
(zur)g(Realisierung)f(eines)i(RSA-Kryptosystems,)e(Diploma)d(Thesis,)k
(Dept.)g(of)127 193 y(Computer)e(Science,)i(Swiss)f(F)m(ederal)g
(Institute)h(of)e(T)m(ec)o(hnology)m(,)f(Marc)o(h)i(1990.)42
282 y([87])19 b(H.C.)13 b(William)o(s,)e(A)k Fg(p)9 b
Fi(+)h(1)j(metho)q(d)h(of)f(factoring,)g Fe(Mathematics)j(of)f
(Computation)p Fi(,)f(V)m(ol.)f(39,)g(No.)g(159,)127
334 y(pp.)g(225-234,)f(1982.)42 424 y([88])19 b(H.C.)11
b(William)o(s)f(and)i(B.)h(Sc)o(hmid,)d(Some)h(remarks)h(concerning)h
(the)g(M.I.T.)e(public-k)o(ey)h(cryptosystem,)127 476
y Fe(BIT)p Fi(,)h(V)m(ol.)g(19,)f(pp.)i(525-538,)e(1979.)42
566 y([89])19 b(K.)e(W)m(o)q(oldridge,)e(V)m(alues)i(tak)o(en)g(man)o
(y)e(times)h(b)o(y)h(Euler's)g(phi-function,)f Fe(Pr)n(o)n(c)n(e)n(e)n
(dings)i(of)f(the)h(AMS)p Fi(,)127 618 y(V)m(ol.)12 b(76,)h(pp.)h
(229-234,)d(1979.)42 708 y([90])19 b(Sp)q(eci\014cations)g(for)e(a)h
(digital)e(signature)i(standard,)h(US)e(F)m(ederal)h(Register,)h(V)m
(ol.)d(56,)i(No.)f(169,)h(Au-)127 760 y(gust)c(30,)f(1991.)899
2699 y FE(35)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF