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FA(mo)r(d)23 b Fy(n)g Fv(==)g Fs(\006)p Fy(1)p FA(\))28 b Fo(c)l(ontinue)p FA(;)f(//)g Fy(n)h FA(pre\262lo)e(testom)i(pre)f(toto)g Fy(b)h FA(a)f(p)r (okra\243ujeme)g(\244alej)3249 2009 y Fj(1)259 2140 y Fo(else)h FA({)425 2240 y Fy(l)c Fv(=)f Fy(b)614 2210 y Fw(t)672 2240 y FA(mo)r(d)g Fy(n)p FA(;)425 2340 y Fo(for)29 b FA(\()p Fy(j)23 b Fv(=)g Fy(1)p FA(;)28 b Fy(j)22 b(<)h(s)p FA(;)28 b Fy(j)p FA(++\){)508 2441 y Fy(l)c Fv(=)f Fy(l)678 2410 y Fw(2)741 2441 y FA(mo)r(d)h Fy(n)p FA(;)508 2541 y Fo(if)29 b FA(\()p Fy(l)24 b Fv(==)e(-)p Fy(1)p FA(\))28 b Fo(br)l(e)l(ak)p FA(;)195 b(//)26 b Fy(n)i FA(pre\262lo)f(testom)g(pre)g(toto)g Fy(b)h FA(a)f(p)r (okra\243ujeme)g(\244alej)425 2641 y(})425 2741 y Fo(if)i FA(\()p Fy(l)24 b Fs(6)p Fv(=)e(-)p Fy(1)p FA(\))29 b Fo(r)l(eturn)d FA(\020)7 b Fy(n)27 b FA(je)h(zlo\272en\351)f (\243\355slo\021;)259 2842 y(})93 2942 y(})93 3042 y Fo(r)l(eturn)g FA(\020)7 b Fy(n)27 b FA(je)h(prv)n(o)r (\243\355slo\021;)-64 3193 y(Matematic)n(ky)34 b(sa)h(d\341)g (vyjadri\264)g(fakt,)i(\272e)f(pre)f(dan\351)g Fy(n)h FA(a)f(zv)n(olen\351)f Fy(b)i FA(\()p Fy(1)g Fs(\024)g Fy(b)g(<)g(n)p FA(\))g(prejde)g Fy(n)f FA(pre)g Fy(b)h FA(testom)-189 3292 y(nasledo)n(vne:)624 3476 y Fy(b)673 3442 y Fw(t)731 3476 y FA(mo)r(d)24 b Fy(n)f Fv(=)g Fs(\006)p Fy(1)64 b Fp(_)h Fs(9)p Fy(j)23 b Fs(2)g Fp(f)p Fy(1;)14 b(:)g(:)g(:)g(;)g(s)k Fv(-)g Fy(1)p Fp(g)23 b Fv(:)42 b Fy(b)2142 3442 y Fw(t)p Fr(\001)p Fw(2)2229 3417 y Fi(j)2285 3476 y FA(mo)r(d)23 b Fy(n)g Fv(=)g(-)p Fy(1:)643 b FA(\(1.2\))-189 3660 y(Pri)39 b(splnen\355)h(vlastnosti)e(\(1.2\))h (sa)f(zvy\243a)5 b(jne)38 b(ho)n(v)n(or\355,)i(\272e)f Fy(n)h FA(je)f(siln\375m)g(pseudoprv)n(o)r(\243\355slom)e(vzh\251adom)i (na)g(b\341zu)-189 3759 y Fy(b)p FA(.)47 b(Miller-Rabino)n(v)30 b(test)h(je)g(pra)n(vdep)r(o)r(dobnostn\375)e(algoritm)n(us,)i (tak\272e)f(niek)n(edy)h(m\364\272e)f(da\264)h(nespr\341)n(vn)n(u)f(o)r (dp)r(o)n(v)n(e\244.)-189 3859 y(K)-7 b(v)i(alita)27 b(tohoto)g(algoritm)n(u)f(vypl\375v)-5 b(a)27 b(z)g(nasledo)n(vnej)g(v) n(et)n(y)g(\(d\364k)-5 b(az)27 b(p)r(ozri)g([K)n(ob87a)m(]\).)-189 4027 y FB(V)-8 b(eta)32 b(1.1.2.)40 b Fo(A)n(k)35 b Fy(n)g Fo(je)h(zlo\272en\351)g(nep)l(\341rne)f(\243\355slo,)k(tak)c (vlastnos\264)43 b FA(\(1.2\))35 b Fo(plat\355)h(pr)l(e)g(najviac)h Fy(25)p Fo(\045)f(v\262etk\375ch)g Fy(b)g Fo(\()p Fy(1)d Fs(\024)-189 4126 y Fy(b)23 b(<)g(n)p Fo(\).)-64 4294 y FA(T)-7 b(ak\272e)26 b(Miller-Rabino)n(v)g(test)i(d\341)n(v)-5 b(a)27 b(tak)n(\351to)f(v\375sledky:)-64 4461 y Fs(\017)41 b FA(ak)27 b(je)g Fy(n)h FA(prv)n(o)r(\243\355slo)e(\025)h(o)r(dp)r(o)n (v)n(e\244)g(je)h(v\272dy)f(spr\341)n(vna:)35 b(\022)7 b Fy(n)27 b FA(je)h(prv)n(o)r(\243\355slo\020)7 b(;)-64 4629 y Fs(\017)41 b FA(ak)27 b(je)i Fy(n)g FA(zlo\272en\351)f (nep\341rne)g(\243\355slo)g(\025)g(pra)n(vdep)r(o)r(dobnos\264)f(om)n (ylu,)h(t.j.)40 b(o)r(dp)r(o)n(v)n(ede)28 b(\022)7 b Fy(n)28 b FA(je)h(prv)n(o)r(\243\355slo\020)7 b(,)27 b(je)i(na)5 b(jviac)48 4696 y Fw(1)p 28 4710 77 4 v 28 4759 a(4)64 4742 y Fi(k)115 4729 y FA(.)-189 4897 y(Dobr\341)24 b(spr\341)n(v)-5 b(a)22 b(je,)k(\272e)e(s)h(rast\372cim)e Fy(k)i FA(pra)n(vdep)r(o)r(dobnos\264)e(om)n(ylu)h(v)n(e\251mi)g (r\375c)n(hlo)g(\(exp)r(onenci\341lne\))g(kles\341.)35 b(M\364\272eme)24 b(si)-189 4996 y(stano)n(vi\264)d(p)r(o\272ado)n(v)-5 b(an)n(\372)19 b(\372ro)n(v)n(e\254)h(om)n(ylu)h(\(nen)n(ulo)n(v\372\)) g(a)g(vyp)r(o)r(\243\355ta\264)g(ho)r(dnotu)h(parametra)d Fy(k)j FA(tak,)h(ab)n(y)d(sme)h(ju)h(dosiahli.)-189 5215 y FB(Mo)s(dul\341rne)31 b(umo)s(c\254o)m(v)-5 b(anie)-189 5369 y FA(Pri)36 b(inicializ\341cii)g(RSA)g(ak)n(o)f(a)5 b(j)36 b(pri)g(samotnom)f(\262ifro)n(v)-5 b(an\355)35 b(a)g(de\262ifro)n(v)-5 b(an\355)35 b(sa)h(p)r(ou\272\355v)-5 b(a)36 b(op)r(er\341cia)e(umo)r(c\254o)n(v)-5 b(ania)35 b(v)-189 5468 y(mo)r(dul\341rnej)29 b(aritmetik)n(e.)41 b(T)-7 b(o)29 b(znamen\341,)g(\272e)g(p)r(otrebujeme)g(efekt\355vn)n(y) g(algoritm)n(us,)f(ktor\375)g(p)r(o)r(\243\355ta)i Fy(v)c Fv(=)f Fy(a)3242 5438 y Fw(b)3311 5468 y FA(mo)r(d)e Fy(n)p FA(,)-189 5568 y(pri\243om)k(\243\355sla)g Fy(a;)14 b(b;)g(n)27 b FA(m\364\272u)g(ma\264)h(niek)n(o\251k)n(o)e(sto)h(aleb)r (o)g(tis\355c)h(bito)n(v.)-64 5668 y(Nec)n(h)k Fy(b)f Fv(=)h(\()p Fy(b)404 5680 y Fw(k)462 5668 y Fy(:)14 b(:)g(:)f(b)621 5680 y Fw(1)662 5668 y Fy(b)711 5680 y Fw(0)751 5668 y Fv(\))783 5680 y Fw(2)857 5668 y FA(je)33 b(reprezen)n(t\341cia)d (\243\355sla)i Fy(b)g FA(v)h(bin\341rnej)f(s\372sta)n(v)n(e.)51 b(Na)5 b(jjedno)r(duc)n(h\262\355)32 b(algoritm)n(us)f(na)-189 5768 y(mo)r(dul\341rne)c(umo)r(c\254o)n(v)-5 b(anie)27 b(\(priro)r(dzene,)g(deterministic)n(k\375)g(a)g(p)r(olynomi\341ln)n (y\),)g(vyzer\341)f(nasledo)n(vne:)p -189 5823 1505 4 v -97 5877 a Fh(1)-62 5900 y Fg(-)p Ff(1)e Fe(je)f(v)h(mo)r (dul\341rnej)f(aritmetik)n(e)28 b(mo)r(d)23 b Ff(n)p Fe(;)g(t.j.)30 b Ff(n)16 b Fg(-)f Ff(1)p eop %%Page: 7 6 7 6 bop 189 100 a Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))1865 b FA(7)418 432 y Fo(Mo)l(dul\341rne)31 b(umo)l(c\254ovanie)471 532 y Fy(v)23 b Fv(=)g Fy(1)p FA(;)471 632 y Fy(p)g Fv(=)g Fy(a)p FA(;)471 731 y Fo(for)29 b FA(\()p Fy(i)23 b Fv(=)g Fy(0)p FA(;)28 b Fy(i)23 b Fs(\024)g Fy(k)p FA(;)28 b Fy(i)p FA(++\))e({)554 831 y Fo(if)j FA(\()p Fy(b)714 843 y Fw(i)768 831 y Fv(==)23 b Fy(1)p FA(\))28 b Fy(v)c Fv(=)e Fy(p)d Fs(\001)f Fy(v)24 b FA(mo)r(d)f Fy(n)p FA(;)582 930 y Fy(p)g Fv(=)g Fy(p)18 b Fs(\001)h Fy(p)k FA(mo)r(d)g Fy(n)p FA(;)471 1030 y(})314 1178 y(Op)r(er\341cie)e(mo)r (d)i(a)f(prev)n(o)r(d)g(do)g(bin\341rnej)h(s\372sta)n(vy)e(je)i (tie\272)g(mo\272n\351)g(robi\264)f(efekt\355vne.)36 b(Algoritm)n(us)21 b(je)i(mo\272\351)g(upra)n(vi\264)189 1277 y(tak,)k(\272e)h(prev)n(o)r(d)e(do)i(bin\341rnej)f(s\372sta)n(vy)f (je)i(ukryt)n(\375)f(v)n(o)g(vn)n(\372tri)g(algoritm)n(u:)418 1477 y Fo(Upr)l(aven\351)j(mo)l(dul\341rne)g(umo)l(c\254ovanie)471 1576 y Fy(v)23 b Fv(=)g Fy(1)p FA(;)471 1676 y Fy(p)g Fv(=)g Fy(a)p FA(;)471 1775 y Fo(while)29 b FA(\()p Fy(b)24 b(>)e(0)p FA(\))29 b({)554 1875 y Fo(if)g FA(\()p Fy(b)f FA(je)g(nep\341rne\))f Fy(v)c Fv(=)g Fy(p)18 b Fs(\001)h Fy(v)k FA(mo)r(d)g Fy(n)p FA(;)582 1975 y Fy(p)g Fv(=)g Fy(p)18 b Fs(\001)h Fy(p)k FA(mo)r(d)g Fy(n)p FA(;)582 2074 y Fy(b)g Fv(=)g Fy(b=2)p FA(;)83 b(//)26 b(celo)r (\243\355seln\351)h(delenie)471 2174 y(})189 2437 y FB(K)m(on\262truk)m (cia)33 b(parametro)m(v)f Fy(e)g FB(a)h Fy(d)189 2590 y FA(V)-7 b(erejn\375)37 b(exp)r(onen)n(t)f Fy(e)i FA(m\364\272eme)e(k) n(on\262truo)n(v)-5 b(a\264)35 b(viacer\375mi)h(sp)r(\364sobmi.)64 b(M\364\272eme)36 b Fy(e)h FA(v)n(oli\264)g(ak)n(o)e(n\341ho)r(dn\351)h (nep\341rne)189 2690 y(\243\355slo)28 b(a)g(testo)n(v)-5 b(a\264,)28 b(\243i)h(nsd)p Fv(\()p Fy(e;)14 b(\036)p Fv(\()p Fy(n)p Fv(\)\))26 b(=)f Fy(1)p FA(,)k(pri\243om)f(p)r(ostup)h (opakujeme,)f(k\375m)g(vho)r(dn\351)h Fy(e)g FA(nen\341)5 b(jdeme.)39 b(Zvy\243a)5 b(jne)27 b(sa)189 2789 y(v)n(erejn\375)f(exp)r 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Fs(2)g Fm(Z)19 b Fv(:)45 b Fy(ub)20 b Fv(+)f Fy(va)26 b Fv(=)f FA(nsd)p Fv(\()p Fy(a;)14 b(b)p Fv(\))p FA(.)42 b(T)-7 b(ak\272e)28 b(ak)g(p)r(olo\272\355me)h Fy(a)c Fv(=)h Fy(\036)p Fv(\()p Fy(n)p Fv(\))k FA(a)f Fy(b)c Fv(=)h Fy(e)p FA(,)k(tak)189 3288 y(dostaneme)i Fy(u;)14 b(v)p FA(:)47 b Fy(ue)22 b Fv(+)g Fy(v\036)p Fv(\()p Fy(n)p Fv(\))32 b(=)f FA(nsd)p Fv(\()p Fy(e;)14 b(\036)p Fv(\()p Fy(n)p Fv(\)\))p FA(.)54 b(T)-7 b(o)32 b(znamen\341,)h(\272e)f Fy(ue)g Fs(\021)f Fy(1)37 b Fv(\()p FA(mo)r(d)28 b Fy(\036)p Fv(\()p Fy(n)p Fv(\)\))p FA(.)53 b(T)-7 b(a)5 b(jn\375)32 b(exp)r(onen)n(t)g Fy(d)189 3387 y FA(dostaneme)27 b(pr\355padnou)g(\372pra)n(v)n(ou)e Fy(u)j FA(\(ak)g(je)g Fy(u)g FA(z\341p)r(orn\351,)e(prip)r(o)r (\243\355tame)h(vho)r(dn\375)h(n\341sob)r(ok)e Fy(\036)p Fv(\()p Fy(n)p Fv(\))p FA(\),)j(inak)e Fy(d)d Fv(=)e Fy(u)p FA(.)189 3602 y FB(R\375c)m(hly)32 b(v\375p)s(o)s(\243et)g (de\262ifro)m(v)-5 b(acej)32 b(funk)m(cie)189 3755 y FA(V)-7 b(zh\251adom)30 b(na)g(mo)r(dul\341rne)f(op)r(er\341cie)g(s)g (dlh\375mi)i(\243\355slami)e(je)h(snaha)f(o)h(ur\375c)n(hlenie)f (\262ifro)n(v)-5 b(acej/de\262ifro)n(v)g(acej)27 b(funk)n(cie)189 3855 y(priro)r(dzen\341.)34 b(Ur\375c)n(hlenie)24 b(\262ifro)n(v)-5 b(acej)23 b(funk)n(cie)h(sa)f(\(okrem)g(in\351ho\))i(dosah)n(uje)e(v)n (o\251b)r(ou)g(mal\351ho)h(v)n(erejn\351ho)f(exp)r(onen)n(tu)189 3955 y Fy(e)p FA(.)36 b(Ke\244\272e)21 b(ta)5 b(jn\375)22 b(exp)r(onen)n(t)g Fy(d)g FA(sa)g(p)r(o)r(\243\355ta)g(z)g Fy(e)g FA(a)g Fy(\036)p Fv(\()p Fy(n)p Fv(\))p FA(,)j(jeho)d (\262trukt\372ru)f(nem\364\272eme)h(pr\355li\262)g(o)n(vplyvni\264)f (\(v)h(skuto)r(\243nosti)189 4054 y(m\364\272eme)d(t)n(\375m,)i(\272e)e (na)5 b(jsk)n(\364r)17 b(zv)n(ol\355me)h Fy(d)h FA(s)g(p)r(o\272ado)n (v)-5 b(anou)18 b(\262trukt\372rou)g(a)g(dop)r(o)r(\243\355tame)h Fy(e)p FA(,)j(\243o)c(sa)h(v\262ak)f(z)h(b)r(ezp)r(e\243nostn\375c)n(h) 189 4154 y(d\364)n(v)n(o)r(do)n(v)25 b(pr\355li\262)j(neo)r(dp)r (or\372\243a\).)314 4253 y(Ur\375c)n(hlenie)37 b(v\375p)r(o)r(\243tu)h Fy(D)p Fv(\()p Fy(c)p Fv(\))h FA(sp)r(o)r(\243\355v)-5 b(a)37 b(zvy\243a)5 b(jne)37 b(v)n(o)g(vyu\272it\355)h(\203\355nsk)n (ej)g(zvy\262k)n(o)n(v)n(ej)d(v)n(et)n(y)j(\(p)r(ozri)f(pr\355loh)n(u)g (A.3\).)189 4353 y(P)n(ou\272\355v)-5 b(ate\251)26 b(si)g(ak)n(o)f(ta)5 b(jn)n(\372)26 b(inform\341ciu)f(pam\344t\341)g(nielen)h Fy(d)p FA(,)h(ale)f(a)5 b(j)25 b(prv)n(o)r(\243\355sla)f Fy(p)p FA(,)j Fy(q)p FA(.)37 b(De\262ifro)n(v)-5 b(anie)25 b(\262ifro)n(v)n(\351ho)f(textu)i Fy(c)189 4453 y FA(prebieha)i(tak,)h (\272e)f(p)r(ou\272\355v)-5 b(ate\251)29 b(vyp)r(o)r(\243\355ta)g Fy(a)1601 4465 y Fw(1)1666 4453 y Fv(=)c Fy(c)1794 4423 y Fw(d)1865 4453 y FA(mo)r(d)e Fy(p)28 b FA(a)h Fy(a)2246 4465 y Fw(2)2311 4453 y Fv(=)c Fy(c)2439 4423 y Fw(d)2509 4453 y FA(mo)r(d)f Fy(q)p FA(.)40 b(Otv)n(oren\375)27 b(text)i Fy(m)g FA(dostane)f(takto)189 4552 y(\(p)r(ozri)f(d\364k)-5 b(az)27 b(\203\355nsk)n(ej)g(zvy\262k)n(o)n(v)n(ej)f(v)n(et)n(y)h (A.3.1\):)1184 4730 y Fy(m)c Fv(=)g Fy(a)1420 4742 y Fw(1)1460 4730 y Fy(q)p Fv(\()p Fy(q)1592 4696 y Fl(-)p Fw(1)1708 4730 y FA(mo)r(d)g Fy(p)p Fv(\))18 b(+)h Fy(a)2122 4742 y Fw(2)2162 4730 y Fy(p)p Fv(\()p Fy(p)2292 4696 y Fl(-)p Fw(1)2406 4730 y FA(mo)r(d)k Fy(q)p Fv(\))h FA(mo)r(d)f Fy(n:)825 b FA(\(1.3\))189 4917 y(Na)24 b(prv\375)g(p)r (oh\251ad)g(sa)g(m\364\272e)g(zda\264,)h(\272e)f(jedno)h(mo)r (dul\341rne)e(umo)r(cnenie)i(sme)f(nahradili)g(dv)n(oma)f(\()p Fy(c)3287 4887 y Fw(d)3358 4917 y FA(mo)r(d)g Fy(p)i FA(a)f Fy(c)3720 4887 y Fw(d)3791 4917 y FA(mo)r(d)189 5017 y Fy(q)p FA(\),)31 b(tak\272e)e(o)h(ur\375c)n(hlen\355)f (nem\364\272e)g(b)n(y\264)h(re\243.)43 b(Opak)29 b(je)h(v\262ak)f(pra)n (vdou.)42 b(Dv)n(e)30 b(realizo)n(v)-5 b(an\351)28 b(umo)r(cnenia)h(ma) 5 b(j\372)30 b(krat\262ie)189 5117 y(mo)r(duly)-7 b(,)44 b(preto)c(preb)r(ehn)n(\372)h(r\375c)n(hlej\262ie)f(ak)n(o)f(jedno)i (umo)r(cnenie)f(s)h(dlh\375m)g(mo)r(dulusom)g Fy(n)p FA(.)76 b(N\341sledn\341)40 b(line\341rna)189 5216 y(k)n(om)n (bin\341cia)19 b(\(1.3\))h(je)g(u\272)h(len)f(elemen)n(t\341rn)n(ym)f (v\375p)r(o)r(\243tom.)35 b(P)n(oznamena)5 b(jme,)20 b(\272e)h(ho)r(dnot)n(y)f Fy(q)3126 5186 y Fl(-)p Fw(1)3240 5216 y FA(mo)r(d)j Fy(p)e FA(a)f Fy(p)3603 5186 y Fl(-)p Fw(1)3717 5216 y FA(mo)r(d)j Fy(q)189 5316 y FA(m\364\272u)k(b)n(y\264) h(predvyp)r(o)r(\243\355tan\351)f(a)g(nie)h(je)g(p)r(otrebn\351)f(ic)n (h)g(pri)h(k)-5 b(a\272dom)27 b(v\375p)r(o)r(\243te)g(de\262ifro)n(v)-5 b(acej)27 b(funk)n(cie)g(zno)n(v)-5 b(a)27 b(p)r(o)r(\243\355ta\264.) 189 5548 y Fn(1.1.3)112 b(Bezp)s(e\243nos\264)189 5701 y FA(Prv\375m)41 b(jedno)r(duc)n(h\375m)h(p)r(ozoro)n(v)-5 b(an\355m)39 b(je,)45 b(\272e)d(b)r(ezp)r(e\243nos\264)f(RSA)h(z\341)n (vis\355)e(na)h(zlo\272itosti)g(probl\351m)n(u)f(faktoriz\341cie.)189 5800 y(Mo)r(dulus)28 b Fy(n)h FA(je)f(v)n(erejne)g(zn\341m)n(y)-7 b(,)28 b(ro)n(vnak)n(o)d(ak)n(o)i(v)n(erejn\375)g(exp)r(onen)n(t)h Fy(e)p FA(.)40 b(Ak)28 b(b)n(y)g(b)r(olo)g(mo\272n\351)g Fy(n)g FA(\251ahk)n(o)f(faktorizo)n(v)-5 b(a\264,)189 5900 y(t.j.)64 b(vyp)r(o)r(\243\355ta\264)37 b(prv)n(o)r(\243\355sla)d Fy(p)j FA(a)f Fy(q)p FA(,)j(tak)d(ktok)n(o\251v)n(ek)f(vie)h(vyp)r(o)r (\243\355ta\264)g(ho)r(dnotu)h(ta)5 b(jn\351ho)36 b(exp)r(onen)n(tu)g Fy(d)h FA(ro)n(vnak\375m)p eop %%Page: 8 7 8 7 bop -189 100 a FA(8)1866 b Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))-189 333 y FA(sp)r(\364sob)r(om,)35 b(ak)n(o)e(to)h(z)g(ho)r(dn\364t)h Fy(e)g FA(a)e Fy(\036)p Fv(\()p Fy(n)p Fv(\))j(=)e(\()p Fy(p)23 b Fv(-)g Fy(1)p Fv(\)\()p Fy(q)h Fv(-)e Fy(1)p Fv(\))36 b FA(urobil)d(p)r(ou\272\355v)-5 b(ate\251)35 b(pri)f(inicializ\341cii)g(syst\351m)n(u.)56 b(So)-189 432 y(znalos\264ou)26 b(ta)5 b(jn\351ho)27 b(k\251\372\243a)h(u\272)f (nie)h(je)g(probl\351mom)f(de\262ifro)n(v)-5 b(a\264.)-64 536 y(T)e(eda)36 b(ak)h(vieme)g(faktorizo)n(v)-5 b(a\264)35 b Fy(n)p FA(,)40 b(tak)d(vieme)g(\022rozbi\264\020)43 b(RSA.)37 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Fv(\()p Fy(n)p Fv(\))h FA(a)f Fy(d)p FA(,)h(preto\272e)e(znalos\264)g(\251ub)r(o)n(v)n (o\251nej)h(z)h(nic)n(h)f(je)g(ekviv)-5 b(alen)n(tn\341)28 b(pr\341)n(v)n(e)f(faktoriz\341cii)h Fy(n)p FA(,)i(ak)n(o)d(uvid\355me) j(v)-189 1134 y(nasledo)n(vnom.)-64 1238 y(P)n(oznamena)5 b(jme,)36 b(\272e)f(ak)g(b)n(y)g(b)r(ol)h(d\364k)-5 b(az)34 b(ekviv)-5 b(alencie)35 b(rozbitia)g(RSA)h(a)f(faktoriz\341cie)e(k)n (on\262trukt\355vn)n(y)-7 b(,)36 b(viedlo)-189 1338 y(b)n(y)c(to)g(k)g (neo)r(dolnosti)g(RSA)h(v)n(o)r(\243i)e(\372toku)h(s)g(mo\272nos\264ou) g(v)n(o\251b)n(y)f(\262ifro)n(v)n(\351ho)f(textu)j(\(CCA\).)g (Existuj\372)g(syst\351m)n(y)-7 b(,)33 b(pre)-189 1437 y(ktor\351)22 b(sa)g(d\341)h(dok)-5 b(\341za\264)22 b(sp)r (om\355nan\341)g(ekviv)-5 b(alencia,)23 b(napr\355klad)e(Rabino)n(v)h (syst\351m)h(\(p)r(ozri)g(\243as\264)f(1.2\))g(aleb)r(o)h(Williamso)n (v)-189 1537 y(syst\351m)k([Wil80],)h(v\262etky)f(s\372)g(v\262ak)g (neo)r(doln\351)g(v)n(o)r(\243i)g(CCA.)-64 1641 y(Zauj\355ma)n(v\375)22 b(v\375sledok)g(prezen)n(to)n(v)-5 b(ali)22 b(Boneh)h(a)g(V)-7 b(enk)i(atesan)23 b([BV98)o(],)i(k)n(e\244)e(uk)-5 b(\341zali,)24 b(\272e)f(rozbitie)g(\022lo)n(w-exp)r(onen)n(t\020)-189 1741 y(RSA)g(\(t.j.)36 b(RSA)23 b(s)g(mal\375m)f(v)n(erejn\375m)g(exp)r (onen)n(tom\))g(nem\364\272e)g(b)n(y\264)h(ekviv)-5 b(alen)n(tn\351)22 b(faktoriz\341cii,)g(p)r(okia\251)g(nie)h(je)g(fakto-)-189 1840 y(riz\341cia)j(\251ahk\375)h(probl\351m.)37 b(Probl\351m)27 b(sp)r(o)r(\243\355v)-5 b(a)27 b(v)h(tom,)g(\272e)f(p)r(ou\272it\341)g (reduk)n(cia)g(m\341)g(p)r(omerne)g(obmedzen\375)g(c)n(harakter.)-189 2079 y FB(F)-8 b(aktoriz\341cia)34 b Fy(n)e FB(pri)f(znalosti)g Fy(\036)p Fv(\()p Fy(n)p Fv(\))-189 2241 y FA(Ak)22 b(ktok)n(o\251v)n (ek)e(p)r(ozn\341)h Fy(\036)p Fv(\()p Fy(n)p Fv(\))p FA(,)k(tak)c(vie)h(n\341)5 b(js\264)21 b(prv)n(o)r(\243\355sla)g Fy(p)p FA(,)i Fy(q)f FA(rie\262en\355m)f(s\372sta)n(vy)g(dv)n(o)r(c)n (h)g(ro)n(vn\355c)f(o)i(dv)n(o)r(c)n(h)f(nezn\341m)n(yc)n(h:)1631 2432 y Fy(p)e Fs(\001)f Fy(q)24 b Fv(=)f Fy(n;)1280 2557 y Fv(\()p Fy(p)c Fv(-)f Fy(1)p Fv(\)\()p Fy(q)i Fv(-)e Fy(1)p Fv(\))24 b(=)f Fy(\036)p Fv(\()p Fy(n)p Fv(\))p Fy(:)-189 2796 y FB(F)-8 b(aktoriz\341cia)34 b Fy(n)e FB(pri)f(znalosti)g Fy(e)i FB(a)f Fy(d)-189 2957 y FA(Znalos\264)27 b Fy(e)h FA(m\364\272eme)f(v\272dy)h(predp)r(oklada\264,)f(leb)r(o)g Fy(e)i FA(je)f(s\372\243as\264ou)e(v)n(erejn\351ho)h(k\251\372\243a)g (ro)n(vnak)n(o)e(ak)n(o)i Fy(n)p FA(.)37 b(Predt)n(\375m,)28 b(ak)n(o)-189 3057 y(p)r(op\355\262eme)f(p)r(ostup)h(faktoriz\341cie,)e (dok)-5 b(\341\272eme)27 b(p)r(omo)r(cn)n(\372)g(lem)n(u.)-189 3236 y FB(Lema)k(1.1.3.)39 b Fo(Ne)l(ch)26 b Fy(n;)14 b(m)23 b Fs(2)h Fm(N)t Fo(.)43 b(Ne)l(ch)26 b(existuje)f(c)l(el\351)h (\243\355slo)h Fy(a)e Fo(tak\351,)i(\272e)f FA(nsd)p Fv(\()p Fy(a;)14 b(n)p Fv(\))23 b(=)g Fy(1)j Fo(a)g Fy(a)2770 3206 y Fw(m)2861 3236 y Fs(6\021)c Fy(1)38 b Fv(\()p FA(mo)r(d)28 b 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y(\(1.5\)\),)31 b(tak\272e)e(vieme)h(zisti\264)g(tretiu) g(mo)r(cnin)n(u)f(otv)n(oren\351ho)f(textu.)44 b(V\375p)r(o)r(\243tom) 30 b(tretej)g(o)r(dmo)r(cnin)n(y)f(n\341sledne)396 2402 y(\251ahk)n(o)e(ur\243\355me)g Fy(m)p FA(.)314 2572 y(Pri)18 b(p)r(ou\272\355v)-5 b(an\355)19 b(mal\375c)n(h)f(v)n(erejn\375c)n(h)f (exp)r(onen)n(to)n(v)h(je)h(preto)e(p)r(otrebn\351)i(zab)r(ezp)r (e\243i\264,)i(ab)n(y)c(k)i(p)r(o)r(dobn\375m)g(probl\351mom)189 2672 y(nedo)r(c)n(h\341dzalo.)189 2894 y FB(\332tok)32 b(na)h(kr\341tky)g(ta)5 b(jn\375)34 b(exp)s(onen)m(t)d Fy(d)189 3050 y FA(V\375ho)r(dami)24 b(kr\341tk)n(eho)f(ta)5 b(jn\351ho)24 b(exp)r(onen)n(tu)g Fy(d)h FA(je)f(r\375c)n(hlej\262ie)g (de\262ifro)n(v)-5 b(anie)23 b(\(resp.)36 b(r\375c)n(hlej\262ie)23 b(digit\341lne)h(\022p)r(o)r(dpiso)n(v)-5 b(a-)189 3150 y(nie\020)7 b(,)26 b(p)r(ozri)g(\243as\264)g FB(??)q FA(\))g(a)g(men\262ie)h(pam\344\264o)n(v)n(\351)e(n\341roky)f(na)j(usc) n(ho)n(v)-5 b(anie)25 b(ta)5 b(jn\351ho)26 b(k\251\372\243a.)36 b(Je)26 b(zn\341m)n(y)g(\372tok)g(na)g Fy(d)h FA(\(p)r(ozri)189 3249 y([Wie90)o(]\),)35 b(ak)d(je)h(nsd)p Fv(\()p Fy(p)22 b Fv(-)g Fy(1;)14 b(q)23 b Fv(-)e Fy(1)p Fv(\))34 b FA(mal\375)e(a)h Fy(e)f(<)g(n)p FA(,)i(ktor\375)e(zist\355)h Fy(d)g FA(a\272)f(do)h (d\250\272ky)2945 3216 y Fw(1)p 2945 3230 37 4 v 2945 3278 a(4)3024 3249 y FA(d\250\272ky)g Fy(n)g FA(\(t.j.)54 b Fy(d)32 b(<)f(n)3759 3219 y Fw(0:25)3894 3249 y FA(\).)189 3349 y(Ned\341)n(vno)f(Boneh)h(a)h(Durfee,)g(p)r(ozri)f([BD99],)h (vylep\262ili)f(ten)n(to)h(v\375sledok,)f(k)n(e\244)g(uk)-5 b(\341zali,)32 b(ak)n(o)e(rek)n(on\262truo)n(v)-5 b(a\264)29 b(ta)5 b(jn\375)189 3448 y(exp)r(onen)n(t)27 b(pre)h Fy(d)23 b(<)g(n)904 3418 y Fw(0:292)1075 3448 y FA(.)189 3671 y FB(T)-8 b(akmer)32 b(CCA)f(\372tok)189 3827 y FA(RSA)40 b(m\341)f(vlastnosti,)j(ktor\351)c(umo\272\254)n(uj\372)i (\022takmer\020)k(CCA)c(\372tok)f(\(\372tok)h(s)f(mo\272nos\264ou)f(v)n (o\251b)n(y)h(\262ifro)n(v)n(\351ho)e(textu\).)189 3926 y(T)-7 b(en)n(to)33 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4624 y Fv(=)c Fy(D)p Fv(\()p Fy(c)2866 4594 y Fr(0)2890 4624 y Fv(\))h(=)f(\()p Fy(c)3114 4594 y Fr(0)3138 4624 y Fv(\))3170 4594 y Fw(d)3241 4624 y FA(mo)r(d)f Fy(n)p FA(,)29 b(vyp)r(o)r(\243\355tame)189 4723 y Fy(m)23 b Fv(=)g Fy(m)458 4693 y Fr(0)481 4723 y Fy(x)525 4693 y Fl(-)p Fw(1)639 4723 y FA(mo)r(d)g Fy(n)g Fv(=)g(\()p Fy(x)1061 4693 y Fw(e)1102 4723 y Fy(c)p Fv(\))1174 4693 y Fw(d)1222 4723 y Fy(x)1266 4693 y Fl(-)p Fw(1)1380 4723 y FA(mo)r(d)g Fy(n)g Fv(=)g Fy(c)1766 4693 y Fw(d)1837 4723 y FA(mo)r(d)g Fy(n)p FA(.)314 4824 y(Priro)r(dzene,)29 b(cel\351)g(toto)g(funguje)h(len)g(v)f(tom)h (pr\355pade,)f(ak)g(prostredie,)g(v)g(ktorom)g(je)g(RSA)i(p)r (ou\272it\351,)f(umo\272\254)n(uje)189 4924 y(de\262ifro)n(v)-5 b(anie)24 b(na)h(p)r(o\272iadanie.)36 b(V)25 b(mnoh\375c)n(h)g (situ\341ciac)n(h)g(je)h(to)f(tak)g(m\364\272e)g(b)n(y\264,)h(napr.)36 b(pri)25 b(p)r(ou\272it\355)h(RSA)g(na)f(auten)n(ti-)189 5024 y(\034k)-5 b(\341ciu,)26 b(pri)f(p)r(ou\272\355v)-5 b(an\355)26 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b(aleb)r(o)h(na)5 b(jni\272\262ieho)31 b(bitu.)49 b(Pr\341)n(v)n(e)31 b(o)g(na)5 b(jni\272\262om)31 b(\(paritnom\))h(bite)g(otv)n(oren\351ho)189 5900 y(textu)c(v)f(RSA)i (syst\351me)e(budeme)h(diskuto)n(v)-5 b(a\264)27 b(v)g(nasledo)n(vnom.) p eop %%Page: 12 11 12 11 bop -189 100 a FA(12)1824 b Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))-64 333 y FA(Predsta)n(vme)34 b(si,)j(\272e)e(m\341me)g(k)g(disp)r (oz\355cii)g(algoritm)n(us)f(\(or\341kulum\),)i(ktor\375)e(pre)h (\251ub)r(o)n(v)n(o\251n\351)g(zadan\351)f Fy(E)p Fv(\()p Fy(x)p Fv(\))i FA(vie)-189 432 y(zisti\264)25 b(paritu)g Fy(x)p FA(.)35 b(Samozrejme,)25 b(predp)r(oklad\341me,)f(\272e)h(ten)n (to)f(algorim)n(us)f(m\364\272e)i(p)r(ou\272\355v)-5 b(a\264)24 b(ho)r(dnot)n(y)h Fy(e)g FA(a)g Fy(n)p FA(.)36 b(Uk)-5 b(\341\272eme)-189 532 y(nasledo)n(vn\375)26 b(v\375sledok:)19 699 y(Ak)33 b(m\341me)g(algoritm)n(us)e (ur\243uj\372ci)i(paritu)g(otv)n(oren\351ho)e(textu)j Fy(x)f 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b(T)-7 b(a)5 b(jn\375m)28 b(k\251\372\243om)f(je)h(faktoriz\341cia)e Fy(n)p FA(,)i(teda)f(prv)n (o)r(\243\355sla)f Fy(p)h FA(a)h Fy(q)p FA(.)p 189 5744 1505 4 v 281 5798 a Fh(4)316 5821 y Fe(Existuj\372)33 b(a)t(j)f(efekt\355vnej\262ie)i(pr\355stup)n(y)g(na)f(zis\264o)n(v)l (anie)h(kv)l(adratic)n(k)n(ej)h(reziduit)n(y)g(mo)r(dulo)d Ff(n)g Fe(\(ak)i(p)r(ozn\341me)f(faktoriz\341ciu\))i(ak)n(o)f(sa)189 5900 y(p)r(ok\372\262a\264)25 b(sp)r(o)r(\243\355ta\264)g(druh)n(\372)g (o)r(dmo)r(cnin)n(u.)31 b(N\341m)22 b(v\262ak)i(momen)n(t\341lne)f(ide) h(na)t(jm\344)e(o)i(princ\355p.)p eop %%Page: 16 15 16 15 bop -189 100 a FA(16)1824 b Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))-189 333 y FB(\222ifro)m(v)-5 b(anie.)82 b FA(Ak)28 b(c)n(hceme)f(\262ifro)n (v)-5 b(a\264)27 b(spr\341)n(vu)f Fy(m)d Fv(=)f(\()p Fy(m)1604 345 y Fw(0)1659 333 y Fy(:)14 b(:)g(:)g(m)1846 345 y Fw(k)p Fl(-)p Fw(1)1977 333 y Fv(\))p FA(,)28 b(kde)g Fv(\()p Fy(m)2323 345 y 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Fv(\()p Fy(Z)2609 345 y Fw(11)2705 333 y Fp(n)18 b(f)p Fy(0)p Fp(g)p Fy(;)c Fs(\001)p Fv(\))p FA(\))1518 524 y Fy(i)p 1588 560 4 120 v 83 w(1)84 b(2)f(3)h(4)125 b(5)83 b(6)h(7)f(8)h(9)f(10)p 1428 563 1490 4 v 1470 647 a(g)1517 617 y Fw(i)p 1588 683 4 120 v 1631 647 a Fy(7)h(5)f(2)h(3)f(10)h(4)g(6) f(9)h(8)125 b(1)396 827 y FA(T)-7 b(eda)28 b(napr\355klad)e (diskr\351tn)n(y)h(logaritm)n(us)f Fy(4)i FA(pri)f(z\341klade)f Fy(7)j FA(je)f Fy(6)p FA(.)290 991 y(2.)41 b(Nec)n(h)28 b Fy(p)23 b Fv(=)g Fy(11)28 b FA(a)g(nec)n(h)f Fy(g)c Fv(=)g Fy(5)p FA(.)1539 1178 y Fy(i)p 1609 1213 V 83 w(1)84 b(2)f(3)h(4)f(5)h(6)f(7)h(8)f(9)h(10)p 1449 1217 1449 4 v 1491 1300 a(g)1538 1270 y Fw(i)p 1609 1336 4 120 v 1652 1300 a Fy(5)g(3)f(4)h(9)f(1)h(5)f(3)h(4)f(9)125 b(1)396 1485 y FA(Napr\355klad)35 b(diskr\351tn)n(y)f(logaritm)n(us)f Fy(9)j FA(pri)f(z\341klade)f Fy(5)i FA(je)f Fy(4)h FA(\(aleb)r(o)f(a)5 b(j)35 b Fy(9)p FA(\),)j(ale)d(diskr\351tn)n(y)f(logaritm)n(us)f Fy(2)j FA(pri)396 1585 y(z\341klade)27 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Fs(2)g Fy(Z)2651 3148 y Fw(p)2714 3136 y Fp(n)15 b(f)p Fy(0)p Fp(g)p FA(.)36 b(O)25 b(v)n(o\251b)r(e)h Fy(g)p FA(,)h(p)r(okia\251)-189 3235 y(m\341)g(b)n(y\264)h(gener\341torom,)d(sme)i(\243iasto)r(\243ne)g (ho)n(v)n(orili)f(pri)h(probl\351me)g(diskr\351tneho)g(logaritm)n(u.) -64 3341 y(Niek)n(edy)40 b(sa)h(pre)f(zv\375\262enie)h(efekt\355vnosti) f(\262ifro)n(v)-5 b(ania)40 b(p)r(ou\272ije)h(predv\375p)r(o)r(\243et)g (ho)r(dn\364t)g Fy(r)g FA(a)f Fy(y)3018 3311 y Fw(k)3086 3341 y FA(mo)r(d)23 b Fy(p)p FA(,)45 b(ktor\351)-189 3441 y(nez\341)n(visia)27 b(na)i(spr\341)n(v)n(e.)40 b(P)n(otom)29 b(\262ifro)n(v)-5 b(anie)27 b(p)r(ozost\341)n(v)-5 b(a)28 b(len)h(z)g(jedn\351ho)g(mo)r(dul\341rneho)g(n\341sob)r(enia)f (a)g(je)i(v)n(e\251mi)f(r\375c)n(h-)-189 3540 y(le.)67 b(Nev\375ho)r(dou)37 b(je)g(p)r(otreba)g(dr\272a\264)g(predvyp)r(o)r (\243\355tan)n(\372)g(ho)r(dnotu)g Fy(y)2043 3510 y Fw(k)2111 3540 y FA(mo)r(d)23 b Fy(p)38 b FA(v)f(ta)5 b(jnosti)37 b(-)g(v)h(pr\355pade)f(prezrade-)-189 3640 y(nia)c(b)n(y)f (protivn\355k)g(v)n(edel)g(de\262ifro)n(v)-5 b(a\264)32 b(spr\341)n(vu)f Fy(m)i FA(jedno)r(duc)n(h\375m)g(predelen\355m)g (\(resp.)52 b(pren\341sob)r(en\355m)31 b Fy(s)i FA(ho)r(dnotou)-189 3739 y Fv(\()p Fy(y)-108 3709 y Fw(k)-63 3739 y Fv(\))-31 3709 y Fl(-)p Fw(1)84 3739 y FA(mo)r(d)23 b Fy(p)p FA(\).)-189 3884 y Fo(Pozn\341mka.)43 b FA(V)29 b(tom)n(to)f(syst\351me)g(nie)g (s\372)g(\262ifro)n(v)-5 b(anie)27 b(a)h(de\262ifro)n(v)-5 b(anie)27 b(na)n(vz\341)5 b(jom)27 b(in)n(v)n(erzn\351)g(bijek)n(cie,)h (ak)n(o)f(tom)n(u)h(b)r(olo)-189 3983 y(v)k(pr\355pade)f(RSA,)h(preto)f (ho)h(nie)g(je)g(mo\272n\351)f(priamo)r(\243iaro)f(p)r(ou\272i\264)i (pre)f(digit\341lne)h(p)r(o)r(dpisy)-7 b(.)49 b(P)n(o)r(drobnej\262ie) 32 b(p)r(ozri)f(v)-189 4083 y(\243asti)c FB(??)-189 4330 y(Bezp)s(e\243nos\264)-189 4494 y FA(Bezp)r(e\243nos\264)h(ElGamalo)n (vho)g(syst\351m)n(u)g(je)h(zalo\272en\341)e(na)i(zlo\272itosti)f (probl\351m)n(u)g(diskr\351tneho)g(logaritm)n(u.)39 b(Ak)29 b(b)n(y)g(b)r(ol)-189 4593 y(protivn\355k)f(sc)n(hopn\375)f(vyp)r(o)r (\243\355ta\264)h Fy(x)p FA(,)g(tak)g(b)n(y)g(v)n(edel)f(de\262ifro)n (v)-5 b(a\264)27 b(\251ub)r(o)n(v)n(o\251n)n(\372)h(spr\341)n(vu.)37 b(Ak)29 b(b)n(y)e(b)r(ol)i(sc)n(hopn\375)e(vyp)r(o)r(\243\355ta\264) -189 4693 y Fy(k)42 b FA(zo)e(vz\264ah)n(u)h Fy(r)46 b Fv(=)g Fy(g)542 4663 y Fw(k)609 4693 y FA(mo)r(d)23 b Fy(p)p FA(,)45 b(tak)c(vie)g(v\375p)r(o)r(\243tom)g Fy(s)28 b Fs(\001)f Fv(\()p Fy(y)1810 4663 y Fw(k)1855 4693 y Fv(\))1887 4663 y Fl(-)p Fw(1)2002 4693 y FA(mo)r(d)c Fy(p)41 b FA(zisti\264)h(spr\341)n(vu)e Fy(m)p FA(.)78 b(T)-7 b(ak\272e,)44 b(ak)d(vieme)-189 4793 y(efekt\355vne)30 b(p)r(o)r(\243\355ta\264)g(diskr\351tn)n(y)f(logaritm)n(us,)f(tak)i (vieme)f(\022rozbi\264\020)36 b(ElGamalo)n(v)28 b(syst\351m.)43 b(Na)30 b(druhej)g(strane,)f(zatia\251)-189 4892 y(sa)18 b(nep)r(o)r(darilo)g(dok)-5 b(\341za\264,)19 b(\272e)g(probl\351m)f (rozbitia)g(toh)n(to)g(syst\351m)n(u)g(je)h(ekviv)-5 b(alen)n(tn\375)18 b(probl\351m)n(u)g(diskr\351tneho)g(logaritm)n(u.) -189 4992 y(M\364\272e)29 b(existo)n(v)-5 b(a\264)29 b(sp)r(\364sob,)g(ak)n(o)f(zo)h(zn\341m)n(yc)n(h)g(ho)r(dn\364t)h Fy(r)c Fv(=)g Fy(g)1771 4962 y Fw(k)1839 4992 y FA(mo)r(d)d Fy(p)29 b FA(a)h Fy(y)c Fv(=)h Fy(g)2383 4962 y Fw(x)2448 4992 y FA(mo)r(d)c Fy(p)30 b FA(vyp)r(o)r(\243\355ta\264)f Fy(g)3131 4962 y Fw(kx)3237 4992 y FA(mo)r(d)23 b Fy(p)30 b FA(\(a)-189 5091 y(n\341sledne)21 b(de\262ifro)n(v)-5 b(a\264\))21 b(b)r(ez)i(toho,)f(ab)n(y)f(sme)h(zistili)g(ho)r(dnotu)g Fy(k)g FA(aleb)r(o)f Fy(x)p FA(,)i(teda)f(b)r(ez)g(v\375p)r(o)r(\243tu) g(diskr\351tneho)f(logaritm)n(u.)-189 5191 y(Zatia\251)h(tak\375)h(sp)r (\364sob)f(nep)r(ozn\341me)g Fy(:)14 b(:)g(:)37 b FA(V)23 b(\262p)r(eci\341ln)n(yc)n(h)f(pr\355pado)r(c)n(h)g(a)g(za)h(ist)n (\375c)n(h)f(predp)r(oklado)n(v)f(ekviv)-5 b(alenciu)22 b(uk)-5 b(\341zal)-189 5291 y(Maurer)26 b(v)i([Mau94)o(].)-64 5396 y(T)-7 b(ak\272e)32 b(v)i(pr\355pade)f(ElGamalo)n(vho)f(syst\351m) 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FA(je)g(prv)n(o)r(\243\355slo)d(a) j Fy(g)g FA(je)f(gener\341tor)f(grup)n(y)g Fv(\()p Fy(Z)2931 5912 y Fw(p)2997 5900 y Fp(n)18 b(f)p Fy(0)p Fp(g)p Fy(;)c Fs(\001)p Fv(\))p FA(.)p eop %%Page: 19 18 19 18 bop 189 100 a Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))1824 b FA(19)189 333 y Fq(1.5)130 b(Knapsac)l(k)45 b(syst\351m)189 514 y FA(Knapsac)n(k)34 b(syst\351m)i(je)h(jedn\375m)f(z)h(na)5 b(jstar\262\355c)n(h)34 b(asymetric)n(k\375c)n(h)g(\262ifro)n(v)-5 b(ac\355c)n(h)35 b(syst\351mo)n(v.)61 b(Z\341kladn\375)35 b(v)-5 b(arian)n(t)35 b(pre-)189 614 y(zen)n(to)n(v)-5 b(ali)34 b(Merkle)h(a)g(Hellman)g([MH78].)60 b(V\375ho)r(dou)35 b(toh)n(to)g(syst\351m)n(u)f(je)i(jeho)f(jedno)r(duc)n(hos\264)h(a)e (\251ahk)n(os\264,)j(s)e(ak)n(ou)189 714 y(ho)c(mo\272no)g(mo)r (di\034k)n(o)n(v)-5 b(a\264)30 b(a)h(vytv)-5 b(\341ra\264)31 b(\244al\262ie)f(v)-5 b(arian)n(t)n(y)e(.)47 b(Nev\375ho)r(dou)31 b(je)h(kryptogra\034c)n(k)-5 b(\341)28 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Fw(n)958 1327 y Fv(\))23 b Fs(2)h Fp(f)p Fy(0;)14 b(1)p Fp(g)1265 1297 y Fw(n)1336 1327 y FA(tak\375)-7 b(,)22 b(\272e)e Fy(Xp)j Fv(=)g Fy(s)p FA(,)f(kde)e Fy(Xp)g FA(je)h(sk)-5 b(al\341rn)n(y)18 b(s\372\243in)i(v)n(ektoro)n(v)e (\(t.j.)36 b Fy(Xp)23 b Fv(=)3519 1265 y Fu(P)3607 1285 y Fw(n)3607 1352 y(i)p Fl(=)p Fw(1)3739 1327 y Fy(x)3783 1339 y Fw(i)3814 1327 y Fy(p)3863 1339 y Fw(i)3894 1327 y FA(\).)189 1427 y(Na)n(vy\262e,)j(ak)h(tak\375)g(v)n(ektor)g (existuje,)g(m\364\272eme)h(ho)f(c)n(hcie\264)g(n\341)5 b(js\264.)189 1588 y FB(Pr\355klad)32 b(1.5.1.)39 b FA(Knapsac)n(k)g (probl\351m)i(pre)f Fy(X)46 b Fv(=)f(\()p Fy(3;)14 b(5;)g(9;)g(10;)g (42)p Fv(\))44 b FA(a)d Fy(s)k Fv(=)g Fy(15)d FA(m\341)e(rie\262enie,)k (ktor\375m)c(je)h(v)n(ektor)189 1687 y Fv(\()p Fy(0;)14 b(1;)g(0;)g(1;)g(0)p Fv(\))p FA(.)39 b(V)28 b(pr\355pade)f Fy(s)c Fv(=)g Fy(35)29 b FA(rie\262enie)d(nem\341.)314 1848 y(P)n(oznamena)5 b(jme,)41 b(\272e)e(knapsac)n(k)f(probl\351m)h (je)g(v)n(o)g(v\262eob)r(ecnosti)f(v)n(e\251mi)h(\264a\272k\375)g(\025) g(je)h(to)f(NP-\372pln\375)h(probl\351m.)189 1948 y(Na)i(druhej)g (strane,)j(v)d(niektor\375c)n(h)f(pr\355pado)r(c)n(h)g(je)i(jeho)f (rie\262enie)f(\251ahk)n(\351.)81 b(Jedn\375m)42 b(z)g(tak\375c)n(h)f (pr\355pado)n(v)g(je)h(a)5 b(j)189 2047 y(tzv.)56 b(sup)r(errast\372ci) 32 b(\(sup)r(er-increasing\))g(v)n(ektor.)54 b(Ho)n(v)n(or\355me,)34 b(\272e)g(v)n(ektor)e Fy(X)i Fv(=)f(\()p Fy(x)2882 2059 y Fw(1)2922 2047 y Fy(;)14 b(:)g(:)g(:)g(;)g(x)3151 2059 y Fw(n)3202 2047 y Fv(\))34 b FA(je)g(sup)r(errast\372ci,)g(ak)189 2096 y Fu(P)277 2117 y Fw(k)277 2183 y(i)p Fl(=)p Fw(1)409 2158 y Fy(x)453 2170 y Fw(i)507 2158 y Fy(<)23 b(x)637 2170 y Fw(k)p Fl(+)p Fw(1)796 2158 y FA(pre)k Fs(8)p Fy(1)c Fs(\024)g Fy(k)g(<)g(n)p FA(.)37 b(Algoritm)n(us)27 b(p)r(otom)h(vyzer\341)e(nasledo)n(vne)g(\(ak)h(m\341)h(dan\341)e (in\262tancia)h(knapsac)n(k)189 2258 y(probl\351m)n(u)g(rie\262enie,)g (tak)g(v)n(ektor)f Fv(\()p Fy(p)1357 2270 y Fw(1)1398 2258 y Fy(;)14 b(:)g(:)g(:)f(;)h(p)1631 2270 y Fw(n)1683 2258 y Fv(\))27 b FA(je)h(t)n(\375m)n(to)f(rie\262en\355m\):)418 2457 y Fo(Knapsack)k(pr)l(obl\351m)f(so)g(sup)l(err)l(ast\372cim)g (vektor)l(om)471 2557 y FA(vstup:)37 b Fy(X)24 b Fv(=)e(\()p Fy(x)972 2569 y Fw(1)1013 2557 y Fy(;)14 b(:)g(:)g(:)f(;)h(x)1241 2569 y Fw(n)1292 2557 y Fv(\))28 b FA(\025)f(sup)r(erzv)-5 b(\344\243\262uj\372ci)28 b(v)n(ektor;)720 2657 y Fy(s)23 b Fs(2)h Fm(N)471 2756 y Fo(for)29 b FA(\()p Fy(i)23 b Fv(=)g Fy(n)p FA(;)28 b Fy(i)23 b Fs(\025)f Fy(1)p FA(;)29 b Fy(i)18 b Fv(-)g(-)p FA(\))28 b({)637 2856 y Fo(if)h FA(\()p Fy(s)23 b Fs(\025)g Fy(2x)982 2868 y Fw(i)1013 2856 y FA(\))28 b Fo(r)l(eturn)e FA(\020nem\341)h (rie\262enie\021;)637 2955 y Fo(if)i FA(\()p Fy(s)23 b Fs(\025)g Fy(x)941 2967 y Fw(i)972 2955 y FA(\))28 b({)803 3055 y Fy(p)852 3067 y Fw(i)907 3055 y Fv(=)22 b Fy(1)p FA(;)803 3155 y Fy(s)h Fv(=)g Fy(s)18 b Fv(-)h Fy(x)1132 3167 y Fw(i)1163 3155 y FA(;)637 3254 y(})637 3354 y Fo(else)28 b Fy(p)845 3366 y Fw(i)900 3354 y Fv(=)23 b Fy(0)p FA(;)471 3454 y(})110 b Fo(if)29 b FA(\()p Fy(s)23 b(>)g(0)p FA(\))28 b Fo(r)l(eturn)f FA(\020nem\341)g(rie\262enie\021;) 314 3601 y(K)n(orektnos\264)f(toh)n(to)j(\(p)r(o)n(v)-5 b(ahou)28 b(greedy\))g(algoritm)n(u)g(vypl\375v)-5 b(a)28 b(z)h(vlastnosti)f(\022sup)r(errast\372costi\020)34 b(vstupn\351ho)29 b(v)n(ek-)189 3700 y(tora.)314 3800 y(My\262lienk)-5 b(a)23 b(knapsac)n(k)e(syst\351m)n(u)i(sp)r(o)r(\243\355v)-5 b(a)24 b(v)f(tom,)h(ak)n(o)f(\251ahk)n(o)f(rie\262ite\251n\375)i (knapsac)n(k)d(probl\351m)i(pretransformo)n(v)-5 b(a\264)189 3899 y(na)27 b(\264a\272k)n(o)g(rie\262ite\251n\375)g(knapsac)n(k)f (probl\351m.)189 4114 y FB(Inicializ\341cia.)83 b FA(P)n(ou\272\355v)-5 b(ate\251)33 b(zv)n(ol\355)f(sup)r(errast\372ci)g(v)n(ektor)g Fy(X)h Fv(=)f(\()p Fy(x)2400 4126 y Fw(1)2440 4114 y Fy(;)14 b(:)g(:)g(:)g(;)g(x)2669 4126 y Fw(n)2720 4114 y Fv(\))p FA(,)35 b(priro)r(dzen\351)d(\243\355sla)g Fy(m)h FA(a)g Fy(t)g FA(tak)n(\351,)h(\272e)189 4214 y Fy(m)25 b(>)g(2x)463 4226 y Fw(n)544 4214 y FA(a)j(nsd)p Fv(\()p Fy(t;)14 b(m)p Fv(\))25 b(=)g Fy(1)p FA(.)42 b(\204alej)29 b(vyp)r(o)r(\243\355ta,)g(napr\355klad)e(p)r(omo)r(cou)i (roz\262\355ren\351ho)d(Euklido)n(vho)j(algoritm)n(u)e(\(p)r(ozri)189 4313 y(\243as\264)37 b(A.1\),)j Fy(u)e FA(tak)n(\351,)i(\272e)d Fy(t)25 b Fs(\001)g Fy(u)40 b Fs(\021)g Fy(1)d Fv(\()p FA(mo)r(d)28 b Fy(n)p Fv(\))p FA(.)67 b(V\244ak)-5 b(a)38 b(nes\372delite\251nosti)g Fy(t)f FA(a)g Fy(m)h FA(in)n(v)n(erzn\375)e (prv)n(ok)g(k)i Fy(t)f FA(mo)r(dulo)189 4413 y Fy(m)j FA(existuje.)73 b(P)n(ou\272\355v)-5 b(ate\251)40 b(transform)n(uje)f (v)n(ektor)f Fy(X)i FA(na)g(v)n(ektor)e Fy(X)2425 4383 y Fr(0)2491 4413 y Fv(=)43 b(\()p Fy(x)2681 4383 y Fr(0)2673 4436 y Fw(1)2714 4413 y Fy(;)14 b(:)g(:)g(:)f(;)h(x)2950 4383 y Fr(0)2942 4434 y Fw(n)2993 4413 y Fv(\))p FA(,)44 b(kde)39 b Fy(x)3310 4383 y Fr(0)3302 4436 y Fw(i)3376 4413 y Fv(=)44 b Fy(t)26 b Fs(\001)g Fy(x)3637 4425 y Fw(i)3691 4413 y FA(mo)r(d)e Fy(m)189 4513 y FA(\()p Fs(8)p Fy(1)f Fs(\024)g Fy(i)g Fs(\024)f Fy(n)p FA(\).)38 b(V)-7 b(erejn\375m)27 b(k\251\372\243om)h(je)g(v)n(ektor)e Fy(X)1789 4482 y Fr(0)1812 4513 y FA(,)i(ta)5 b(jn\375m)27 b(k\251\372\243om)h(je)g(tro)5 b(jica)26 b Fv(\()p Fy(X;)14 b(m;)g(u)p Fv(\))p FA(.)189 4727 y FB(\222ifro)m(v)-5 b(anie.)82 b FA(Priestor)34 b(spr\341)n(v)f(p)r(ozost\341)n(v)-5 b(a)33 b(zo)h(v\262etk\375c)n(h)g Fy(n)h FA(bito)n(v\375c)n(h)f(v)n (ektoro)n(v,)g(t.j.)59 b Fp(f)p Fy(0;)14 b(1)p Fp(g)3248 4697 y Fw(n)3299 4727 y FA(.)58 b(Dlh\262ie)35 b(otv)n(oren\351)189 4827 y(text)n(y)29 b(sa)g(\(tradi\243ne\))h(rozdelia)e(na)h Fy(n)h FA(bito)n(v)n(\351,)f(samostatne)f(\262ifro)n(v)-5 b(an\351)28 b(bloky)-7 b(.)43 b(Nec)n(h)29 b(otv)n(oren\375)f(text)i (je)f Fy(p)e Fs(2)f Fp(f)p Fy(0;)14 b(1)p Fp(g)3875 4797 y Fw(n)3927 4827 y FA(.)189 4927 y(\222ifro)n(v\375)26 b(text)i(je)g(p)r(otom)g(\243\355slo)f Fy(c)c Fv(=)g Fy(pX)1440 4896 y Fr(0)1463 4927 y FA(.)189 5141 y FB(De\262ifro)m(v)-5 b(anie.)82 b FA(P)n(ou\272\355v)-5 b(ate\251,)26 b(p)r(ozna)5 b(j\372ci)25 b(ta)5 b(jn\375)26 b(k\251\372\243,)g(vie)f(in\262tanciu)h (\264a\272k)n(\351ho)e(knapsac)n(k)g(probl\351m)n(u)h Fv(\()p Fy(X)3648 5111 y Fr(0)3671 5141 y Fy(;)14 b(c)p Fv(\))26 b FA(pre-)189 5241 y(vies\264)g(na)g(in\262tanciu)f(\251ahk)n (\351ho)h(knapsac)n(k)e(probl\351m)n(u)h(so)g(sup)r(errast\372cim)g(v)n (ektorom.)35 b(V)-7 b(yp)r(o)r(\243\355ta)26 b Fy(c)3321 5211 y Fr(0)3368 5241 y Fv(=)c Fy(u)16 b Fs(\001)f Fy(c)24 b FA(mo)r(d)f Fy(m)j FA(a)189 5340 y(rie\262i,)h(napr\355klad)f (vy\262\262iep)r(op\355san\375m)h(algoritmom,)e(probl\351m)i(pre)h(v)n (ektor)e Fy(X)i FA(a)f(ho)r(dnotu)h Fy(c)3111 5310 y Fr(0)3134 5340 y FA(.)189 5502 y FB(Pr\355klad)k(1.5.2.)39 b FA(Zv)n(o\251me)25 b(si)g(sup)r(errast\372ci)f(v)n(ektor)g Fy(X)f Fv(=)g(\()p Fy(1;)14 b(2;)g(4;)g(8;)g(17;)g(35)p Fv(\))p FA(,)30 b(nec)n(h)25 b Fy(m)e Fv(=)g Fy(71)j FA(a)f(nec)n(h)g Fy(t)e Fv(=)g Fy(22)p FA(.)37 b(P)n(otom)189 5601 y Fy(u)23 b Fv(=)g Fy(42)p FA(,)g(leb)r(o)e Fy(42)t Fs(\001)t Fy(22)k Fs(\021)e Fy(1)37 b Fv(\()p FA(mo)r(d)28 b Fy(71)p Fv(\))p FA(.)36 b(V)-7 b(erejn\375m)21 b(k\251\372\243om)f (je)h(v)n(ektor)e Fy(X)2481 5571 y Fr(0)2527 5601 y Fv(=)k Fy(22)t Fs(\001)t Fy(X)i FA(mo)r(d)e Fy(71)h Fv(=)f(\()p Fy(22;)14 b(44;)g(17;)g(34;)g(19;)g(60)p Fv(\))p FA(.)189 5701 y(T)-7 b(a)5 b(jn\375)27 b(k\251\372\243)h(je)g(tro)5 b(jica)26 b Fv(\(\()p Fy(1;)14 b(2;)g(4;)g(8;)g(17;)g(35)p Fv(\))p Fy(;)g(71;)g(42)p Fv(\))p FA(.)314 5800 y(Nec)n(h)33 b(v)n(ektor)f Fy(p)h Fv(=)g(\()p Fy(1;)14 b(0;)g(0;)g(1;)g(1;)g(1)p Fv(\))37 b FA(je)d(otv)n(oren\375)e(text.)55 b(P)n(otom)33 b(k)h(nem)n(u)g(prisl\372c)n(ha)5 b(j\372ci)32 b(\262ifro)n(v\375)g (text)i(je)g(\243\355slo)189 5900 y Fy(c)23 b Fv(=)g Fy(pX)450 5870 y Fr(0)496 5900 y Fv(=)g Fy(22)d Fv(+)e Fy(34)h Fv(+)g Fy(19)g Fv(+)f Fy(60)24 b Fv(=)f Fy(135)p FA(.)p eop %%Page: 20 19 20 19 bop -189 100 a FA(20)1824 b Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))-64 333 y FA(Pred)22 b(de\262ifro)n(v)-5 b(an\355m)22 b(\262ifro)n(v)n (\351ho)f(textu)i Fy(c)h Fv(=)f Fy(135)h FA(ho)f(na)5 b(jsk)n(\364r)20 b(upra)n(v\355me:)34 b Fy(c)2303 303 y Fr(0)2349 333 y Fv(=)23 b Fy(42)9 b Fs(\001)g Fy(135)25 b FA(mo)r(d)f Fy(71)g Fv(=)e Fy(61)p FA(.)37 b(P)n(o)22 b(rie\262en\355)-189 432 y(knapsac)n(k)31 b(probl\351m)n(u)h(pre)g(v)n (ektor)f Fy(X)h Fv(=)g(\()p Fy(1;)14 b(2;)g(4;)g(8;)g(17;)g(35)p Fv(\))36 b FA(a)c(ho)r(dnotu)h Fy(c)2207 402 y Fr(0)2262 432 y Fv(=)e Fy(61)j FA(dost\341)n(v)-5 b(ame)31 b(op\344\264)i(otv)n (oren\375)d(text)-189 532 y Fv(\()p Fy(1;)14 b(0;)g(0;)g(1;)g(1;)g(1)p Fv(\))p FA(.)-64 721 y(Pred)35 b(zv)n(erejnen\355m)g Fy(X)673 691 y Fr(0)732 721 y FA(je)g(dobr\375m)g(pra)n(vidlom)f(ho)r (dnot)n(y)h(v)g(\254om)g(utriedi\264.)61 b(P)n(otom)35 b(si)h(p)r(erm)n(ut\341ciu)f(ho)r(dn\364t)-189 821 y(m)n(us\355me)28 b(pam\344ta\264)f(sp)r(olu)g(s)g(ta)5 b(jn\375m)28 b(k\251\372\243om,)g (ab)n(y)e(sme)i(v)n(edeli)f(k)n(orektne)f(de\262ifro)n(v)-5 b(a\264.)-64 928 y(K)n(orektnos\264)25 b(knapsac)n(k)h(syst\351m)n(u)h (vypl\375v)-5 b(a)27 b(z)h(nasledo)n(vnej)e(lem)n(y)-7 b(.)-189 1118 y FB(Lema)31 b(1.5.1)f(\(K)m(orektnos\264)i(knapsac)m(k)i (syst\351m)m(u\).)39 b Fo(Ne)l(ch)j Fy(X)p Fo(,)j Fy(X)2089 1088 y Fr(0)2112 1118 y Fo(,)g Fy(m)p Fo(,)g Fy(u)d Fo(s\372)f (de\034novan\351)h(ako)h(vy\262\262ie,)i(ne)l(ch)-189 1218 y Fy(c)31 b Fs(2)f Fm(N)44 b Fo(a)34 b Fy(c)185 1187 y Fr(0)239 1218 y Fv(=)c Fy(u)22 b Fs(\001)f Fy(c)j FA(mo)r(d)f Fy(m)p Fo(.)50 b(Potom)35 b(in\262tancie)f(knapsack)h(pr)l (obl\351mu)f(pr)l(e)g Fy(X)g Fo(a)h Fy(c)2551 1187 y Fr(0)2608 1218 y Fo(a)f(pr)l(e)g Fy(X)2892 1187 y Fr(0)2949 1218 y Fo(a)g Fy(c)g Fo(maj\372)g(najviac)-189 1317 y(je)l(dno,)d(r)l (ovnak\351)g(rie\262enie.)-189 1507 y(D\364kaz.)43 b FA(Je)37 b(\251ahk)n(\351)f(nahliadn)n(u\264,)k(\272e)d(ak)g(m\341)g 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b Fs(\001)g Fy(9)1425 5616 y Fl(-)p Fw(1)1541 5646 y Fv(=)22 b Fy(5)p FA(.)37 b(P)n(otom)23 b(dost\341)n(v)-5 b(ame)23 b Fy(x)g Fv(=)g Fy(\013)2600 5616 y Fw(2)2651 5646 y Fv(-)11 b Fy(11)g Fv(-)g Fy(7)25 b Fv(=)d Fy(25)11 b Fv(-)g Fy(11)g Fv(-)g Fy(7)26 b Fv(=)d Fy(7)-189 5746 y FA(a)k Fy(y)d Fv(=)e Fy(\013)p Fv(\()p Fy(11)e Fv(-)e Fy(x)p Fv(\))g(-)h Fy(4)k Fv(=)g Fy(5)c Fs(\001)f Fy(4)h Fv(-)g Fy(4)k Fv(=)g Fy(3)p FA(.)37 b(T)-7 b(eda)28 b Fv(\()p Fy(11;)14 b(4)p Fv(\))20 b(+)e(\()p Fy(7;)c(10)p Fv(\))25 b(=)e(\()p Fy(7;)14 b(3)p Fv(\))p FA(.)-64 5900 y(S)27 b(tro)r(c)n(hou)g(\372silia)g(je)h (mo\272n\351)f(dok)-5 b(\341za\264)27 b(nasledo)n(vn)n(\372)f(lem)n(u.) p eop %%Page: 23 22 23 22 bop 189 100 a Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))1824 b FA(23)189 333 y FB(Lema)31 b(1.6.1.)39 b Fo(Mno\272ina)26 b(b)l(o)l(dov)f(eliptickej)i(krivky)f(sp)l(olu)f(s)f(op)l(er)l (\341ciou)i(s\243\355tania)g(tvor\355)f(komutat\355vnu)f(grupu,)i Fv(\()p Fs(E)7 b Fy(;)14 b Fv(+\))p Fo(.)314 507 y FA(Z\341kladnou)29 b(op)r(er\341ciou,)i(ktor\341)f(umo\272\254)n(uje)h(vybudo)n(v)-5 b(a\264)31 b(anal\363gie)e(asymetric)n(k\375c)n(h)g(\262ifro)n(v)-5 b(ac\355c)n(h)30 b(syst\351mo)n(v)g(zalo-)189 607 y(\272en\375c)n(h)h (na)g(probl\351me)f(diskr\351tneho)h(logaritm)n(u,)g(je)g(n\341sob)r (enie)g(b)r(o)r(du)h(na)f Fs(E)38 b FA(k)n(on\262tan)n(tou.)47 b(Nec)n(h)31 b Fy(P)e Fs(2)h(E)39 b FA(a)31 b Fy(n)e Fs(2)h Fm(N)t FA(,)189 707 y(p)r(otom)k Fy(n)23 b Fs(\001)f Fy(P)34 b Fv(=)f Fy(P)19 b Fv(+)f Fy(P)h Fv(+)f Fs(\001)c(\001)g(\001)k Fv(+)g Fy(P)759 747 y Fu(|)p 796 747 199 9 v 199 w({z)p 1069 747 V 199 w(})1009 821 y Fw(n)1305 707 y FA(.)56 b(P)n(o)r(dobne,)36 b(ak)n(o)c(pri)i(efekt\355vnom)g(v\375p)r(o)r (\243te)f(mo)r(dul\341rneho)g(umo)r(cnenia)h(\(p)r(ozri)189 900 y(\243as\264)27 b(1.1.2\),)g(a)5 b(j)27 b(v)g(tom)n(to)g (pr\355pade)g(vieme)h(ten)n(to)f(s\372\243et)h(efekt\355vne)g(p)r(o)r (\243\355ta\264,)g(t.j.)37 b(p)r(ou\272it\355m)28 b Fy(O)p Fv(\()p FA(log)14 b Fy(n)p Fv(\))28 b FA(op)r(er\341ci\355.)418 1105 y Fo(V\375p)l(o)l(\243)l(et)i(n\341sob)l(enia)g(kon\262tantou)g (na)f(eliptickej)j(krivke)471 1211 y FA(vstup:)37 b Fy(P)28 b FA(\025)f(b)r(o)r(d)h(na)f(eliptic)n(k)n(ej)h(krivk)n(e)e Fs(E)7 b FA(;)720 1313 y Fy(n)23 b Fs(2)h Fm(N)471 1416 y Fy(V)f Fv(=)g FB(O)p FA(;)471 1518 y Fy(Q)g Fv(=)g Fy(P)r FA(;)471 1621 y Fo(while)29 b FA(\()p Fy(n)24 b(>)f(0)p FA(\))28 b({)554 1723 y Fo(if)h FA(\()p Fy(n)f FA(je)g(nep\341rne\))f Fy(V)c Fv(=)f Fy(Q)d Fv(+)f Fy(V)5 b FA(;)582 1826 y Fy(Q)23 b Fv(=)g Fy(Q)18 b Fv(+)g Fy(Q)p FA(;)582 1928 y Fy(n)23 b Fv(=)g Fy(n=2)p FA(;)83 b(//)27 b(celo)r(\243\355seln\351)g(delenie)471 2031 y(})471 2133 y Fo(r)l(eturn)g Fy(V)5 b FA(;)314 2291 y(V)31 b(tom)n(to)g (algoritme)f(ozna\243uje)h Fv(+)g FA(op)r(er\341ciu)f(s\243\355tania)h (b)r(o)r(do)n(v)g(na)g(eliptic)n(k)n(ej)g(krivk)n(e.)47 b(V\375sledok)30 b(s\372\243in)n(u)i Fy(nP)f FA(je)189 2391 y(b)r(o)r(d)d Fy(V)6 b FA(.)189 2622 y FB(Probl\351m)28 b(diskr\351tneho)h(logaritm)m(u)g(na)h(eliptic)m(k\375c)m(h)h(krivk)-5 b(\341c)m(h.)84 b FA(Dan\341)26 b(je)g(eliptic)n(k)-5 b(\341)26 b(krivk)-5 b(a)25 b Fs(E)33 b FA(a)26 b(dv)-5 b(a)25 b(b)r(o)r(dy)189 2722 y Fy(P)-7 b(;)14 b(Q)21 b Fs(2)i(E)7 b FA(.)37 b(T)-7 b(reba)27 b(n\341)5 b(js\264)27 b Fy(x)c Fs(2)g Fm(N)38 b FA(tak)n(\351,)27 b(ak)g(existuje,)h(\272e)f Fy(x)18 b Fs(\001)h Fy(P)k Fv(=)g Fy(Q)p FA(.)314 2824 y(Predp)r(oklad\341)33 b(sa,)k(\272e)e(ten)n(to)f(probl\351m)h(je)g (\264a\272k\375)-7 b(.)59 b(Na)n(vy\262e,)35 b(v)g(s\372\243asnosti)f (na)5 b(jlep\262ie)35 b(algoritm)n(y)e(pre)h(probl\351m)189 2924 y(diskr\351tneho)25 b(logaritm)n(u)g(nie)h(je)h(mo\272n\351)f (aplik)n(o)n(v)-5 b(a\264)25 b(na)h(probl\351m)f(diskr\351tneho)h (logaritm)n(u)e(na)i(eliptic)n(k\375c)n(h)g(krivk)-5 b(\341c)n(h.)189 3024 y(V\244ak)g(a)27 b(tom)n(u)h(ro)n(vnak\372)d (\372ro)n(v)n(e\254)h(b)r(ezp)r(e\243nosti)i(vieme)f(dosiahn)n(u\264)g (s)h(men\262\355m)g(p)r(o)r(\243tom)f(bito)n(v)g(ta)5 b(jn\351ho)27 b(k\251\372\243a.)314 3126 y(V)32 b(nasledo)n(vnej)g (\243asti)g(uv)n(edieme)h(anal\363giu)e(ElGamalo)n(vho)g(syst\351m)n(u) h(na)g(eliptic)n(k\375c)n(h)h(krivk)-5 b(\341c)n(h)31 b(\025)h(budeme)h(ju)189 3226 y(ozna\243o)n(v)-5 b(a\264)26 b(EC-ElGamalo)n(v)g(syst\351m.)189 3473 y Fn(1.6.1)112 b(EC-ElGamalo)m(v)36 b(syst\351m)189 3632 y FB(Inicializ\341cia.)83 b FA(Na)5 b(jsk)n(\364r)23 b(zv)n(ol\355me)h(eliptic)n(k\372)h(krivku)f Fs(E)33 b FA(a)24 b(b)r(o)r(d)i Fy(G)d Fs(2)g(E)7 b FA(.)36 b(Ho)r(dnot)n(y)25 b Fs(E)32 b FA(a)25 b Fy(G)g FA(m\364\272u)g(b)n (y\264)g(sp)r(olo)r(\243n\351)f(pre)189 3732 y(viacer\375c)n(h)k(p)r (ou\272\355v)-5 b(ate\251o)n(v.)45 b(P)n(ou\272\355v)-5 b(ate\251)30 b(zv)n(ol\355)f(n\341ho)r(dn\351)h Fy(x)e Fs(2)f Fm(N)41 b FA(a)30 b(vyp)r(o)r(\243\355ta)f(b)r(o)r(d)i Fy(Y)c Fv(=)h Fy(xG)p FA(.)44 b(V)-7 b(erejn\375m)30 b(k\251\372\243om)g(je)189 3831 y(p)r(otom)e(tro)5 b(jica)26 b Fv(\()p Fy(Y)-7 b(;)14 b Fs(E)7 b Fy(;)14 b(G)p Fv(\))23 b FA(a)k(ta)5 b(jn\375m)27 b(k\251\372\243om)h(je)g(ho)r(dnota)f Fy(x)p FA(.)189 4063 y FB(\222ifro)m(v)-5 b(anie.)82 b FA(Priestor)27 b(spr\341)n(v)f(je)i(mno\272ina)f(b)r(o)r(do)n(v)g Fs(E)7 b FA(.)37 b(Nec)n(h)28 b Fy(M)22 b Fs(2)i(E)35 b FA(je)28 b(b)r(o)r(d,)g(ktor\375)e(c)n(hceme)h(\262ifro)n(v)-5 b(a\264.)290 4237 y(1.)41 b(Zv)n(ol\355me)27 b(n\341ho)r(dn\351)g Fy(k)d Fs(2)f Fm(N)t FA(.)290 4415 y(2.)41 b(\222ifro)n(v\375)27 b(text)h(je)g(dv)n(o)5 b(jica)26 b(b)r(o)r(do)n(v)h Fs(h)p Fy(R;)14 b(S)p Fs(i)p FA(,)28 b(kde)g Fy(R)22 b Fv(=)h Fy(kG)k FA(a)h Fy(S)23 b Fv(=)g Fy(M)18 b Fv(+)g Fy(kY)5 b FA(.)189 4646 y FB(De\262ifro)m(v)-5 b(anie.)82 b FA(P)n(ou\272\355v) -5 b(ate\251,)21 b(ktor\375)e(p)r(ozn\341)g(ta)5 b(jn\375)20 b(k\251\372\243)g Fy(x)p FA(,)h(vie)f(de\262ifro)n(v)-5 b(a\264)18 b(spr\341)n(vu)h(\(b)r(o)r(d\))i(v\375p)r(o)r(\243tom)e Fy(M)k Fv(=)g Fy(S)s Fv(-)s Fy(xR)p FA(.)189 4785 y Fo(Pozn\341mka.)43 b FA(V\262imnime)27 b(si,)g(\272e)g(v)f(EC-ElGamalo)n(v)n(om)f (syst\351me)h(je)h(oproti)f(p)r(\364)n(v)n(o)r(dn\351m)n(u)g(syst\351m) n(u)g(d\364sledne)g(nahra-)189 4884 y(den\351)33 b(n\341sob)r(enie)g (s\243\355tan\355m.)53 b(V)33 b(skuto)r(\243nosti)g(je)g(rozdiel)f (medzi)i(s\243\355tan\355m)f(a)f(n\341sob)r(en\355m)h(iba)g (terminologic)n(k\375)-7 b(.)51 b(V)189 4984 y(ob)r(o)r(c)n(h)28 b(pr\355pado)r(c)n(h)f(ide)h(o)g(op)r(er\341cie)f(v)h(\(k)n (one\243n\375c)n(h\))f(grup\341c)n(h.)37 b(Ozna\243enie)27 b(op)r(er\341cie)g(na)h(eliptic)n(k\375c)n(h)f(krivk)-5 b(\341c)n(h)27 b(ak)n(o)189 5084 y(s\243\355tanie)g(m\341)h(len)f (historic)n(k)n(\351)f(d\364)n(v)n(o)r(dy)-7 b(.)314 5222 y(K)n(orektnos\264)25 b(EC-ElGamalo)n(vho)h(syst\351m)n(u)h (vypl\375v)-5 b(a)27 b(z)h(nasledo)n(vnej)e(iden)n(tit)n(y:)1166 5411 y Fy(S)19 b Fv(-)f Fy(xR)23 b Fv(=)g Fy(M)18 b Fv(+)g Fy(kY)g Fv(-)g Fy(xkG)23 b Fv(=)g Fy(M)18 b Fv(+)g Fy(kxG)g Fv(-)g Fy(xkG)23 b Fv(=)f Fy(M:)189 5642 y FB(Realizo)m(v)-5 b(ate\251nos\264)31 b(a)h(b)s(ezp)s(e\243nos\264)189 5800 y FA(S)26 b(praktic)n(k\375m)e(vyu\272it\355m)h(EC-ElGamalo)n(vho) f(syst\351m)n(u,)i(ale)f(a)5 b(j)25 b(in\375c)n(h)g(syst\351mo)n(v)f (zalo\272en\375c)n(h)g(na)h(eliptic)n(k\375c)n(h)g(kriv-)189 5900 y(k)-5 b(\341c)n(h,)27 b(je)h(sp)r(o)5 b(jen\375c)n(h)27 b(viacero)f(ot\341zok,)g(ktor\351)g(treba)h(zo)r(dp)r(o)n(v)n(eda\264.) p eop %%Page: 24 23 24 23 bop -189 100 a FA(24)1824 b Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))-64 333 y Fs(\017)41 b Ft(Ak)n(o)e(zv)n(oli\264)h Fs(E)48 b Ft(a)40 b Fy(G)p Ft(?)75 b FA(Eliptic)n(k\372)41 b(krivku)f (m\364\272eme)g(v)n(oli\264)f(n\341ho)r(dne.)76 b(Zv)n(ol\355me)39 b(ho)r(dnot)n(y)h Fy(x)3084 345 y Fw(0)3124 333 y Fy(;)14 b(y)3210 345 y Fw(0)3251 333 y Fy(;)g(a)44 b Fs(2)3438 345 y Fw(R)3530 333 y Fy(F)19 432 y FA(\(predp)r(oklad\341me)20 b(k)n(one\243n\351)g(p)r(ole)h Fy(F)p FA(\))h(a)f(vyp)r(o)r (\243\355tame)g Fy(b)i Fv(=)g Fy(y)1910 402 y Fw(2)1910 456 y(0)1956 432 y Fv(-)6 b Fy(x)2069 402 y Fw(3)2069 456 y(0)2114 432 y Fv(-)g Fy(ax)2278 444 y Fw(0)2318 432 y FA(.)35 b(Sk)n(on)n(trolujeme,)21 b(\243i)g Fy(4a)3111 402 y Fw(3)3157 432 y Fv(+)6 b Fy(27b)3357 402 y Fw(2)3421 432 y Fs(6)p Fv(=)23 b Fy(0)p FA(.)19 532 y(Ak)j(\341no,)h(m\341me)f (eliptic)n(k\372)h(krivku)f Fy(y)1219 502 y Fw(2)1283 532 y Fv(=)d Fy(x)1413 502 y Fw(3)1470 532 y Fv(+)16 b Fy(ax)g Fv(+)h Fy(b)p FA(.)36 b(V)27 b(opa\243nom)f(pr\355pade)g (generujeme)g(ho)r(dnot)n(y)h(zno)n(v)-5 b(a.)19 632 y(Bo)r(d)26 b Fy(G)h FA(je)g(dan\375)f(dv)n(o)5 b(jicou)26 b Fv(\()p Fy(x)988 644 y Fw(0)1028 632 y Fy(;)14 b(y)1114 644 y Fw(0)1155 632 y Fv(\))p FA(.)37 b(Priro)r(dzene,)26 b(takto)h(zv)n(olen\375)e(b)r(o)r(d)j Fy(G)e FA(nem)n(us\355)h(b)n (y\264)g(gener\341torom)d(grup)n(y)19 731 y Fv(\()p Fs(E)7 b Fy(;)14 b Fv(+\))p FA(.)37 b(Ba)27 b(\243o)g(viac,)g(grupa)f Fv(\()p Fs(E)7 b Fy(;)14 b Fv(+\))28 b FA(v)n(\364b)r(ec)f(nem)n (us\355)h(b)n(y\264)f(cyklic)n(k)-5 b(\341.)19 880 y(Ak)29 b(sa)f(nec)n(hceme,)h(napr\355klad)f(z)i(b)r(ezp)r(e\243nostn\375c)n(h) e(d\364)n(v)n(o)r(do)n(v,)g(usp)r(ok)n(o)5 b(ji\264)28 b(s)h(\243isto)g(n\341ho)r(dnou)f(v)n(o\251b)r(ou)h Fs(E)36 b FA(a)29 b Fy(G)p FA(,)19 980 y(m\364\272eme)k(vyp)r(o)r (\243\355ta\264)i Fy(N)f FA(\025)g(p)r(o)r(\243et)g(b)r(o)r(do)n(v)g (eliptic)n(k)n(ej)g(krivky)f Fs(E)42 b FA(\(in\375mi)35 b(slo)n(v)-5 b(ami)33 b(r\341d)h(grup)n(y\).)56 b(Na)34 b(v\375p)r(o)r(\243et)19 1080 y(mo\272no)39 b(p)r(ou\272i\264)i (napr\355klad)e(Sc)n(ho)r(ofo)n(v)g(algoritm)n(us)f([Sc)n(h85)o(],)43 b(ktor\375)d(je)g(deterministic)n(k\375)g(a)g(p)r(olynomi\341ln)n(y)-7 b(.)19 1179 y(N\341sledne)33 b(o)n(v)n(er\355me,)h(\272e)g Fy(N)h FA(m\341)f(a)5 b(j)33 b(v)n(e\251k\375)h(prv)n(o)r (\243\355seln\375)f(faktor)g(\025)h(in\375mi)g(slo)n(v)-5 b(ami,)35 b(nie)f(je)h(rozlo\272ite\251n\351)e(nad)19 1279 y(mno\272inou)23 b(mal\375c)n(h)h(prv)n(o)r(\243\355sel.)34 b(T\341to)24 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(ky)e(siln\375)k(gener\341tor)d(n\341ho)r(dn\375c)n(h)i(\243\355sel.) 314 5213 y(Ro)n(vnak)n(o,)19 b(ak)n(o)f(v)i(in\375c)n(h)f(oblastiac)n (h,)h(ni\243)g(nie)g(je)g(\243ierno-biele)f(a)g(v)h(k)n(onkr\351tnej)f (situ\341cii)g(treba)g(zv)-5 b(a\272o)n(v)g(a\264)19 b(mno\272stv)n(o)189 5313 y(\244al\262\355c)n(h)27 b(faktoro)n(v.)p eop %%Page: 26 25 26 25 bop -189 962 a Fz(Do)5 b(datok)62 b(A)-189 1384 y FC(T)-18 b(e\363ria)72 b(\243\355sel)-189 1823 y FA(K)26 b(te\363rii)g(\243\355sel)g(existuje)g(v)n(e\251a)g(dobr\375c)n(h)f (kn\355h)h(\(napr.)36 b([K)n(ob87a)n(,)26 b(Sc)n(h86)o(,)g(Ros92)o (]\),)h(p)r(omo)r(cou)f(ktor\375c)n(h)f(\243itate\251)h(m\364\272e)-189 1923 y(roz\262\355ri\264)37 b(sv)n(o)5 b(je)37 b(znalosti)g(v)i(tejto)f (oblasti.)69 b(T\341to)37 b(pr\355loha)g(obsah)n(uje)h(len)g(p)r (oznatky)g(nevyhn)n(utne)g(p)r(otrebn\351)g(pre)-189 2022 y(p)r(o)r(c)n(hop)r(enie)27 b(nami)h(preb)r(eranej)f(problematiky) -7 b(.)-189 2334 y Fq(A.1)131 b(Roz\262\355ren\375)45 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Fy(va)82 b Fv(=)h FA(nsd)p Fv(\()p Fy(a;)14 b(b)p Fv(\))p Fy(:)1237 b FA(\(A.2\))-189 5414 y Fo(D\364kaz.)43 b FA(Vlastnos\264)21 b(\(A.1\))i(plat\355)f (trivi\341lne)f(\(zo)r(dp)r(o)n(v)n(ed\341)g(klasic)n(k)n(\351m)n(u)f (Euklido)n(vm)n(u)i(algoritm)n(u)e(a)i(vypl\375v)-5 b(a)21 b(z)h(toho,)h(\272e)-189 5514 y(nsd)p Fv(\()p Fy(a;)14 b(b)p Fv(\))24 b(=)g FA(nsd)p Fv(\()p Fy(b;)14 b(s)529 5526 y Fw(2)569 5514 y Fv(\))24 b(=)g FA(nsd)p Fv(\()p Fy(s)907 5526 y Fw(2)947 5514 y Fy(;)14 b(s)1022 5526 y Fw(3)1063 5514 y Fv(\))24 b(=)g Fs(\001)14 b(\001)g(\001)24 b Fv(=)f FA(nsd)p Fv(\()p Fy(s)1608 5526 y Fw(n)p Fl(-)p Fw(2)1747 5514 y Fy(;)14 b(s)1822 5526 y Fw(n)p Fl(-)p Fw(1)1960 5514 y Fv(\))25 b(=)e Fy(s)2141 5526 y Fw(n)p Fl(-)p Fw(1)2280 5514 y FA(\).)39 b(Preto)28 b(sa)f(v)h(\244al\262om)f (zameriame)g(na)-189 5613 y(\(A.2\).)37 b(Na)5 b(jsk)n(\364r)26 b(matematic)n(k)n(ou)g(induk)n(ciou)h(uk)-5 b(\341\272eme,)28 b(\272e)f(pre)g(v\262etky)g Fy(k)c Fs(\024)g Fy(n)g Fv(:)42 b(\(-)p 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b(krok:)644 528 y Fv(\(-)p Fy(1)p Fv(\))812 493 y Fw(k)p Fl(-)p Fw(1)944 528 y Fy(u)997 540 y Fw(k)1042 528 y Fy(b)18 b Fv(+)h(\(-)p Fy(1)p Fv(\))1359 493 y Fw(k)1404 528 y Fy(v)1445 540 y Fw(k)1489 528 y Fy(a)k Fv(=)1563 665 y(=)f(\(-)p Fy(1)p Fv(\))1816 631 y Fw(k)p Fl(-)p Fw(1)1949 665 y Fv(\()p Fy(q)2031 677 y Fw(k)2075 665 y Fy(u)2128 677 y Fw(k)p Fl(-)p Fw(1)2279 665 y Fv(+)c Fy(u)2413 677 y Fw(k)p Fl(-)p Fw(2)2545 665 y Fv(\))p Fy(b)g Fv(+)h(\(-)p Fy(1)p Fv(\))2894 631 y Fw(k)2938 665 y Fv(\()p Fy(q)3020 677 y Fw(k)3065 665 y Fy(v)3106 677 y Fw(k)p Fl(-)p Fw(1)3256 665 y Fv(+)g Fy(v)3379 677 y Fw(k)p Fl(-)p Fw(2)3510 665 y Fv(\))p Fy(a)k Fv(=)1563 803 y(=)f(-)p Fy(q)1761 815 y Fw(k)1806 803 y Fv(\(\(-)p Fy(1)p Fv(\))2006 769 y Fw(k)p Fl(-)p Fw(2)2138 803 y Fy(u)2191 815 y Fw(k)p Fl(-)p Fw(1)2323 803 y Fy(b)c Fv(+)h(\(-)p Fy(1)p Fv(\))2640 769 y Fw(k)p Fl(-)p Fw(1)2772 803 y Fy(v)2813 815 y Fw(k)p Fl(-)p Fw(1)2944 803 y Fy(a)p Fv(\))g(+)1641 941 y(+)f(\(-)p Fy(1)p Fv(\))1890 906 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Fw(n)p Fl(-)p Fw(1)1135 1692 y Fu(|)p 1172 1692 132 9 v 132 w({z)p 1378 1692 V 132 w(})1318 1767 y Fw(u)1561 1638 y Fy(b)18 b Fv(+)g(\(-)p Fy(1)p Fv(\))1877 1604 y Fw(n)p Fl(-)p Fw(1)2017 1638 y Fy(v)2058 1650 y Fw(n)p Fl(-)p Fw(1)1709 1692 y Fu(|)p 1746 1692 169 9 v 169 w({z)p 1989 1692 V 169 w(})1935 1767 y Fw(v)2210 1638 y Fy(a)23 b Fv(=)g Fy(s)2408 1650 y Fw(n)p Fl(-)p Fw(1)2569 1638 y Fv(=)g FA(nsd)p Fv(\()p Fy(a;)14 b(b)p Fv(\))p Fy(:)p 3889 1935 4 57 v 3893 1882 50 4 v 3893 1935 V 3942 1935 4 57 v 189 2151 a Fo(Pozn\341mka.)43 b FA(Ni\243)32 b(n\341m)g(nebr\341ni)f(uv)-5 b(a\272o)n(v)g(a\264)31 b(o)h(cel\375c)n(h)f(\243\355slac)n(h,)h(t.j.)50 b Fy(a;)14 b(b)30 b Fs(2)g Fm(Z)c FA(\()p Fp(j)p Fy(a)p Fp(j)j Fy(>)h Fp(j)p Fy(b)p Fp(j)p FA(\).)49 b(Ke\244\272e)32 b(pre)f(\251ub)r(o)n(v) n(o\251n)n(\372)189 2250 y(dv)n(o)5 b(jicu)20 b(cel\375c)n(h)g (\243\355sel)g Fy(x;)14 b(y)20 b FA(plat\355)h(nsd)p Fv(\()p Fy(x;)14 b(y)p Fv(\))23 b(=)g FA(nsd)p Fv(\(-)p Fy(x;)14 b(y)p Fv(\))23 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Fv(=)25 b(-)p Fy(3)p FA(.)40 b(Ke\244\272e)396 3840 y Fy(a)29 b(<)h(0)i FA(a)f Fy(b)e(>)g(0)p FA(,)k(p)r(olo\272\355me)e Fy(v)f Fv(=)f(-)p Fy(v)1609 3810 y Fr(0)1661 3840 y Fv(=)g Fy(3)j FA(a)f Fy(u)f Fv(=)f Fy(u)2135 3810 y Fr(0)2188 3840 y Fv(=)g Fy(4)p FA(.)49 b(P)n(otom)31 b Fy(ub)21 b Fv(+)g Fy(va)29 b Fv(=)g Fy(4)22 b Fs(\001)f Fy(21)h Fv(+)e Fy(3)i Fs(\001)f Fv(\(-)p Fy(27)p Fv(\))31 b(=)e Fy(3)h Fv(=)396 3940 y FA(nsd)p Fv(\(-)p Fy(27;)14 b(21)p Fv(\))p FA(.)290 4130 y(3.)41 b(Nec)n(h)28 b Fy(a)23 b Fv(=)f Fy(40)p FA(,)29 b Fy(b)23 b Fv(=)g Fy(3)p FA(.)458 4284 y Fy(i)p 536 4320 V 113 w(s)639 4296 y Fw(i)814 4284 y Fy(q)864 4296 y Fw(i)995 4284 y Fy(u)1048 4296 y Fw(i)1179 4284 y Fy(v)1220 4296 y Fw(i)p 396 4323 906 4 v 446 4407 a Fy(0)p 536 4443 4 120 v 100 w(40)101 b Fv(--)142 b Fy(0)131 b(1)446 4527 y(1)p 536 4563 V 142 w(3)100 b Fv(--)142 b Fy(1)131 b(0)446 4646 y(2)p 536 4682 V 142 w(1)143 b(13)102 b(13)131 b(1)446 4766 y(3)p 536 4802 V 142 w(0)184 b(3)102 b(40)131 b(3)396 4926 y FA(T)-7 b(eda)25 b(nsd)o Fv(\()p Fy(a;)14 b(b)p Fv(\))23 b(=)g Fy(1)p FA(,)j Fy(u)d Fv(=)g(\(-)p Fy(1)p Fv(\))1453 4896 y Fw(3)1507 4926 y Fs(\001)12 b Fy(13)24 b Fv(=)f(-)p Fy(13)p FA(,)j Fy(v)d Fv(=)g(\(-)p Fy(1)p Fv(\))2246 4896 y Fw(2)2299 4926 y Fs(\001)12 b Fy(1)24 b Fv(=)f Fy(1)p FA(.)36 b(Sk\372\262k)-5 b(a:)35 b Fy(ub)12 b Fv(+)g Fy(va)23 b Fv(=)g(-)p Fy(13)12 b Fs(\001)g Fy(3)g Fv(+)g Fy(1)g Fs(\001)g Fy(40)26 b Fv(=)d Fy(1)p FA(.)314 5111 y(Zlo\272itos\264)j(roz\262\355ren\351ho)g(Euklido)n(vho)h (algoritm)n(u)g(je)h(asymptotic)n(ky)e(ro)n(vnak)-5 b(\341)26 b(ak)n(o)h(zlo\272itos\264)g(klasic)n(k)n(\351ho)e(\(\022delia-)189 5210 y(ceho\020)7 b(\))22 b(Euklido)n(vho)f(algoritm)n(u.)34 b(V\262imnime)23 b(si,)g(\272e)f(ak)g(sa)f(nebudeme)i(zaob)r(era\264)e (\243\355slami)h Fy(u)3111 5222 y Fw(i)3165 5210 y FA(a)g Fy(v)3270 5222 y Fw(i)3301 5210 y FA(,)i(dostaneme)d(pr\341)n(v)n(e)189 5310 y(ten)n(to)32 b(algoritm)n(us.)47 b(Nie)32 b(je)h(\264a\272k)n (\351)e(uk)-5 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y(D\364kaz.)144 b FA(1.)41 b(Nec)n(h)e Fy(x)568 1540 y Fw(2)650 1570 y Fs(\021)i Fy(y)805 1540 y Fw(2)883 1570 y Fv(\()p FA(mo)r(d)28 b Fy(p)p Fv(\))p FA(,)42 b(pre)c(nejak)n(\351)g Fy(x;)14 b(y)42 b Fs(2)g Fp(f)p Fy(1;)14 b(:)g(:)g(:)g(;)g(p)25 b Fv(-)h Fy(1)p Fp(g)p FA(.)71 b(P)n(otom)38 b Fy(p)k Fp(j)g Fv(\()p Fy(x)3020 1540 y Fw(2)3086 1570 y Fv(-)26 b Fy(y)3224 1540 y Fw(2)3264 1570 y Fv(\))42 b Fu(\))g Fy(p)g Fp(j)19 1670 y Fv(\()p Fy(x)20 b Fv(-)g Fy(y)p Fv(\)\()p Fy(x)h Fv(+)g Fy(y)p Fv(\))p FA(.)47 b(M\364\272u)30 b(nasta\264)h(dv)-5 b(a)30 b(pr\355pady)-7 b(.)46 b(Prv\375)-7 b(,)31 b(k)n(e\244)g Fy(p)d Fp(j)g Fv(\()p Fy(x)20 b Fv(-)h Fy(y)p Fv(\))29 b Fu(\))f Fy(x)g Fv(=)g Fy(y)p FA(,)k(t.j.)47 b Fy(x)31 b FA(a)f Fy(y)h FA(s\372)g(ro)n(vnak)n(\351.)19 1769 y(Druh\375)k(pr\355pad)g(nastane,)i(k)n(e\244)e Fy(p)h Fp(j)g Fv(\()p Fy(x)23 b Fv(+)h Fy(y)p Fv(\))36 b Fu(\))h Fy(x)e Fv(=)h Fy(p)24 b Fv(-)g Fy(y)p FA(,)37 b(t.j.)62 b Fy(x)35 b FA(a)g Fy(y)h FA(s\372)f(na)n(vz\341)5 b(jom)34 b(r\364zne)g(\(leb)r(o)i Fy(p)f FA(je)19 1869 y(nep\341rne\).)47 b(T)-7 b(eda)30 b(na)h(kv)-5 b(adratic)n(k)n(\351)29 b(rez\355duum)i(mo)r(dulo)g Fy(p)h FA(sa)e(v\272dy)h(zobrazia)e(\(umo)r (cnen\355m)j(na)f(druh)n(\372\))g(dv)-5 b(a)19 1969 y(prvky)26 b(z)i Fp(f)p Fy(1;)14 b(:)g(:)g(:)f(;)h(p)19 b Fv(-)f Fy(1)p Fp(g)p FA(.)37 b(Preto)27 b(je)h(pr\341)n(v)n(e)e(p)r(olo)n (vica)g(\243\355sel)h Fp(f)p Fy(1;)14 b(:)g(:)g(:)g(;)g(p)k Fv(-)h Fy(1)p Fp(g)27 b Fy(QR)2513 1981 y Fw(p)2559 1969 y FA(.)-88 2133 y(2.)42 b(P)n(o)r(\243\355ta)5 b(jme:)36 b Fv(\()p Fy(p)19 b Fv(-)f Fy(x)p Fv(\))703 2103 y Fw(2)767 2133 y FA(mo)r(d)23 b Fy(p)g Fv(=)g(\()p Fy(p)1188 2103 y Fw(2)1247 2133 y Fv(-)18 b Fy(2px)h Fv(+)f Fy(x)1606 2103 y Fw(2)1646 2133 y Fv(\))23 b FA(mo)r(d)g Fy(p)g Fv(=)g Fy(x)2085 2103 y Fw(2)2149 2133 y FA(mo)r(d)g Fy(p)g Fv(=)g Fy(a)p FA(,)k(\243o)g(b)r(olo)g(treba)g(dok)-5 b(\341za\264.)p 3511 2232 4 57 v 3515 2180 50 4 v 3515 2232 V 3564 2232 4 57 v -64 2397 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3 v 159 2866 a(4)271 2876 y FA(mo)r(d)g Fy(p)p Fo(.)-189 3038 y(D\364kaz.)43 b FA(Ke\244\272e)27 b Fy(p)c Fs(\021)f Fy(3)38 b Fv(\()p FA(mo)r(d)28 b Fy(4)p Fv(\))p FA(,)g(tak)1093 3001 y Fw(p)p Fl(+)p Fw(1)p 1093 3019 130 4 v 1140 3066 a(4)1260 3038 y FA(je)g(cel\351)g(\243\355slo.)36 b(P)n(o)r(\243\355ta)5 b(jme:)886 3242 y Fy(x)930 3208 y Fw(2)994 3242 y Fs(\021)22 b Fv(\()p Fy(a)1174 3181 y Fi(p)p Fa(+)p Fi(1)p 1174 3195 116 3 v 1216 3228 a(4)1304 3242 y Fv(\))1336 3208 y Fw(2)1400 3242 y Fs(\021)h Fy(a)1549 3181 y Fi(p)p Fa(+)p Fi(1)p 1548 3195 V 1590 3228 a(2)1701 3242 y Fs(\021)g Fy(a)18 b Fs(\001)g Fy(a)1960 3181 y Fi(p)p Fa(-)p Fi(1)p 1960 3195 V 2001 3228 a(2)2173 3242 y Fv(\()p FA(mo)r(d)28 b Fy(p)p Fv(\))p Fy(:)884 b FA(\(A.4\))-189 3429 y(Vieme,)28 b(\272e)f Fy(a)h FA(je)g Fy(QR)480 3441 y Fw(p)525 3429 y FA(,)g(preto)f(existuje)h Fy(b)f FA(tak)n(\351,)g(\272e)h Fy(b)1528 3399 y Fw(2)1591 3429 y Fs(\021)23 b Fy(a)36 b Fv(\()p FA(mo)r(d)28 b Fy(p)p Fv(\))p FA(.)37 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b(testo)n(v)-5 b(anie,)35 b(ktor\351)f(je)g(v\262eob)r(ecn\351)g(\(plat\355)g(pre)g(\251ub)r(o)n (v)n(o\251n\351)g(prv)n(o)r(\243\355slo)f Fy(p)p FA(\),)j(je)f(nasledo) n(vn\351:)48 b Fy(a)34 b FA(je)-189 4816 y Fy(QR)-69 4828 y Fw(p)9 4816 y Fu(,)e Fy(a)185 4760 y Fi(p)p Fa(-)p Fi(1)p 185 4774 V 226 4806 a(2)346 4816 y Fs(\021)e Fy(1)37 b Fv(\()p FA(mo)r(d)28 b Fy(p)p Fv(\))33 b FA(\(toto)f(tvrdenie)g(sa)g (naz\375v)-5 b(a)31 b(Eulero)n(v)n(o)f(krit\351rium\).)51 b(Jedna)31 b(implik)-5 b(\341cia)32 b(vypl\375v)-5 b(a)32 b(z)-189 4916 y(malej)27 b(F)-7 b(ermato)n(v)n(ej)26 b(v)n(et)n(y)-7 b(,)27 b(druh)n(\372)g(je)h(mo\272n\351)f(uk)-5 b(\341za\264)27 b(cez)g(reprezen)n(t\341ciu)f(prvk)n(o)n(v)f(v)j Fy(Z)2564 4928 y Fw(p)2638 4916 y FA(p)r(omo)r(cou)f(gener\341tora)d (tejto)-189 5016 y(m)n(ultiplik)-5 b(at\355vnej)28 b(grup)n(y)-7 b(.)-189 5146 y Fo(Pozn\341mka.)43 b FA(Rie\262enie)30 b(ro)n(vnice)e Fy(x)907 5116 y Fw(2)974 5146 y Fs(\021)f Fy(a)36 b Fv(\()p FA(mo)r(d)28 b Fy(p)p 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Fy(q)30 b Fo(s\372)f(nep)l(\341rne)h(prvo)l(\243\355sla.)40 b(Nasle)l(dovn\351) 31 b(tvr)l(denia)g(s\372)e(ekvivalentn\351:)-91 5900 y(1.)43 b(Existuje)29 b(pr)l(avdep)l(o)l(dobnostn\375)j(p)l (olynomi\341lny)g(algoritmus)e(pr)l(e)g(faktoriz\341ciu)i Fy(n)p Fo(.)p eop %%Page: 31 30 31 30 bop 189 100 a Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))1824 b FA(31)287 333 y Fo(2.)42 b(Existuje)33 b(pr)l(avdep)l(o)l (dobnostn\375)h(p)l(olynomi\341lny)h(algoritmus)e(pr)l(e)g(v\375p)l(o)l (\243)l(et)f(druh\375ch)i(o)l(dmo)l(cn\355n)f(kvadr)l(atick\375ch)396 432 y(r)l(ez\355du\355)e(mo)l(dulo)f Fy(n)p Fo(.)189 598 y(D\364kaz.)43 b FA(\(1)37 b Fu(\))g FA(2\))g(Ma)5 b(jme)37 b(k)g(disp)r(oz\355cii)g(pra)n(vdep)r(o)r(dobnostn\375)e(p)r (olynomi\341ln)n(y)h(algoritm)n(us)g(pre)h(faktoriz\341ciu)e Fy(n)p FA(.)189 698 y(P)n(ou\272ijeme)h(ho)f(a)g(zo)g(zadan\351ho)f Fy(n)i FA(vyp)r(o)r(\243\355tame)f Fy(p)g FA(a)g Fy(q)p FA(.)61 b(Chceme)36 b(rie\262i\264)f(ro)n(vnicu)f Fy(x)3007 668 y Fw(2)3084 698 y Fs(\021)h Fy(a)i Fv(\()p FA(mo)r(d)28 b Fy(n)p Fv(\))p FA(,)38 b(kde)d Fy(a)g FA(je)189 798 y Fy(QR)309 810 y Fw(n)360 798 y FA(.)h(Ak)26 b(b)n(y)f Fy(a)g FA(b)r(olo)g Fy(QNR)1109 810 y Fw(n)1159 798 y FA(,)h(tak)g(jedno)r(duc)n(ho)f(vyp)r(o)r(\243\355tame)g(nek)n(orektn)n (\372)f(druh)n(\372)h(o)r(dmo)r(cnin)n(u)g(\(leb)r(o)h(neexistuje\),) 189 897 y(\243o)33 b(vieme)h(o)n(v)n(eri\264)e(a)h(ni\243)h(sa)f (nestane.)55 b(Na)5 b(jsk)n(\364r)32 b(n\341)5 b(jdeme)33 b(druh\351)h(o)r(dmo)r(cnin)n(y)f Fy(a)g FA(mo)r(dulo)h Fy(p)g FA(a)f(mo)r(dulo)g Fy(q)h FA(\(ak)g(s\372)189 997 y Fy(p;)14 b(q)31 b Fs(\021)f Fy(3)37 b Fv(\()p FA(mo)r(d)28 b Fy(4)p Fv(\))p FA(,)35 b(tak)c(m\364\272eme)h(p)r(ou\272i\264)h(lem)n (u)f(A.4.2,)h(v)f(opa\243nom)f(pr\355pade)g(treba)h(p)r(ou\272i\264)g (n\341ro)r(\243nej\262\355)f(p)r(ostup,)189 1096 y(p)r(ozri)c(napr.)36 b([K)n(ob87a)n(,)27 b(stran)n(y)g(47\02548]\):)1749 1279 y Fy(u)1802 1245 y Fw(2)1866 1279 y Fs(\021)c Fy(a)83 b Fv(\()p FA(mo)r(d)28 b Fy(p)p Fv(\))1762 1417 y Fy(v)1803 1382 y Fw(2)1866 1417 y Fs(\021)23 b Fy(a)83 b Fv(\()p FA(mo)r(d)28 b Fy(q)p Fv(\))189 1612 y FA(P)n(ou\272it\355m)g (\203\355nsk)n(ej)f(zvy\262k)n(o)n(v)n(ej)f(v)n(et)n(y)g(dostaneme)h (h\251adan\351)h(rie\262enie)e(ak)n(o)h(line\341rn)n(u)f(k)n(om)n (bin\341ciu)h Fy(u)h FA(a)f Fy(v)p FA(:)1180 1795 y Fy(x)22 b Fv(=)h Fy(q)p Fv(\()p Fy(q)1464 1760 y Fl(-)p Fw(1)1639 1795 y FA(mo)r(d)28 b Fy(p)p Fv(\))p Fy(u)19 b Fv(+)f Fy(p)p Fv(\()p Fy(p)2190 1760 y Fl(-)p Fw(1)2365 1795 y FA(mo)r(d)28 b Fy(q)p Fv(\))p Fy(v)c FA(mo)r(d)f Fy(n:)189 1977 y FA(T)-7 b(rivi\341lne)27 b(plat\355)i Fy(x)24 b Fs(\021)g Fy(u)37 b Fv(\()p FA(mo)r(d)28 b Fy(p)p Fv(\))h FA(a)f Fy(x)c Fs(\021)g Fy(v)37 b Fv(\()p FA(mo)r(d)28 b Fy(q)p Fv(\))p FA(.)40 b(Odtia\251)28 b(dost\341)n(v)-5 b(ame)27 b Fy(x)2700 1947 y Fw(2)2765 1977 y Fs(\021)d Fy(a)36 b Fv(\()p FA(mo)r(d)28 b Fy(p)p Fv(\))h FA(a)f Fy(x)3384 1947 y Fw(2)3449 1977 y Fs(\021)c Fy(a)36 b Fv(\()p FA(mo)r(d)28 b Fy(q)p Fv(\))p FA(.)189 2077 y(Vieme,)g(\272e)f Fy(p;)14 b(q)28 b FA(s\372)g(nes\372delite\251n\351,)g(teda)f(v\244ak) -5 b(a)27 b(d\364sledku)g(A.3.2)g(m\341me)h Fy(x)2653 2047 y Fw(2)2716 2077 y Fs(\021)22 b Fy(a)37 b Fv(\()p FA(mo)r(d)28 b Fy(n)p Fv(\))p FA(.)314 2226 y(\(2)i Fu(\))g FA(1\))g(Ma)5 b(jme)30 b(k)g(disp)r(oz\355cii)g(algoritm)n(us,)f (ktor\375)g(rie\262i)h(ro)n(vnice)f Fy(x)2529 2196 y Fw(2)2596 2226 y Fs(\021)f Fy(a)36 b Fv(\()p FA(mo)r(d)28 b Fy(n)p Fv(\))p FA(.)45 b(Uk)-5 b(\341\272eme,)31 b(ak)n(o)e(faktori-) 189 2326 y(zo)n(v)-5 b(a\264)27 b Fy(n)p FA(,)h(t.j.)37 b(zisti\264)28 b(prv)n(o)r(\243\355sla)e Fy(p)p FA(,)i Fy(q)p FA(.)290 2492 y(1.)41 b(Zv)n(ol\355me)27 b(n\341ho)r(dne)g Fy(m)c Fs(2)1198 2504 y Fw(R)1270 2492 y Fp(f)p Fy(2;)14 b(3)g(:)g(:)g(:)f(;)h(n)19 b Fv(-)f Fy(1)p Fp(g)p FA(.)37 b(P)n(olo\272\355me)27 b Fy(a)c Fv(=)g Fy(m)2450 2462 y Fw(2)2513 2492 y FA(mo)r(d)g Fy(n)p FA(.)290 2658 y(2.)41 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1831 V 101 w(3)f(1)208 b(1)623 b(1)1210 1915 y(9)p 1292 1950 V 84 w(9105)85 b(3952)p 1790 1950 V 1807 1950 V 101 w(4)665 b(1)291 b(1)189 2092 y FA(T)-7 b(abu\251k)i(a)38 b(obsah)n(uje)g(prvky)g Fy(\013)1185 2104 y Fw(i;j)1263 2092 y FA(,)j(pre)d Fy(i)k Fv(=)f Fy(1;)14 b(:)g(:)g(:)g(;)g(9)39 b FA(a)f Fy(j)j Fv(=)g Fy(1;)14 b(:)g(:)g(:)g(;)g(8)p FA(.)70 b(Pr\341zdne)38 b(miesta)g(ozna\243uj\372)g(ho)r(dnotu)h Fy(0)p FA(.)189 2191 y(Jednotliv)n(\351)19 b(v)n(ektory)g Fy(\015)923 2203 y Fw(i)953 2191 y FA(,)j(medzi)e(ktor\375mi)g(budeme)g (h\251ada\264)g(lie\341rne)f(z\341)n(vislosti,)h(s\372)g(v)g(nasledo)n (vnej)f(tabu\251k)n(e)h(\(pr\341zdne)189 2291 y(miesta)27 b(op\344to)n(vne)f(ozna\243uj\372)h(ho)r(dnotu)h Fy(0)p FA(\):)1259 2479 y Fy(i)p 1329 2515 V 167 w(a)1507 2491 y Fw(i)p 1578 2515 V 1595 2515 V 1812 2479 a Fy(2)83 b(3)h(5)f(7)h(11)g(13)g(17)g(19)p 1206 2519 1727 4 v 1248 2602 a(1)p 1329 2638 4 120 v 125 w(105)p 1578 2638 V 1595 2638 V 101 w(\015)1689 2614 y Fw(1)1812 2602 y 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Fw(2)3336 4203 y Fv(\()p FA(mo)r(d)28 b Fy(n)p Fv(\))p FA(.)67 b(Boh)n(u-)396 4302 y(\272ia\251,)28 b(nsd)p Fv(\()p Fy(8091)21 b Fv(+)d Fy(2288;)c(n)p Fv(\))25 b(=)e Fy(10379)j Fv(=)c Fy(n)p FA(,)28 b(tak\272e)f(zno)n(v)-5 b(a)27 b(sme)g(ni\243)h(zm)n(yslupln\351)f(nezistili.)290 4466 y(3.)41 b(\204alej)g(vyp)r(o)r(\243\355tame,)j(\272e)c(v)n(ektory) f Fy(\015)1592 4478 y Fw(4)1659 4466 y Fv(+)27 b Fy(\015)1800 4478 y Fw(7)1867 4466 y Fv(+)g Fy(\015)2008 4478 y Fw(9)2092 4466 y Fv(=)45 b FB(0)p FA(.)75 b(T)-7 b(eda)40 b(dost\341)n(v)-5 b(ame)39 b(k)n(ongruenciu)g Fv(\()p Fy(a)3545 4478 y Fw(4)3586 4466 y Fy(a)3637 4478 y Fw(7)3677 4466 y Fy(a)3728 4478 y Fw(9)3768 4466 y Fv(\))3800 4435 y Fw(2)3885 4466 y Fs(\021)396 4565 y Fv(\()p Fy(2)469 4535 y Fw(6=2)584 4565 y Fy(5)625 4535 y Fw(4=2)738 4565 y Fy(7)779 4535 y Fw(2=2)893 4565 y Fy(13)975 4535 y Fw(2=2)1090 4565 y Fy(19)1172 4535 y Fw(2=2)1286 4565 y Fv(\))1318 4535 y Fw(2)1396 4565 y Fv(\()p FA(mo)r(d)28 b Fy(n)p Fv(\))p FA(,)34 b(t.j.)54 b Fy(3400)2080 4535 y Fw(2)2154 4565 y Fs(\021)31 b Fy(3293)2414 4535 y Fw(2)2493 4565 y Fv(\()p FA(mo)r(d)d Fy(n)p Fv(\))p FA(.)53 b(Odtia\251)33 b(v\375p)r(o)r (\243tom)g(nsd)p Fv(\()p Fy(3400)24 b Fv(+)396 4665 y Fy(3293;)14 b(n)p Fv(\))36 b(=)d Fy(97)i FA(dostaneme)e(netrivi\341ln)n (y)g(faktor)g Fy(n)h FA(a)f(predelen\355m)h(a)5 b(j)33 b(druh\375)-7 b(.)55 b(T)-7 b(eda)34 b Fy(n)g FA(je)g(s\372\243inom)f (dv)n(o)r(c)n(h)396 4764 y(prv)n(o)r(\243\355sel:)j Fy(10379)25 b Fv(=)e Fy(97)d Fs(\001)e Fy(107)p FA(.)189 4979 y FB(\203aso)m(v)-5 b(\341)34 b(zlo\272itos\264)f(algoritm)m(u.)81 b FA(Prista)n(vme)29 b(sa)f(na)5 b(jsk)n(\364r)28 b(pri)h(dv)n(o)r(c)n(h)f(probl\351mo)r(c)n (h)h(met\363)r(dy)g(kv)-5 b(adratic)n(k\375c)n(h)28 b(re-)189 5079 y(z\355du\355.)290 5239 y(1.)41 b(P)n(o)r(\243as)36 b(pr\341ce)f(algoritm)n(u)g(p)r(otrebujeme)h(v)n(e\251a)g(kr\341t)f (testo)n(v)-5 b(a\264)36 b(rozklad)e Fy(b)2794 5251 y Fw(i)2862 5239 y FA(nad)i(b\341zou)f(prv)n(o)r(\243\355sel.)62 b(T)-7 b(oto)35 b(je)396 5338 y(mo\272n\351)h(robi\264)f(na)5 b(jjedno)r(duc)n(h\262\355m)36 b(sp)r(\364sob)r(om)f(\025)h(p)r (ostupne)g(prec)n(h\341dza\264)e(b\341zou)h Fy(p)3087 5350 y Fw(1)3128 5338 y Fy(;)14 b(:)g(:)g(:)f(;)h(p)3361 5350 y Fw(h)3446 5338 y FA(a)35 b(sk\372\262a\264)g(deli\264)396 5438 y(prv)n(o)r(\243\355slami,)27 b(p)r(okia\251)g(to)g(ide.)290 5601 y(2.)41 b(Ak)n(o)36 b(v)n(oli\264)h(b\341zu)f(prv)n(o)r (\243\355sel?)63 b(T)-7 b(reba)36 b(si)g(uv)n(edomi\264,)j(\272e)d(sa)g (dost\341)n(v)-5 b(ame)36 b(do)g(dv)n(o)r(c)n(h)g(protic)n(ho)r (dn\375c)n(h)f(tlak)n(o)n(v.)396 5701 y(Ak)d(zv)n(ol\355me)e(b\341zu)h (prv)n(o)r(\243\355sel)f(mal\372)h(\(teda)h Fy(h)f FA(je)g(mal\351\),)h (tak)f(bude)h(v)n(e\251mi)f(dlho)g(trv)-5 b(a\264,)33 b(k\375m)e(vygenerujeme)396 5800 y(dostato)r(\243n\375)h(p)r(o)r (\243et)i Fy(b)1107 5812 y Fw(i)1171 5800 y FA(rozlo\272ite\251n\375c)n (h)d(nad)i(b\341zou)f(prv)n(o)r(\243\355sel.)51 b(\203\355m)34 b(viac)e(prv)n(o)r(\243\355sel,)h(t)n(\375m)f(v)-5 b(\344\243\262ia)32 b(pra)n(vde-)396 5900 y(p)r(o)r(dobnos\264,)d(\272e)g(sa)f(n\341ho)r (dn\351)h(kv)-5 b(adratic)n(k)n(\351)27 b(rez\355duom)h(mo)r(dulo)h(n)g (bude)g(da\264)g(nad)g(nimi)g(rozlo\272i\264.)40 b(Na)29 b(druhej)p eop %%Page: 36 35 36 35 bop -189 100 a FA(36)1824 b Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))19 333 y FA(strane,)32 b(ak)g(je)g(b\341za)g(prv)n(o)r(\243\355sel)f(priv) n(e\251k)-5 b(\341)31 b(\()p Fy(h)h FA(je)h(v)n(e\251k)n(\351\),)g(tak) f(sa)g(v\375p)r(o)r(\243to)n(v)n(o)e(n\341ro)r(\243n\375mi)h(op)r (er\341ciami)g(stan)n(\372)19 432 y(rozklad)c Fy(B)367 444 y Fw(i)428 432 y FA(nad)i(b\341zou,)f(Gausso)n(v)-5 b(a)28 b(elimin\341cia)g(v)h(matici)g(v)n(ektoro)n(v)e Fy(\015)2336 444 y Fw(1)2376 432 y Fy(;)14 b(\015)2464 444 y Fw(2)2504 432 y Fy(;)g(:)g(:)g(:)42 b FA(a)29 b(zv\375\262ia)f (sa)g(a)5 b(j)29 b(n\341roky)e(na)19 532 y(pam\344\264.)36 b(Vho)r(dnou)28 b(v)n(o\251b)r(ou)f Fy(h)g FA(je)h(asymptotic)n(ky)e Fy(\002)p Fv(\()1725 469 y Fs(p)p 1795 469 488 4 v 1795 532 a FA(log)14 b Fy(n)g FA(log)g(log)g Fy(n)p Fv(\))p FA(.)-64 698 y(D\341)35 b(sa)g(uk)-5 b(\341za\264,)37 b(\272e,)h(ak)d(je)g Fy(h)h FA(zv)n(olen\351)e(ak)n(o)g Fy(C)1442 710 y Fw(1)1483 635 y Fs(p)p 1552 635 V 63 x FA(log)14 b Fy(n)g FA(log)g(log)h Fy(n)p FA(,)37 b(kde)f Fy(C)2322 710 y Fw(1)2398 698 y FA(je)g(nejak)-5 b(\341)35 b(kladn\341)g(k)n(on\262tan)n(ta,)h(tak)-189 798 y(zlo\272itos\264)c (met\363)r(dy)h(kv)-5 b(adratic)n(k\375c)n(h)30 b(rez\355du\355)j(je)g Fy(e)1398 767 y Fl(\()p Fw(C)p Fl(+)p Fw(O)p Fl(\()p Fw(1)p Fl(\)\))1703 724 y Fr(p)p 1758 724 398 3 v 43 x Fj(log)12 b Fw(n)g Fj(log)g(log)g Fw(n)2192 798 y FA(\()p Fy(C)34 b FA(je)f(k)n(on\262tan)n(ta,)f(ktor\341)f(v)i(z\341)n (vislosti)e(na)-189 897 y(implemen)n(t\341cii)d(le\272\355)f(medzi)731 827 y Fs(p)p 800 827 42 4 v 70 x Fy(2)h FA(a)f Fy(2)p FA(\).)-64 997 y(V)-7 b(ylep\262enia)27 b(met\363)r(dy)g(kv)-5 b(adratic)n(k\375c)n(h)26 b(rez\355du\355)h(sp)r(o)r(\243\355v)-5 b(a)5 b(j\372)28 b(v)f(dv)n(o)r(c)n(h)g(oblastiac)n(h:)-64 1163 y Fs(\017)41 b FA(Lep\262ie)33 b(genero)n(v)-5 b(anie)31 b(kv)-5 b(adratic)n(k\375c)n(h)31 b(rez\355du\355.)54 b(K)-7 b(v)i(adratic)n(k)n(\351)32 b(rez\355du\341)g(sa)h (negeneruj\372)g(n\341ho)r(dne)g(\(vtedy)h(s\372)19 1262 y(asymptotic)n(ky)18 b(ro)n(vnak)n(o)f(dlh\351)i(ak)n(o)f Fy(n)p FA(\))j(ale)e(tak,)h(ab)n(y)f(b)r(oli)h(krat\262ie)e(\(vtedy)i (je)g(pra)n(vdep)r(o)r(dobnos\264)e(ic)n(h)h(\372sp)r(e\262n\351ho)19 1362 y(rozlo\272enia)25 b(nad)i(b\341zou)g(prv)n(o)r(\243\355sel)g(v)-5 b(\344\243\262ia\))26 b(aleb)r(o)h(s)h(predv\355date\251nou)f (\262trukt\372rou)f(faktoro)n(v.)-64 1528 y Fs(\017)41 b FA(Efekt\355vnej\262\355)24 b(rozklad)e(nad)h(b\341zou)f(prv)n(o)r (\243\355sel.)35 b(Pri)23 b(kv)-5 b(adratic)n(k\375c)n(h)21 b(rez\355du\341c)n(h)i(s)g(predv\355date\251nou)g(\262trukt\372rou)19 1628 y(faktoro)n(v)i(p)r(omo)r(cou)i(tzv.)37 b(sita.)-64 1794 y(Met\363)r(dami)24 b(realizuj\372cimi)g(uv)n(eden\351)g (vylep\262enia)g(s\372)g(napr\355klad)g(Con)n(tin)n(ued)g(F)-7 b(ractions)23 b(aleb)r(o)h(Quadratic)f(Siev)n(e.)-189 1893 y(Ic)n(h)39 b(p)r(ekn\375)g(p)r(opis)g(mo\272no)f(n\341)5 b(js\264)39 b(v)g([W)-7 b(un88].)71 b(P)n(oznamena)5 b(jme,)41 b(\272e)e(jeden)h(z)f(v)-5 b(arian)n(to)n(v)37 b(Quadratic)g(Siev)n(e)i(m\341)-189 1993 y(heuristic)n(k\372)19 b(\243aso)n(v\372)e(zlo\272itos\264)i(\(t.j.)35 b(\243aso)n(v\372)18 b(zlo\272itos\264)h(sa)f(nep)r(o)r(darilo)h(form\341lne)f(dok)-5 b(\341za\264,)20 b(b)r(ez)g(ist)n(\375c)n(h)f(predp)r(oklado)n(v)-189 2102 y(a)27 b(h)n(yp)r(ot\351z\))h Fy(O)p Fv(\()p Fy(e)364 2072 y Fl(\()p Fw(1)p Fl(+)p Fw(")p Fl(\))542 2028 y Fr(p)p 597 2028 398 3 v 44 x Fj(log)12 b Fw(n)g Fj(log)g(log)g Fw(n)999 2102 y Fv(\))p FA(,)28 b(pre)f(\251ub)r(o)n(v)n(o\251n\351)g Fy(")c(>)g(0)p FA(.)p eop %%Page: 37 36 37 36 bop 189 999 a FC(Literat\372ra)189 1432 y FA([BBS86])98 b(Blum,)27 b(L.)h(\025)e(Blum,)i(M.)f(\025)g(Sh)n(ub,)g(M.:)37 b(A)28 b(simple)f(unpredictable)g(pseudo-random)e(n)n(um)n(b)r(er)i (generator.)581 1532 y Fo(SIAM)i(Journal)h(on)g(Computing)p FA(,)e(15\(2\):364\025383,)23 b(1986.)189 1705 y([BD99])140 b(Boneh,)26 b(D.)h(\025)e(Durfee,)i(G.:)36 b(Cryptanalysis)25 b(of)h(RSA)h(with)f(priv)-5 b(ate)26 b(k)n(ey)f Fy(d)i FA(less)e(than)i Fy(N)3396 1675 y Fw(0:292)3567 1705 y FA(,)f Fo(A)l(dvanc)l(es)581 1805 y(in)k(cryptolo)l(gy)h(\025)f(EUR)n (OCR)-6 b(YPT'99)p FA(,)28 b(LNCS)g(1592,)e(pp.)i(1\02511,)e (Springer-V)-7 b(erlag,)25 b(1999.)189 1978 y([Ble96])143 b(Bleic)n(hen)n(bac)n(her,)21 b(D.:)34 b(Generating)21 b(ElGamal)h(signatures)e(without)i(kno)n(wing)e(the)i(secret)f(k)n(ey) -7 b(,)22 b Fo(A)l(dvanc)l(es)581 2078 y(in)30 b(Cryptolo)l(gy)h(\025)g (EUR)n(OCR)-6 b(YPT'96)p FA(,)28 b(LNCS)f(1070,)f(pp.)i(10\02518,)d (Springer-V)-7 b(erlag,)25 b(1996.)189 2251 y([Ble98])143 b(Bleic)n(hen)n(bac)n(her,)21 b(D.:)35 b(Chosen)21 b(ciphertext)h (attac)n(ks)f(against)g(proto)r(cols)f(based)i(on)f(the)i(RSA)f (encryption)581 2351 y(standard)k(PK)n(CS)h(#1,)g Fo(A)l(dvanc)l(es)i (in)g(Cryptolo)l(gy)i(\025)f(CR)-6 b(YPTO'98)p FA(,)28 b(LNCS)f(????,)i(pp.)e(1\02512,)e(Springer-)581 2450 y(V)-7 b(erlag,)26 b(1998.)189 2624 y([BM79])127 b(Birkho\033,)27 b(G.)h(\025)f(MacLane,)f(S.:)38 b Fo(Pr)l(eh\251ad)31 b(mo)l(dernej)g(algebry)p FA(,)e(Alfa,)f(1979.)189 2797 y([BV98])141 b(Boneh,)28 b(D.)h(\025)f(V)-7 b(enk)i(atesan,)28 b(R.:)38 b(Breaking)27 b(RSA)h(ma)n(y)g(not)g(b)r(e)h(equiv)-5 b(alen)n(t)28 b(to)g(factoring,)f Fo(A)l(dvanc)l(es)j(in)581 2897 y(cryptolo)l(gy)h(\025)g(EUR)n(OCR)-6 b(YPT'98)p FA(,)28 b(LNCS)f(1403,)f(pp.)i(59\02571,)d(Springer-V)-7 b(erlag,)25 b(1998.)189 3070 y([DDN91])74 b(Dolev,)41 b(D.)e(\025)f(Dw)n(ork,)i(C.)e(\025)g(Naor,)i(M.:)59 b(Non-malleable)37 b(cryptograph)n(y)-7 b(,)39 b Fo(Pr)l(o)l(c)l(e)l(e) l(dings)h(of)h(the)f(23r)l(d)581 3170 y(annual)30 b(Symp)l(osium)g(on)f (The)l(ory)i(of)g(Computing)p FA(,)d(pp.)g(542\025552,)c(A)n(CM,)k (1991.)189 3343 y([DH76])137 b(Di\036e,)24 b(W.)f(\025)f(Hellman,)h (M.:)35 b(New)22 b(directions)g(in)h(cryptograph)n(y)-7 b(.)19 b Fo(IEEE)26 b(T)-6 b(r)l(ansactions)26 b(on)f(Information)581 3443 y(The)l(ory)31 b(IT-22)p FA(,)e(pp.)f(644\025654,)c(1976.)189 3616 y([ElG85])118 b(ElGamal,)22 b(T.:)34 b(A)21 b(public-k)n(ey)f (cryptosystem)g(and)h(a)f(signature)g(sc)n(heme)h(based)f(on)h (discrete)f(logarithms,)581 3716 y Fo(A)l(dvanc)l(es)30 b(in)g(Cryptolo)l(gy)h(\025)f(CR)-6 b(YPTO'84)p FA(,)29 b(pp.)f(10\02518,)d(Springer-V)-7 b(erlag,)25 b(1985.)189 3889 y([GKP94])76 b(Graham,)26 b(R.)h(\025)g(Kn)n(uth,)g(D.)h(\025)e(P) n(atashnik,)h(O.:)36 b Fo(Concr)l(ete)30 b(mathematics:)39 b(a)29 b(foundation)i(for)f(c)l(omputer)581 3989 y(scienc)l(e)p FA(,)e(A)n(ddison-W)-7 b(esley)g(,)27 b(2nd)g(edition,)h(1994.)189 4162 y([GM82])121 b(Goldw)n(asser,)30 b(S.)h(\025)g(Micali,)g(S.:)44 b(Probabilistic)30 b(encryption)h(and)g(ho)n(w)f(to)h(pla)n(y)f(men)n (tal)h(p)r(ok)n(er)f(k)n(eeping)581 4262 y(secret)38 b(all)h(partial)g(information,)i Fo(Pr)l(o)l(c)l(e)l(e)l(dings)g(of)g (the)g(14th)g(A)n(CM)g(Symp)l(osium)g(on)f(the)h(The)l(ory)h(of)581 4362 y(Computing)p FA(,)28 b(pp.)g(365\025377,)c(1982.)189 4535 y([GM84])121 b(Goldw)n(asser,)22 b(S.)i(\025)f(Micali,)h(S.:)35 b(Probabilistic)23 b(encryption.)g Fo(Journal)j(of)h(Computer)f(and)g (System)g(Scien-)581 4635 y(c)l(es)p FA(,)i(28\(2\):270\025299,)23 b(1984.)189 4808 y([Gor92])122 b(Gordon,)32 b(D.:)45 b(Designing)32 b(and)f(detecting)h(trap)r(do)r(ors)e(for)h(discrete)h (log)e(cryptosystems,)i Fo(A)l(dvanc)l(es)h(in)581 4908 y(Cryptolo)l(gy)f(\025)e(CR)-6 b(YPTO'92)p FA(,)28 b(LNCS)g(740,)e(pp.) i(66\02575,)e(Springer-V)-7 b(erlag,)25 b(1992.)189 5081 y([HW54])115 b(Hardy)-7 b(,)26 b(G.)g(\025)g(W)-7 b(righ)n(t,)27 b(E.:)36 b Fo(A)n(n)28 b(Intr)l(o)l(duction)g(to)h(the)f(The)l(ory)i (of)g(Numb)l(ers)p FA(,)c(Oxford)f(Univ)n(ersit)n(y)g(Press,)581 5181 y(3rd)i(edition,)g(1954.)189 5354 y([K)n(ob87a])69 b(K)n(oblitz,)27 b(N.:)37 b Fo(A)29 b(c)l(ourse)h(in)g(numb)l(er)f(the) l(ory)h(and)g(crypto)l(gr)l(aphy)p FA(,)g(Springer-V)-7 b(erlag,)25 b(1987.)189 5527 y([K)n(ob87b])65 b(K)n(oblitz,)42 b(N.:)62 b(Elliptic)41 b(curv)n(e)d(cryptosystems.)h Fo(Mathematics)j(of)g(Computation)p FA(,)i(48\(177\):203\025209,)581 5627 y(1987.)189 5800 y([LKBS92])c(Lo)n(xton,)24 b(J.)h(\025)g(Kho)r (o,)f(D.)i(\025)f(Bird,)g(G.)g(\025)g(Seb)r(erry)-7 b(,)24 b(J.:)36 b(A)25 b(cubic)g(RSA)h(co)r(de)f(equiv)-5 b(alen)n(t)24 b(to)h(factorization.)581 5900 y Fo(Journal)k(of)i(Cryptolo)l(gy)p FA(,)f(5:139\025150,)24 b(1992.)2028 6110 y(37)p eop %%Page: 38 37 38 37 bop -189 100 a FA(38)1824 b Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))-189 333 y FA([LL93])158 b(Lenstra,)25 b(A.)g(\025)g(Lenstra,)g(H.)h (\(editors\):)35 b Fo(The)29 b(development)f(of)h(the)f(numb)l(er)e (\034eld)i(sieve)p FA(,)f(Lecture)e(Notes)203 432 y(in)j(Mathematics)f (1554,)f(Springer-V)-7 b(erlag,)25 b(1993.)-189 603 y([LN94])148 b(Lidl,)41 b(R.)d(\025)f(Niederreiter,)i(H.:)58 b Fo(Intr)l(o)l (duction)39 b(to)g(\034nite)f(\034elds)i(and)g(their)f(applic)l(ations) p FA(,)k(Cam)n(bridge)203 702 y(Univ)n(ersit)n(y)27 b(Press,)g(revised) f(edition,)i(1994.)-189 872 y([Mau94])98 b(Maurer,)46 b(U.:)69 b(T)-7 b(o)n(w)n(ards)41 b(the)j(equiv)-5 b(alence)43 b(of)h(breaking)e(the)i(di\036e-hellman)f(proto)r(col)f(an)h(compu-)203 972 y(ting)35 b(discrete)f(logarithms,)g Fo(A)l(dvanc)l(es)j(in)f (Cryptolo)l(gy)i(\025)f(CR)-6 b(YPTO'94)p FA(,)37 b(LNCS)e(????,)i(pp.) e(271\025281,)203 1072 y(Springer-V)-7 b(erlag,)25 b(1994.)-189 1242 y([MB75])127 b(Morisson,)26 b(M.)h(\025)g(Brillhart,)g(J.:)36 b(A)28 b(metho)r(d)g(of)f(factoring)f(and)i(the)f(factorization)f(of)i Fy(F)3020 1254 y Fw(7)3061 1242 y FA(.)f Fo(Mathematics)203 1341 y(of)k(Computation)p FA(,)d(\(29\):183\025197,)c(1975.)-189 1512 y([MH78])124 b(Merkle,)31 b(R.)g(\025)g(Hellman,)g(M.:)44 b(Hiding)31 b(information)f(and)h(signatures)e(in)i(trap)r(do)r(or)f (knapsac)n(ks.)f Fo(IEEE)203 1611 y(T)-6 b(r)l(ansactions)30 b(on)g(Information)h(The)l(ory)p FA(,)e(24\(5\):525\025530,)23 b(1978.)-189 1782 y([Mil86])140 b(Miller,)25 b(V.:)36 b(Use)26 b(of)f(elliptic)g(curv)n(es)f(in)i(cryptograph)n(y)-7 b(,)23 b Fo(A)l(dvanc)l(es)28 b(in)f(Cryptolo)l(gy)j(\025)e(CR)-6 b(YPTO'85)p FA(,)26 b(pp.)203 1881 y(417\025426,)e(Springer-V)-7 b(erlag,)25 b(1986.)-189 2051 y([MRS88])79 b(Micali,)32 b(S.)f(\025)f(Rac)n(k)n(o\033,)h(C.)g(\025)f(Sloan,)h(R.:)44 b(Notions)30 b(of)h(securit)n(y)f(of)h(public-k)n(ey)f(cryptosystems.)g Fo(SIAM)203 2151 y(Journal)g(on)f(Computing)p FA(,)g (17\(2\):412\025426,)23 b(1988.)-189 2321 y([Nat94])126 b(National)20 b(In)n(tsitute)h(of)f(Standards)f(and)i(T)-7 b(ec)n(hnology:)31 b(Digital)20 b(Signature)g(Standard,)h(NIST)g(FIPS)g (PUB)203 2421 y(186,)26 b(1994.)-189 2591 y([Nat95])126 b(National)35 b(In)n(tsitute)i(of)e(Standards)g(and)h(T)-7 b(ec)n(hnology:)52 b(Secure)35 b(Hash)h(Standard,)h(NIST)f(FIPS)h(PUB) 203 2691 y(180,)26 b(1995.)-189 2861 y([Nat98])126 b(National)20 b(In)n(tsitute)h(of)f(Standards)f(and)i(T)-7 b(ec)n(hnology:)31 b(Digital)20 b(Signature)g(Standard,)h(NIST)g(FIPS)g(PUB)203 2961 y(186-1,)k(1998.)-189 3131 y([PH78])144 b(P)n(ohlig,)22 b(S.)f(\025)f(Hellman,)j(M.:)33 b(An)22 b(impro)n(v)n(ed)d(algorithm)h (for)g(computing)h(logarithms)e(o)n(v)n(er)g Fy(GF)p Fv(\()p Fy(p)p Fv(\))j FA(and)f(its)203 3230 y(cryptographic)k (signi\034cance.)i Fo(IEEE)k(T)-6 b(r)l(ansactions)30 b(on)g(Information)g(The)l(ory)p FA(,)f(24\(1\):106\025110,)24 b(1978.)-189 3401 y([P)n(ol75])143 b(P)n(ollard,)27 b(J.:)36 b(A)28 b(Mon)n(te)f(Carlo)g(metho)r(d)h(for)f(factorization.)f Fo(BIT)p FA(,)i(\(15\):331\025334,)c(1975.)-189 3571 y([P)n(om85])97 b(P)n(omerance,)29 b(C.:)42 b(The)30 b(quadratic)f(siev)n(e)g(factoring)g(algorithm,)g Fo(A)l(dvanc)l(es)j (in)g(cryptolo)l(gy)i(\025)f(EUR)n(OC-)203 3671 y(R)-6 b(YPT'84)p FA(,)29 b(LNCS)e(209,)g(pp.)h(169\025182,)c(Springer-V)-7 b(erlag,)25 b(1985.)-189 3841 y([Rab79])113 b(Rabin,)31 b(M.:)41 b(Digital)30 b(signatures)f(and)g(public-k)n(ey)h(functions)g (as)f(in)n(tractable)g(as)g(factorization,)g(T)-7 b(ec)n(h-)203 3940 y(nical)27 b(rep)r(ort,)g(MIT)h(Lab)r(oratory)d(for)i(Computer)g (Science,)h(MIT/LCS/TR-212,)c(1979.)-189 4111 y([Ros92])126 b(Rosen,)38 b(K.:)53 b Fo(Elementary)38 b(Numb)l(er)f(The)l(ory)h(and)g (Its)f(Applic)l(ations)p FA(,)j(A)n(ddison-W)-7 b(esley)g(,)37 b(3rd)f(edition,)203 4210 y(1992.)-189 4381 y([RSA78])93 b(Riv)n(est,)25 b(R.)g(\025)f(Shamir,)h(A.)g(\025)f(A)n(dleman,)i(L.:) 35 b(A)25 b(metho)r(d)g(for)g(obtaining)f(digital)g(signatures)f(and)i (public-)203 4480 y(k)n(ey)i(cryptosystems.)f Fo(Communic)l(ation)k(of) h(the)f(A)n(CM)p FA(,)e(21:120\025126,)23 b(1978.)-189 4650 y([Rue86])118 b(Ruepp)r(el,)28 b(R.:)37 b Fo(A)n(nalysis)31 b(and)f(Design)g(of)g(Str)l(e)l(am)f(Ciphers)p FA(,)h(Springer-V)-7 b(erlag,)25 b(1986.)-189 4821 y([Sal96])151 b(Salomaa,)26 b(A.:)37 b Fo(Public-key)31 b(crypto)l(gr)l(aphy)p FA(,)f(Springer-V)-7 b(erlag,)25 b(revised)i(edition,)g(1996.)-189 4991 y([Sc)n(h85])135 b(Sc)n(ho)r(of,)41 b(R.:)60 b(Elliptic)41 b(curv)n(es)c(o)n(v)n(er)g (\034nite)j(\034elds)f(and)g(the)g(computation)g(of)g(square)f(ro)r (ots)f(mo)r(d)j Fy(p)p FA(.)203 5090 y Fo(Mathematics)31 b(of)g(Computation)p FA(,)d(44:483\025494,)c(1985.)-189 5261 y([Sc)n(h86])135 b(Sc)n(hro)r(eder,)24 b(M.:)36 b Fo(Numb)l(er)27 b(The)l(ory)h(in)g(Scienc)l(e)g(and)f(Communic)l (ation)p FA(,)g(Springer-V)-7 b(erlag,)23 b(2nd)i(edition,)203 5360 y(1986.)-189 5531 y([Sc)n(h96])135 b(Sc)n(hneier,)30 b(B.:)41 b Fo(Applie)l(d)34 b(Crypto)l(gr)l(aphy:)45 b(pr)l(oto)l(c)l(ols,)34 b(algorithms)f(and)g(sour)l(c)l(e)e(c)l(o)l (de)i(in)f(C)p FA(,)e(John)g(Wiley)203 5630 y(&)d(Sons,)g(second)g (edition,)h(1996.)-189 5800 y([Sha82])128 b(Shamir,)31 b(A.:)43 b(A)30 b(p)r(olynomial)g(time)h(algorithm)e(for)h(breaking)f (the)h(basic)g(Merkle-Hellman)g(cryptosys-)203 5900 y(tem,)e Fo(Pr)l(o)l(c)l(e)l(e)l(dings)i(23r)l(d)h(F)n(OCS)f(Symp)l(osium)p FA(,)e(pp.)g(145\025152,)c(1982.)p eop %%Page: 39 38 39 38 bop 189 100 a Ft(\(28.)27 b(septem)n(bra)g(2000,)e (Neo\034ci\341lna)i(predb)r(e\272n\341)g(v)n(erzia\))1824 b FA(39)189 333 y([Wie90])117 b(Wiener,)23 b(M.:)34 b(Cryptanalysis)20 b(of)i(short)f(RSA)h(secret)f(exp)r(onen)n(ts.)g Fo(IEEE)k(T)-6 b(r)l(ansactions)25 b(on)g(Information)581 432 y(The)l(ory)p FA(,)k(36\(3\):553\025558,)23 b(1990.)189 598 y([Wil80])131 b(Williams,)27 b(H.:)37 b(A)26 b(mo)r(di\034cation)h(of)f(the)h(RSA)g (public-k)n(ey)f(pro)r(cedure.)g Fo(IEEE)j(T)-6 b(r)l(ansactions)30 b(on)e(Infor-)581 698 y(mation)i(The)l(ory)h(IT-26)p FA(,)e(\(6\):726\025729,)24 b(1980.)189 864 y([Wil85])131 b(Williams,)28 b(H.:)38 b(An)28 b Fy(M)1322 834 y Fw(3)1390 864 y FA(public-k)n(ey)f(encryption)g(sc)n(heme,)g Fo(A)l(dvanc)l(es)j (in)g(Cryptolo)l(gy)i(\025)f(CR)-6 b(YPTO'85)p FA(,)581 964 y(LNCS)28 b(218,)e(pp.)i(358\025368,)c(Springer-V)-7 b(erlag,)25 b(1985.)189 1130 y([W)-7 b(un88])92 b(W)-7 b(underlic)n(h,)43 b(M.:)61 b(Computational)38 b(metho)r(ds)i(for)f (factoring)g(large)f(in)n(tegers.)g Fo(A)n(b)l(acus)p FA(,)k(5\(2\):19\02533,)581 1229 y(1988.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF