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A pr\303\255mek el oszl\303\241sa, szit\303\241l\303\241s" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 205 63 "2. Egyszer\305\261 faktoriz\303\241l\303\241si m\303\263 dszerek" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart; with(numtheory);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7QI&GIgcdG6\"I)bigomegaGF$I&cfracGF$I)cfracpolGF$I+cyclot omicGF$I)divisorsGF$I)factorEQGF$I*factorsetGF$I'fermatGF$I)imagunitGF $I&indexGF$I/integral_basisGF$I)invcfracGF$I'invphiGF$I*issqrfreeGF$I' jacobiGF$I*kroneckerGF$I'lambdaGF$I)legendreGF$I)mcombineGF$I)mersenne GF$I(migcdexGF$I*minkowskiGF$I(mipolysGF$I%mlogGF$I'mobiusGF$I&mrootGF $I&msqrtGF$I)nearestpGF$I*nthconverGF$I)nthdenomGF$I)nthnumerGF$I'nthp owGF$I&orderG%*protectedGI)pdexpandGF$I$phiGF$I#piGF$I*pprimrootGF$I)p rimrootGF$I(quadresGF$I+rootsunityGF$I*safeprimeGF$I&sigmaGF$I*sq2fact orGF$I(sum2sqrGF$I$tauGF$I%thueGF$" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 7 "2.1. Pr" }{TEXT 206 8 "\303\263" }{TEXT 206 6 "baoszt" }{TEXT 206 9 "\303\241s" }{TEXT 206 1 "." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 34 "# This is a simple factorization\n" }{MPLTEXT 1 0 35 "# procedure using tria l division.\n" }{MPLTEXT 1 0 37 "# The result is a sequence of pairs\n " }{MPLTEXT 1 0 37 "# [p,e] where the p's are the prime\n" }{MPLTEXT 1 0 42 "# factors and the e's are the exponents.\n" }{MPLTEXT 1 0 47 " # The factors are anyway in increasing order.\n" }{MPLTEXT 1 0 41 "# O nly primes <= P are tried, hence the\n" }{MPLTEXT 1 0 38 "# last \"fac tor\" may composite, if \n" }{MPLTEXT 1 0 27 "# it is greater then P^2 ;\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 55 "tria ldiv:=proc(n::posint,P::posint) local L,p,i,d,nn;\n" }{MPLTEXT 1 0 15 "L:=[]; nn:=n;\n" }{MPLTEXT 1 0 32 "if type(nn,even) and 2<=P then\n" }{MPLTEXT 1 0 53 " for i from 0 while type(nn,even) do nn:=nn/2; od; \n" }{MPLTEXT 1 0 15 " L:=[[2,i]];\n" }{MPLTEXT 1 0 5 "fi;\n" } {MPLTEXT 1 0 29 "if nn mod 3=0 and 3<=P then\n" }{MPLTEXT 1 0 50 " fo r i from 0 while nn mod 3=0 do nn:=nn/3; od;\n" }{MPLTEXT 1 0 21 " L: =[op(L),[3,i]];\n" }{MPLTEXT 1 0 5 "fi;\n" }{MPLTEXT 1 0 13 "d:=2; p:= 5;\n" }{MPLTEXT 1 0 27 "while p<=P and nn>=p^2 do\n" }{MPLTEXT 1 0 22 " if nn mod p=0 then\n" }{MPLTEXT 1 0 52 " for i from 0 while nn m od p=0 do nn:=nn/p; od;\n" }{MPLTEXT 1 0 23 " L:=[op(L),[p,i]];\n" }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 19 " p:=p+d; d:=6-d;\n" } {MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 1 0 36 "if nn>1 then L:=[op(L),[nn,1] ] fi;\n" }{MPLTEXT 1 0 4 "L;\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"nG6\"I'posintG%*protectedG'I\"PGF&F'6'I\"LGF& I\"pGF&I\"iGF&I\"dGF&I#nnGF&F&F&C+>F,7\">F0F%@$3-I%typeGF(6$F0I%evenGF (1\"\"#F*C$?(F.\"\"!\"\"\"F&F7>F0,$*&#F@FF,7#7$FF0,$*&#F@FNF@F0F@F@>F,7$-I#opGF(6#F ,7$FNF.>F/F<>F-\"\"&?(F&F@F@F&31F-F*1*$)F-FF0*&F0F@F-!\"\">F,7$FX7$F-F.>F-,&F-F@F/F@>F/,&\"\"'F@F/Fho@$ 2F@F0>F,7$FX7$F0F@F,F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "trialdiv(2^32+1,1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7$\"$T' \"\"\"7$\"( " 0 "" {MPLTEXT 1 0 3 "#\n" } {MPLTEXT 1 0 38 "# This is a simple primality testing\n" }{MPLTEXT 1 0 35 "# procedure using trial division.\n" }{MPLTEXT 1 0 49 "# The res ult is true if n is prime, else false.\n" }{MPLTEXT 1 0 3 "#\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 47 "trialprime:=proc(n::posint) loca l L,p,i,d,nn;\n" }{MPLTEXT 1 0 31 "if n=1 then RETURN(false) fi;\n" } {MPLTEXT 1 0 44 "if n=2 or n=3 or n=5 then RETURN(true) fi;\n" } {MPLTEXT 1 0 40 "if type(n,even) then RETURN(false) fi;\n" }{MPLTEXT 1 0 37 "if n mod 3=0 then RETURN(false) fi;\n" }{MPLTEXT 1 0 13 "d:=2; p:=5;\n" }{MPLTEXT 1 0 17 "while n>=p^2 do\n" }{MPLTEXT 1 0 39 " if \+ n mod p=0 then RETURN(false) fi;\n" }{MPLTEXT 1 0 19 " p:=p+d; d:=6-d ;\n" }{MPLTEXT 1 0 14 "od; true; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"nG6\"I'posintG%*protectedG6'I\"LGF&I\"pGF&I\"iGF&I\"dGF&I#nnG F&F&F&C*@$/F%\"\"\"-I'RETURNGF(6#I&falseGF(@$55/F%\"\"#/F%\"\"$/F%\"\" &-F46#I%trueGF(@$-I%typeGF(6$F%I%evenGF(F3@$/-I$modGF&6$F%F=\"\"!F3>F- F;>F+F??(F&F2F2F&1*$)F+F;F2F%C%@$/-FK6$F%F+FMF3>F+,&F+F2F-F2>F-,&\"\"' F2F-!\"\"FBF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "trialp rime(2^32+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 13 "2.2. Feladat." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 13 "2.3. Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(verbose proc=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "print(ifactor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"6%I$solGF%I\"rGF%I#t1GF%6%I)rememberGF%I'systemG%*prot ectedGIaoCopyright~(c)~1991~by~the~University~of~Waterloo.~All~rights~ reserved.GF%E\\s*\"-eLwt!z**(-I!GF%6#\"\"#\"\"\"-F36#\"'()H%)F6-F36#\" '<2eF6\"-SEmBx)**4)F2\"\"%F6-F36#\"\"$F6-F36#\"\"&F6-F36#\"#6F6-F36#\" #8F6-F36#\"#F6-F36#\"$n\"F6\"._7/-N?\"*2)F2F 5F6)FAF@F6)FgnF5F6-F36#\"#HF6-F36#\"#JF6-F36#\"#PF6-F36#\"#VF6-F36#\"# `F6\"2!Gyl;-]%*=*8)F2FCF6FenF6FDF6FJF6FPF6-F36#\"#fF6-F36#\"#hF6-F36# \"#nF6-F36#\"#rF6-F36#\"#tF6-F36#\"#zF6\"-g$H*4x)**4F?F6FAF6FDF6FgnF6F GF6FMF6-F36#\"#ZF6-F36#\"$^\"F6-F36#\"$V%F6\"-3!oU;-**&F`qF6-F36#\"-,N `qF6F6\"-++;uw)**0FboF6FAF6)FDF@F6FGF6FMF6FaqF6-F36#\"$t$F6C*@&2%&narg sGF6YQ2argument~requiredF%32F6F\\t4-I%typeGF-6$&%%argsGF4.I%nameGF-YQ? second~argument~must~be~a~nameF%@--Fct6$F$.I(integerGF-@'2\"\"!F$C$>F' F6>F(F$2F$FbuC$>F'!\"\">F(,$F$FiuOFbu-Fct6$F$.I)fractionGF-O*&-%)procn ameG6$-I#opGF-6$F6F$&Fft6#;F5FiuF6-Fdv6$-Fgv6$F5F$FivFiu-Fct6$F$<&.I\" *GF-.I%listGF-.I$setGF-.I)relationGF-O-I$mapGF-6%FdvF$Fiv3-Fct6$F$I\"^ GF--Fct6$F^wF^uO)FcvF^w-Fct6$F$-F36#F^uOFcvYQ2invalid~argumentsF%@$-I) assignedGF-6#&I7ifactor/from_signatureGF%6#F(OFby>F)-I%igcdGF-6$F(\"'? 2s?(F%F6F6F%0F)F6C%>F'*&F'F6-I1ifactor/ifact235GF%6#F)F6>F(-I%iquoGF-6 $F(F)>F)-Fhy6$F)F(>F)-Fhy6$F(\"igmL#4(*G]KKT_/&o:-6-\\ULnO63K//a2+Jg1( yv%))=%>78\">_=)RBQ#>VWyw'G*R%G\\?$f7!e@ZzI(*yye56c\"y)**o*)QLhqw-?=(z uT%e<6)fZ@Q*[2)3#>wd'*\\:G(*)z=]^VV.h9MkueL(z![$)**Hq1x_w')Rd#QeVva%>V J\"))ez>!*He%\\xPDRmf-60&3OlWa]rHz(zX$[VqF@7Hs,A!zAnr6'>C%\\$fw>r,cG\" 31+7sAdQ@A(zC>XzNl(*4)\\0aFc@**)y;u2(R&)H\\..*H7\"yV-VoEZ>IAzQ%H3>#=%y %))*Q?>l?')f!pq1Vkpj(4R=cVxQ,'o1aK,npXBR,W[tUA)H'4-FhUPX3y&*3(>b')GHWs iV[_.pt5=bqyi!=TRbUc#=@W,P9^Ha;\"?(F%F6F6F%F\\zC%>F'*&F'F6-I1ifactor/i fact1stGF%FbzF6FczFgz@%0F(F6C$@%/F\\tF6@%3/I2_Env_ifactor_easyGF%I%tru eGF-2\"#I-I'lengthGF-FdyC$>I/ifactor/bottomGF%I-ifactor/easyGF%>F)-I1i factor/ifact0thGF%FdyC$>Fd\\lI.ifactor/mixedGF%Ff\\lC$>Fd\\l-I$catGF-6 $I)ifactor/GF%Fet>F)-Fh\\l6$F(&Fft6#;FCF\\t@%0F)I%FAILGF-*&F'F6F)F6Fj] lF'F%6#Fd\\lF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "print(`if actor/ifact235`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"6#I\"qG F%6%I)rememberGF%I'systemG%*protectedGIaoCopyright~(c)~1991~by~the~Uni versity~of~Waterloo.~All~rights~reserved.GF%E\\sB\"\"\"F.\"\"#-I!GF%6# F/\"\"$-F16#F3\"$_#*()F0F/F.)F4F/F.-F16#\"\"(F.\"\"&-F16#F=\"%SE**)F0 \"\"%F.F4F.F>F.-F16#\"#6F.FC*$F8F.\"#5*&F0F.F>F.\"\")*$)F0F3F.\"\"**$F 9F.\"#:*&F4F.F>F.\"#!**(F0F.F9F.F>F.\"%?z**FBF.F9F.F>F.FDF.\"$S#*(FBF. F4F.F>F.\"'?2s*.FBF.F9F.F>F.F:F.FDF.-F16#\"#8F.\"#?*&F8F.F>F.\"&?V$*,F BF.F4F.F>F.FDF.FYF.\"#=*&F0F.F9F.\"#I*(F0F.F4F.F>F.\"$!=*(F8F.F9F.F>F. \"%!o%**FLF.F9F.F>F.FYF.\"%S]**FBF.F9F.F>F.F:F.\"#S*&FLF.F>F.\"$?(*(FB F.F9F.F>F.\"#X*&F9F.F>F.\"$?\"*(FLF.F4F.F>F.\"#O*&F8F.F9F.\"#j*&F9F.F: F.\"$g$*(FLF.F9F.F>F.\"#g*(F8F.F4F.F>F.\"&Sa&*,FBF.F9F.F>F.F:F.FDF.\"% !o\"**FBF.F4F.F>F.F:F.\"&![=*,FBF.F4F.F>F.F:F.FDF.@//F$F.F./-I%iremGF+ 6%F$Fen.F'\"\"!*&-I1ifactor/ifact235GI(_syslibGF%F&F.FYF./-F^q6%F$FFF` qFaq*&FcqF.FDF./-F^q6%F$FF%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "print(`ifactor/ifact0th`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"6$I$solGF%I\"pGF%6#IaoCopyright~(c)~1991~by~t he~University~of~Waterloo.~All~rights~reserved.GF%F%C,@$2F$\"(\"o?HO-I !GF%F#@$-I(isprimeGF%F#F/>F'-I.ifactor/powerGF%6$F$.F(@$0F'I%FAILG%*pr otectedGO)-I1ifactor/ifact0thGI(_syslibGF%6#F'F(>F'-I/ifactor/pollp1GF %6$F$\"&.I\"@$/F'I*_tryagainGF%>F'-FF6$F$\"&/I\"@$FJ>F'F<@$/F'F<>F'-I1 ifactor/pp100000GF%F#@$FSC$>F'-I/ifactor/bottomGF%6#%%argsG@$4-I%typeG F=6$F'.I(integerGF=OF'*&-FA6$*&F$\"\"\"F'!\"\"&Fgn6#;\"\"#%&nargsGFdo- FA6$F'FfoFdoF%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "print (`ifactor/ifact1st`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"6'I \"iGF%I#igGF%I\"rGF%I$solGF%I#t1GF%6%I)rememberGF%I'systemG%*protected GIaoCopyright~(c)~1991~by~the~University~of~Waterloo.~All~rights~reser ved.GF%E\\s)\"'\"4u#*(-I!GF%6#\"#<\"\"\"-F56#\"#BF8-F56#\"$,(F8\")xj!e (*,-F56#\"#HF8-F56#\"#JF8-F56#\"#PF8-F56#\"#VF8-F56#\"#`F8\")F)z(G**F4 F8-F56#\"#TF8-F56#\"$d\"F8-F56#\"$j#F8F7F4\"&TR&*(F4F8-F56#\"#>F8-F56# \"$n\"F8\")2vW`**F4F8-F56#\"#ZF8-F56#\"$^\"F8-F56#\"$V%F8\".@gQx![S*0F RF8-F56#\"#fF8-F56#\"#hF8-F56#\"#nF8-F56#\"#rF8-F56#\"#tF8-F56#\"#zF8 \"'>TP*(F4F8FjoF8-F56#\"$t$F8C&>F*F8>F)F$?(F'F8F8\"\"(0F)F8C$>F(-I%igc dGF/6$F)&7)\"$B$\"'\\]w\".*=Sw'*e5\"=f;keB!QSk)[oMo\"*\"jn*)Q)fV\"f7sR %>OC%Qc5Ha=4+Wo$Q&y@J463'eRLU2C3>ZQ7W@n9:A\"))GK!>*))4Fz\"jfl\\m=,#HR1()** 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"f*6$I\"nG6\"I%seedGF%6)I'primesGF%I/factorizationsGF% I\"wGF%I\"iGF%I#t1GF%I%pairGF%I\"dGF%6#IaoCopyright~(c)~1990~by~the~Un iversity~of~Waterloo.~All~rights~reserved.GF%F%C'>F(7V\"$7&\"+D1S,F\", (*3Dp`\"\"/6y%eA>?#\".x8FEAJ$\"-8lv!Qt&\".P>V:f*)*\".P\\9K5!**\"/807M$ en#\"0fit_cM@#\"0rjGNj*=N\"0P2lTF`-'\"0pNY8e!fq\"1rJad-'o9\"\"1>w;QM#z >\"\"1`3NAG=,T\"2H!*[p@j$4=\"2*>\\BLTty:\"28T$o`)pT0$\"2xFV5b'GFK\"2*G [fQBK>!*\"2$f>Bjz#f[*\"3H%[H%p&y&4A\"3$*\\Gf*znU<#\"3@2>\"=A&=QN\"3*[% )GYX**R>%\"3dMJn!*=Wsd\"3B7w-)[kcw&\"38ej;$R/,N'\"4H1$Rd1'3K;\"\"4`Jrg KaFXF\"\"3xY'GW$[,U'*\"4\")>%3ca!\\&*=\"\"4LNqO#H5mI<\"4hP!3'o*f_CB\"4 Zl(*=\\2z$\\<\"4hZZ'=)4=$zB\"4j=-Ni+5j!R\"4LkaoH$p.+W\"4T=HP@(yN'[$\"4 28JID&3b^S\"48Hb8>?CH1(\"4d'Gn*yQ/i%**\"4@S#3!G$*>/!)*\"5J:KQJX;k,9\"5 t-*[jcqa^%=\"5HN!oH\">R3x?\"5*)e[!)Q(yBiK#\"5p(G%y_YAHGG\"5V9q;)H+UzP# \"5.+A/+OFk+H\"8\\#HkoRH87%HK'>F)7V7#7$\"\"#\"\"*7&7$\"\"$\"\"'7$\"\"& \"\"%7$\"\"(Ffo7$\"#6Ffo7'7$F`p\"\"\"7$FbpFep7$\"#8Ffo7$\"#JFfo7$\"%H7 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\"$t(Fep7$\"$@)Fep7(7$\"$*\\Fep7$\"$.&Fep7$\"$4&Fep7$\"$@&Fep7$\"$,(Fe p7$\"%L>Fep7(7$\"$B&Fep7$\"$T&Fep7$\"$Z&Fep7$\"$d&Fep7$\"$\"))Fep7$\"% \\7Fep7(7$\"$(eFep7$\"$$fFep7$\"$,'Fep7$\"$2'Fep7$\"$H*Fep7$\"%t=Fep7( 7$\"$8'Fep7$\"$>'Fep7$\"$J'Fep7$\"$T'Fep7$\"$r*Fep7$\"%f9Fep7(7$\"$V'F ep7$\"$Z'Fep7$\"$`'Fep7$\"$f'Fep7$\"%85Fep7$\"%^>Fep7(7$\"$h'Fep7$\"$t 'Fep7$\"$$oFep7$\"$\"pFep7$\"%\"4\"Fep7$\"%J=Fep7(7$\"$4(Fep7$\"$>(Fep 7$\"$F(Fep7$\"$L(Fep7$\"%j5Fep7$\"%**>Fep7(7$\"$R(Fep7$\"$V(Fep7$\"$^( Fep7$\"$d(Fep7$\"$$)*Fep7$\"%z=Fep7(7$\"$h(Fep7$\"$p(Fep7$\"$(yFep7$\" $(zFep7$\"$n*Fep7$\"%*y\"Fep7(7$\"$4)Fep7$\"$6)Fep7$\"$B)Fep7$\"$F)Fep 7$\"%28Fep7$\"%$*>Fep7(7$\"$H)Fep7$\"$R)Fep7$\"$`)Fep7$\"$d)Fep7$\"%F8 Fep7$\"%*)=Fep7(7$\"$f)Fep7$\"$j)Fep7$\"$x)Fep7$\"$$))Fep7$\"%\"H\"Fep 7$\"%,8Fep7(7$\"$())Fep7$\"$2*Fep7$\"$>*Fep7$\"$P*Fep7$\"%$4\"Fep7$\"% r:Fep7(7$\"$T*Fep7$\"$Z*Fep7$\"$`*Fep7$\"$x*Fep7$\"%@8Fep7$\"%z:Fep7(7 $\"$\"**Fep7$\"$(**Fep7$\"%45Fep7$\"%>5Fep7$\"%B7Fep7$\"%r=Fep7(7$\"%@ 5Fep7$\"%J5Fep7$\"%L5Fep7$\"%\\5Fep7$\"%<6Fep7$\"%t8Fep7(7$\"%R5Fep7$ \"%^5Fep7$\"%h5Fep7$\"%p5Fep7$\"%P7Fep7$\"%`:Fep7(7$\"%(3\"Fep7$\"%(4 \"Fep7$\"%.6Fep7$\"%B6Fep7$\"%49Fep7$\"%x=Fep7(7$\"%46Fep7$\"%H6Fep7$ \"%^6Fep7$\"%j6Fep7$\"%\"Q\"Fep7$\"%,>Fep7(7$\"%`6Fep7$\"%r6Fep7$\"%\" =\"Fep7$\"%(=\"Fep7$\"%*G\"Fep7$\"%H9Fep7(7$\"%$>\"Fep7$\"%,7Fep7$\"%8 7Fep7$\"%<7Fep7$\"%$G\"Fep7$\"%$\\\"Fep7(7$\"%J7Fep7$\"%x7Fep7$\"%(H\" Fep7$\"%.8Fep7$\"%V:Fep7$\"%B8Fep7$\"%h8Fep7$\"%n8Fep7$\"% L9Fep7$\"%>;Fep7$\"%ZFep7$\"%Z=Fep7(7$\"%*[\"Fep7$\"%J:Fep7$\"%\\:Fep7$\"%n;Fep7$\"%z>Fep 7$\"%$y\"Fep7(7$\"%f:Fep7$\"%n:Fep7$\"%$e\"Fep7$\"%$p\"Fep7$\"%B=Fep7$ \"%\\>Fep7(7$\"%2;Fep7$\"%4;Fep7$\"%8;Fep7$\"%(*>Fep7$\"%()>Fep7$\"%4< Fep7(7$\"%@;Fep7$\"%F;Fep7$\"%P;Fep7$\"%@Fep7)7$\"%* p\"Fep7$\"%2>Fep7$\"%LFep7$\"%6=Fep7$\"%TF*F &?(F+FepFep-I%nopsG%*protectedG6#F(I%trueGF`imC$>F,-I%modpGF`im6$-I&po werGF%6$F*&F(6#F+F$@%0-I%igcdGF`im6$,&F,FepFep!\"\"F$FepC&@$0F,FepOF_j m>F.&F)F\\jm?&F-F.Fbim?(F%FepFep&F-6#FfoFbimC$>F*-Ffim6$-Fiim6$F*&F-6# FepF$@&/F*Fep@$2FepF+O-%)procnameG6$F$-Ffim6$-Fiim6$F&Fd[nF$0-F`jm6$,& F*FepFepFcjmF$FepOFc\\nOI*_tryagainGF%>F*F,I%FAILGF`imF%F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "print(`ifactor/pp100000`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"6$I\"iGF%I\"tGF%6#IaoCopyr ight~(c)~1990~by~the~University~of~Waterloo.~All~rights~reserved.GF%F% C$?(F'\"%+S\"%+g\"&+S*I%trueG%*protectedGC$>F(-I%igcdGF16$(I&_prprGF%F 'F$@$0F(\"\"\"@%0F(F$-I'RETURNGF16#F(-F?6#-I2ifactor/wheelfactGI(_sysl ibGF%6$F$,&F'F;F;F;I%FAILGF1F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "print(`ifactor/easy`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%6#IaoCopyright~(c)~1990~by~the~University~of~Waterlo o.~All~rights~reserved.GF%F%-&I0tools/genglobalGF%6#\"\"\"6#-I$catG%*p rotectedG6&I!GF%.I#_cGF%-I'lengthGF0F#.I\"_GF%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "print(`ifactor/ifact235`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"6#I\"qGF%6%I)rememberGF%I'systemG%*prot ectedGIaoCopyright~(c)~1991~by~the~University~of~Waterloo.~All~rights~ reserved.GF%E\\sB\"\"\"F.\"\"#-I!GF%6#F/\"\"$-F16#F3\"$_#*()F0F/F.)F4F /F.-F16#\"\"(F.\"\"&-F16#F=\"%SE**)F0\"\"%F.F4F.F>F.-F16#\"#6F.FC*$F8F .\"#5*&F0F.F>F.\"\")*$)F0F3F.\"\"**$F9F.\"#:*&F4F.F>F.\"#!**(F0F.F9F.F >F.\"%?z**FBF.F9F.F>F.FDF.\"$S#*(FBF.F4F.F>F.\"'?2s*.FBF.F9F.F>F.F:F.F DF.-F16#\"#8F.\"#?*&F8F.F>F.\"&?V$*,FBF.F4F.F>F.FDF.FYF.\"#=*&F0F.F9F. \"#I*(F0F.F4F.F>F.\"$!=*(F8F.F9F.F>F.\"%!o%**FLF.F9F.F>F.FYF.\"%S]**FB F.F9F.F>F.F:F.\"#S*&FLF.F>F.\"$?(*(FBF.F9F.F>F.\"#X*&F9F.F>F.\"$?\"*(F LF.F4F.F>F.\"#O*&F8F.F9F.\"#j*&F9F.F:F.\"$g$*(FLF.F9F.F>F.\"#g*(F8F.F4 F.F>F.\"&Sa&*,FBF.F9F.F>F.F:F.FDF.\"%!o\"**FBF.F4F.F>F.F:F.\"&![=*,FBF .F4F.F>F.F:F.FDF.@//F$F.F./-I%iremGF+6%F$Fen.F'\"\"!*&-I1ifactor/ifact 235GI(_syslibGF%F&F.FYF./-F^q6%F$FFF`qFaq*&FcqF.FDF./-F^q6%F$FF% F%F%" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 9 "2.4. A pr" }{TEXT 206 8 "\303\255" }{TEXT 206 5 "moszt" }{TEXT 206 8 "\303\263" }{TEXT 206 8 "k eloszl" }{TEXT 206 8 "\303\241" }{TEXT 206 1 "s" }{TEXT 206 8 "\3 03\241" }{TEXT 206 1 "r" }{TEXT 206 8 "\303\263" }{TEXT 206 2 "l." }}} {SECT 1 {PARA 4 "" 0 "" {TEXT 206 9 "2.5. A pr" }{TEXT 206 8 "\303\255 " }{TEXT 206 5 "moszt" }{TEXT 206 8 "\303\263" }{TEXT 206 4 "k sz" } {TEXT 206 8 "\303\241" }{TEXT 206 1 "m" }{TEXT 206 8 "\303\241" }{TEXT 206 7 "nak hat" }{TEXT 206 8 "\303\241" }{TEXT 206 7 "reloszl" }{TEXT 206 8 "\303\241" }{TEXT 206 3 "sa." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 13 "2.6. Fermat m" }{TEXT 206 8 "\303\263" }{TEXT 206 7 "dszere." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 48 "# This procedure prepare the sieve table S for\n" }{MPLTEXT 1 0 56 "# Fermat's factorization procedure. Parame ter n is the\n" }{MPLTEXT 1 0 52 "# integer to factor and m is the vec tor of moduli.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 53 "preparefermatsieve:=proc(n,S,m) local i,j,k,x2,x2n;\n" } {MPLTEXT 1 0 24 "x2:=table; x2n:=table;\n" }{MPLTEXT 1 0 21 "for i to \+ nops(m) do\n" }{MPLTEXT 1 0 44 " for j from 0 to m[i]-1 do S[i,j]:=0; od;\n" }{MPLTEXT 1 0 29 " for j from 0 to m[i]-1 do\n" }{MPLTEXT 1 0 26 " x2[j]:=j^2 mod m[i];\n" }{MPLTEXT 1 0 29 " x2n[j]:=j^2-n \+ mod m[i];\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 29 " for j from \+ 0 to m[i]-1 do\n" }{MPLTEXT 1 0 31 " for k from 0 to m[i]-1 do\n" } {MPLTEXT 1 0 42 " if x2n[j]=x2[k] then S[i,j]:=1 fi;\n" }{MPLTEXT 1 0 9 " od;\n" }{MPLTEXT 1 0 15 " od; \n" }{MPLTEXT 1 0 8 "od; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"nG6\"I\"SGF%I\"mGF%6 'I\"iGF%I\"jGF%I\"kGF%I#x2GF%I$x2nGF%F%F%C%>F,I&tableG%*protectedG>F-F 0?(F)\"\"\"F4-I%nopsGF16#F'I%trueGF1C%?(F*\"\"!F4,&&F'6#F)F4F4!\"\"F8> &F&6$F)F*F;?(F*F;F4F&F,6#F*-I$modGF%6$*$)F*\"\"#F4F=>&F-FG-FI6$, &FKF4F$F?F=?(F*F;F4FFAF4F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 40 "# This proced ure do factorization with\n" }{MPLTEXT 1 0 35 "# Fermat's method. Para meter n is\n" }{MPLTEXT 1 0 57 "# the odd number to factor and m is th e list of moduli.\n" }{MPLTEXT 1 0 41 "# Returns with u where u is the largest\n" }{MPLTEXT 1 0 46 "# factor of n less then or equal to sqrt (n).\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 54 "f ermatfactorization:=proc(n::posint,m::list(posint))\n" }{MPLTEXT 1 0 22 "local k,x,y,i,S,r,f;\n" }{MPLTEXT 1 0 63 "if type(n,even) then err or \"first argument must be odd\" fi;\n" }{MPLTEXT 1 0 52 "S:=table(); preparefermatsieve(n,S,m); r:=nops(m);\n" }{MPLTEXT 1 0 30 "k:=array( 1..r); x:=isqrt(n);\n" }{MPLTEXT 1 0 38 "for i to r do k[i]:=-x mod m[ i]; od;\n" }{MPLTEXT 1 0 15 "while true do\n" }{MPLTEXT 1 0 12 " f:=t rue;\n" }{MPLTEXT 1 0 63 " for i to r do if S[i,k[i]]<>1 then f:=fals e; break; fi; od;\n" }{MPLTEXT 1 0 13 " if f then\n" }{MPLTEXT 1 0 22 " y:=isqrt(x^2-n);\n" }{MPLTEXT 1 0 39 " if y^2=x^2-n then RE TURN(x-y) fi;\n" }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 11 " x:=x+1; \n" }{MPLTEXT 1 0 44 " for i to r do k[i]:=k[i]-1 mod m[i]; od;\n" } {MPLTEXT 1 0 8 "od; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"nG6 \"I'posintG%*protectedG'I\"mGF&-I%listGF(6#F'6)I\"kGF&I\"xGF&I\"yGF&I \"iGF&I\"SGF&I\"rGF&I\"fGF&F&F&C*@$-I%typeGF(6$F%I%evenGF(YQ;first~arg ument~must~be~oddF&>F3-I&tableGF(F&-I3preparefermatsieveGF&6%F%F3F*>F4 -I%nopsGF(6#F*>F/-I&arrayGF(6#;\"\"\"F4>F0-I&isqrtGF(6#F%?(F2FMFMF4I%t rueGF(>&F/6#F2-I$modGF&6$,$F0!\"\"&F*FV?(F&FMFMF&FSC'>F5FS?(F2FMFMF4FS @$0&F36$F2FUFMC$>F5I&falseGF([@$F5C$>F1-FP6#,&*$)F0\"\"#FMFMF%Fen@$/*$ )F1F[pFMFho-I'RETURNGF(6#,&F0FMF1Fen>F0,&F0FMFMFM?(F2FMFMF4FS>FU-FX6$, &FUFMFMFenFfnF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "debu g(fermatfactorization);\n" }{MPLTEXT 1 0 40 "fermatfactorization(13*17 ,[3,5,7,8,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I4fermatfactorization G6\"" }}{PARA 9 "" 1 "" {TEXT 207 61 "\{--> enter fermatfactorization, args = 221, [3, 5, 7, 8, 11]" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"I&f alseG%*protectedGE\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"&E\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protected G" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 9 "" 1 "" {TEXT 207 54 "<-- exit fermatfactorization (now at top level) = 13\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "undebug(fermatfactorization);\n" }{MPLTEXT 1 0 40 "fermatfactoriza tion(11111,[3,5,7,8,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I4fermatfac torizationG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#T" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 13 "2.7. Feladat." }}}{SECT 1 {PARA 4 "" 0 " " {TEXT 206 13 "2.8. Feladat." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 31 "2.9. Pollard \317\261 m\303\263" }{TEXT 206 7 "dszere." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 7 "Az n sz" }{TEXT 201 8 "\303\241" }{TEXT 201 5 "m has" }{TEXT 201 8 "\303\255" }{TEXT 201 1 "t" }{TEXT 201 8 "\303\241" }{TEXT 201 16 "sa az x->x^2+c f" } {TEXT 201 8 "\303\274" }{TEXT 201 3 "ggv" }{TEXT 201 8 "\303\251" } {TEXT 201 11 "ny felhaszn" }{TEXT 201 8 "\303\241" }{TEXT 201 1 "l" } {TEXT 201 8 "\303\241" }{TEXT 201 1 "s" }{TEXT 201 8 "\303\241" }{TEXT 201 15 "val; g egy iter" }{TEXT 201 8 "\303\241" }{TEXT 201 2 "ci" } {TEXT 201 8 "\303\263" }{TEXT 201 7 "csoport" }}{PARA 0 "" 0 "" {TEXT 201 1 "m" }{TEXT 201 8 "\303\251" }{TEXT 201 5 "rete " }{TEXT 201 8 " \303\251" }{TEXT 201 32 "s legfeljebb maxgs csoport fog v" }{TEXT 201 8 "\303\251" }{TEXT 201 7 "grehajt" }{TEXT 201 8 "\303\263" }{TEXT 201 4 "dni." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "pollardrhosplit:=proc(n::posint,c::posint,g::posin t,maxgs::posint)\n" }{MPLTEXT 1 0 37 "local x,xx,xp,xo,xpo,i,j,k,ko,l, lo;\n" }{MPLTEXT 1 0 47 "x:=1+c mod n; xp:=1; i:=0; k:=1; l:=1; xx:=1; \n" }{MPLTEXT 1 0 35 "while igcd(xx,n)=1 and iF0-I$modGF&6$,&\"\"\"FAF*FAF%>F2FA>F5\"\"!>F7FA>F9FA>F1 FA?(F&FAFAF&3/-I%igcdGF(6$F1F%FA2F5F.C*>F3F0>F4F2>F8F7>F:F9>F6FDFG?(F& FAFAF&2F6F,C'>F1-F>6$*&F1FA,&F2FAF0!\"\"FAF%>F7,&F7FAFAFgn@$/F7FDC%>F2 F0>F7F9>F9,$*&\"\"#FAF9FAFA>F0-F>6$,&*$)F0FboFAFAF*FAF%>F6,&F6FAFAFA>F 5,&F5FAFAFA@$2FKF%OFK>F0F3>F2F4>F7F8>F9F:FT?(F&FAFAF&3/-FL6$FfnF%FAFVC &FhnFjnFcoFioFgpF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p ollardrhosplit(999863*999883,1,2^4,2^5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"'j)***" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 14 "2.10. Feladat. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 14 "2.11. Fermat t" }{TEXT 206 8 "\303\251" }{TEXT 206 5 "tele." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 13 "2.12. Euler t" }{TEXT 206 8 "\303\251" }{TEXT 206 5 "tele." }} }{SECT 0 {PARA 4 "" 0 "" {TEXT 206 7 "2.13. K" }{TEXT 206 8 "\303\255" }{TEXT 206 9 "nai marad" }{TEXT 206 8 "\303\251" }{TEXT 206 2 "kt" } {TEXT 206 8 "\303\251" }{TEXT 206 4 "tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "chrem([1,2,2],[2, 3,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#B" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 7 "2.14. T" }{TEXT 206 8 "\303\251" }{TEXT 206 4 "tel." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 16 "2.15. Gyors hatv" }{TEXT 206 8 "\303\241" }{TEXT 206 4 "nyoz" }{TEXT 206 8 "\303\241" }{TEXT 206 2 "s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 37 "# Calculation of modular power \+ of a\n" }{MPLTEXT 1 0 41 "# with the left-to-right binary method.\n" } {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 3 " \n" }{MPLTEXT 1 0 60 "left2righ t:=proc(a,e::posint,mult::procedure) local b,x,n;\n" }{MPLTEXT 1 0 29 "b:=convert(e,base,2); x:=a;\n" }{MPLTEXT 1 0 36 "for n from nops(b)-1 by -1 to 1 do\n" }{MPLTEXT 1 0 17 " x:=mult(x,x);\n" }{MPLTEXT 1 0 36 " if b[n]>0 then x:=mult(x,a); fi;\n" }{MPLTEXT 1 0 11 "od; x; end ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"aG6\"'I\"eGF%I'posintG%*prot ectedG'I%multGF%I*procedureGF)6%I\"bGF%I\"xGF%I\"nGF%F%F%C&>F.-I(conve rtGF)6%F'I%baseGF%\"\"#>F/F$?(F0,&-I%nopsGF)6#F.\"\"\"F>!\"\"F?F>I%tru eGF)C$>F/-F+6$F/F/@$2\"\"!&F.6#F0>F/-F+6$F/F$F/F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "debug(left2right); left2right(2,11, (x,y)->x*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I+left2rightG6\"" }} {PARA 9 "" 1 "" {TEXT 207 87 "\{--> enter left2right, args = 2, 11, pr oc (x, y) options operator, arrow; x*y end proc" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"\"F#\"\"!F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%C5" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%[?" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%[?" }}{PARA 9 "" 1 "" {TEXT 207 47 "<-- exit l eft2right (now at top level) = 2048\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%[?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 35 "# Calculation of a modular power \n" }{MPLTEXT 1 0 42 "# with th e left-to-right 2^m-ary method.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 54 "fastexp:=proc(a,e::posint,m::posint,mult::p rocedure)\n" }{MPLTEXT 1 0 25 "local i,j,k,P,x,b,aa,n;\n" }{MPLTEXT 1 0 66 "b:=convert(e,base,2); n:=nops(b)-1; x:=a; P:=[a]; aa:=mult(a,a); \n" }{MPLTEXT 1 0 64 "for j from 2 to 2^(m-1) do P:=[op(P),mult(P[nops (P)],aa)]; od;\n" }{MPLTEXT 1 0 15 "while true do\n" }{MPLTEXT 1 0 29 " if n=0 then return(x) fi;\n" }{MPLTEXT 1 0 50 " if b[n]=0 then x:= mult(x,x); n:=n-1; next; fi;\n" }{MPLTEXT 1 0 43 " i:=1; j:=1; k:=0; \+ x:=mult(x,x); n:=n-1;\n" }{MPLTEXT 1 0 26 " while n>0 and k+j0 do x:=mult(x,x); k:=k-1; od;\n" }{MPLTEXT 1 0 15 " n:=n-1 ;\n" }{MPLTEXT 1 0 9 " fi;\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 20 " x:=mult(x,P[i]);\n" }{MPLTEXT 1 0 42 " while k>0 do x:=mult (x,x); k:=k-1; od;\n" }{MPLTEXT 1 0 8 "od; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I\"aG6\"'I\"eGF%I'posintG%*protectedG'I\"mGF%F('I%mul tGF%I*procedureGF)6*I\"iGF%I\"jGF%I\"kGF%I\"PGF%I\"xGF%I\"bGF%I#aaGF%I \"nGF%F%F%C)>F5-I(convertGF)6%F'I%baseGF%\"\"#>F7,&-I%nopsGF)6#F5\"\" \"FD!\"\">F4F$>F37#F$>F6-F-6$F$F$?(F1F>FD)F>,&F+FDFDFEI%trueGF)>F37$-I #opGF)6#F3-F-6$&F36#-FBFTF6?(F%FDFDF%FOC,@$/F7\"\"!OF4@$/&F56#F7FhnC%> F4-F-6$F4F4>F7,&F7FDFDFE\\>F0FD>F1FD>F2FhnF_oFbo?(F%FDFDF%32FhnF72,&F2 FDF1FDF+@%F[oC$>F2,&F2FDFDFDFboC'F_p>F1F\\p>F0*&F0FD)F>F2FD?(F%FDFDF%2 FhnF2C$F_o>F2,&F2FDFDFEFbo>F4-F-6$F4&F36#F0FfpF%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "debug(fastexp); fastexp(2,11,1,(x,y)->x*y );" }}{PARA 11 "" 1 "" {XPPMATH 20 "I(fastexpG6\"" }}{PARA 9 "" 1 "" {TEXT 207 87 "\{--> enter fastexp, args = 2, 11, 1, proc (x, y) option s operator, arrow; x*y end proc" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\" \"\"F#\"\"!F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#;" } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%C5" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"%[?" }}{PARA 9 "" 1 "" {TEXT 207 44 "< -- exit fastexp (now at top level) = 2048\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%[?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "debug (fastexp); fastexp(2,11,2,(x,y)->x*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I(fastexpG6\"" }}{PARA 9 "" 1 "" {TEXT 207 87 "\{--> enter fastexp, \+ args = 2, 11, 2, proc (x, y) options operator, arrow; x*y end proc" }} {PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"\"F#\"\"!F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" } }{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$c#" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%[?" }}{PARA 9 "" 1 "" {TEXT 207 44 "<-- exit fastexp (now at top level) = 2048\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%[?" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 14 "2.16. Feladat." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 19 "2.17. Pollard p-1 \+ m" }{TEXT 206 8 "\303\263" }{TEXT 206 7 "dszere." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" } {MPLTEXT 1 0 46 "# This procedure is Pollard's p-1 method for\n" } {MPLTEXT 1 0 47 "# factorization. The base is a, and powers of\n" } {MPLTEXT 1 0 46 "# primes up to P are considered so that they\n" } {MPLTEXT 1 0 34 "# are not less then the bound B.\n" }{MPLTEXT 1 0 48 "# The result is the power x of a mod n, where\n" }{MPLTEXT 1 0 61 "# n isthe number to factorize, so the factor is gcd(x-1,n).\n" } {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 45 "pollardpsp lit:=proc(n,a,B,P) local e,d,p,x;\n" }{MPLTEXT 1 0 13 "x:=a mod n;\n" }{MPLTEXT 1 0 44 "if igcd(x-1,n)>1 or P<2 then return(x) fi;\n" } {MPLTEXT 1 0 27 "if P<2 then return(x) fi;\n" }{MPLTEXT 1 0 33 "e:=1; \+ while 2^e1 or P=2 then return(x) fi;\n" } {MPLTEXT 1 0 37 "while 3^e>3*B and e>1 do e:=e-1 od;\n" }{MPLTEXT 1 0 20 "x:=x&^(3^e) mod n;\n" }{MPLTEXT 1 0 13 "d:=2; p:=5;\n" }{MPLTEXT 1 0 15 "while true do\n" }{MPLTEXT 1 0 46 " if igcd(x-1,n)>1 or P

p*B and e>1 do e:=e -1 od;\n" }{MPLTEXT 1 0 22 " x:=x&^(p^e) mod n;\n" }{MPLTEXT 1 0 19 " p:=p+d; d:=6-d;\n" }{MPLTEXT 1 0 11 "od; x; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I\"nG6\"I\"aGF%I\"BGF%I\"PGF%6&I\"eGF%I\"dGF%I\"pGF% I\"xGF%F%F%C/>F--I$modGF%6$F&F$@$52\"\"\"-I%igcdG%*protectedG6$,&F-F6F 6!\"\"F$2F(\"\"#OF-@$F=F?>F*F6?(F%F6F6F%2)F>F*F'>F*,&F*F6F6F6>F--F16$- I#&^GF%6$F-FDF$@$5F5/F(F>F??(F%F6F6F%32,$*&\"\"$F6F'F6F6)FUF*2F6F*>F*, &F*F6F6F<>F--F16$-FK6$F-FVF$>F+F>>F,\"\"&?(F%F6F6F%I%trueGF9C'@$5F52F( F,F??(F%F6F6F%32*&F,F6F'F6)F,F*FWFX>F--F16$-FK6$F-FfoF$>F,,&F,F6F+F6>F +,&\"\"'F6F+F " 0 "" {MPLTEXT 1 0 31 "p ollardpsplit(25852,2,100,100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&CL# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "igcd(%-1,25852);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"$\"G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "pollardpsplit(999863*999917*999961,23,2000,1000);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"2:jYQ.\"HD;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "igcd(%-1,999863*999917*999961);" }}{PARA 11 "" 1 " " {XPPMATH 20 "\"'<****" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 14 "2.1 8. Feladat." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 19 "2.19. Pollard p- 1 m" }{TEXT 206 8 "\303\263" }{TEXT 206 9 "dszere, m" }{TEXT 206 8 "\3 03\241" }{TEXT 206 7 "sodik l" }{TEXT 206 8 "\303\251" }{TEXT 206 3 "p cs" }{TEXT 206 9 "\305\221." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 65 "# This procedure is the second step of Pollard's p-1 method for\n" }{MPLTEXT 1 0 56 "# factorization. The base is a, and primes from list P\n" } {MPLTEXT 1 0 64 "# are considered. The result is the power x of a mod \+ n, where\n" }{MPLTEXT 1 0 62 "# n is the number to factorize, so the \+ factor is gcd(x-1,n).\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 73 "pollardp2split:=proc(n::posint,a::posint,N::posint,m: :posint,M::posint)\n" }{MPLTEXT 1 0 26 "local x,i,j,E,aa,p,pp,d;\n" } {MPLTEXT 1 0 42 "E:=Array(1..N); aa:=a*a mod n; E[1]:=aa;\n" }{MPLTEXT 1 0 42 "for j from 2 to N do E[j]:=E[j-1]*aa od;\n" }{MPLTEXT 1 0 32 "p:=ithprime(m); x:=a&^p mod n;\n" }{MPLTEXT 1 0 43 "for i from m+1 to M while gcd(x-1,n)=1 do\n" }{MPLTEXT 1 0 37 " pp:=nextprime(p); d:=p p-p; p:=pp;\n" }{MPLTEXT 1 0 37 " if d<=2*N then x:=x*E[d/2] mod n;\n " }{MPLTEXT 1 0 31 " else x:=x*(a&^d) mod n; fi;\n" }{MPLTEXT 1 0 11 "od; x; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6''I\"nG6\"I'posintG%* protectedG'I\"aGF&F''I\"NGF&F''I\"mGF&F''I\"MGF&F'6*I\"xGF&I\"iGF&I\"j GF&I\"EGF&I#aaGF&I\"pGF&I#ppGF&I\"dGF&F&F&C*>F5-I&ArrayGF(6#;\"\"\"F,> F6-I$modGF&6$*&F*F@F*F@F%>&F56#F@F6?(F4\"\"#F@F,I%trueGF(>&F56#F4*&&F5 6#,&F4F@F@!\"\"F@F6F@>F7-I)ithprimeGF&6#F.>F2-FC6$-I#&^GF&6$F*F7F%?(F3 ,&F.F@F@F@F@F0/-I$gcdGF&6$,&F2F@F@FSF%F@C&>F8-I*nextprimeGF&6#F7>F9,&F 8F@F7FS>F7F8@%1F9,$*&FJF@F,F@F@>F2-FC6$*&F2F@&F56#,$*&#F@FJF@F9F@F@F@F %>F2-FC6$*&F2F@-Ffn6$F*F9F@F%F2F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "pollardpsplit(8174912477117*23528569104401,3,1000,100 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" " 0 "" {MPLTEXT 1 0 39 "igcd(%-1,8174912477117*235285691044 01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "pollardp2split(8174912477117*23528569104401,%%,10 0,100,10000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\":mYD\"f.W>>M`9R" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "igcd(%-1,8174912477117*23528 569104401);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"/,W5p&GN#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ifactor(%-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*.)-I!G6\"6#\"\"#\"\"%\"\"\")-F%6#\"\"&F(F*-F%6#\"#nF*-F% 6#\"$2\"F*-F%6#\"$*>F*-F%6#\"&J7%F*" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 14 "2.20. Feladat." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 64 "3. \+ Egyszer\305\261 pr\303\255mtesztel\303\251si m\303\263dszerek" }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 18 "4. Lucas-sorozatok" }}{PARA 0 "" 0 "" {TEXT 201 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 23 "5. Alkalmaz\303\241sok " }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 36 "6. Sz\303\241mok \303\251s polinomok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 45 "7. Gyors Fourier-transzf orm\303\241ci\303\263" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 38 "8. Elliptikus f\303\274ggv\303\251nyek" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 59 "9. Sz\303\241mol\303\241s elliptikus g\303\266rb\303\251ken" }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 65 "10. Faktoriz\303\241l\303\241s elliptikus g\303\266rb\303\251kkel" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 55 "11. Pr\303\255mteszt elliptikus g\303\266rb\303\251kkel" }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 37 "12. Polinomfaktoriz\303\241l\303\241s" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 16 "13. Az AKS teszt" }} }{SECT 1 {PARA 3 "" 0 "" {TEXT 205 36 "14. A szita m\303\263dszerek al apjai" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }