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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 203 66 "Sz\303\241m\303\255t\303
\263g\303\251pes sz\303\241melm\303\251let" }}}{EXCHG {PARA 19 "" 0 ""
 {TEXT 204 18 "J\303\241rai Antal" }}}{EXCHG {PARA 19 "" 0 "" {TEXT 
204 68 "Ezek a programok csak szeml\303\251ltet\303\251sre szolg\303\2
41lnak" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 59 "1. A pr\303\255mek el
oszl\303\241sa, szit\303\241l\303\241s" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT 205 63 "2. Egyszer\305\261 faktoriz\303\241l\303\241si m\303\263
dszerek" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 64 "3. Egyszer\305\261 p
r\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" }}}{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 F
ourier-transzform\303\241ci\303\263" }}{PARA 0 "" 0 "" {TEXT 201 0 "" 
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 38 "8. Elliptikus f\303\274ggv\30
3\251nyek" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 205 59 "9. Sz\303\241mol\3
03\241s elliptikus g\303\266rb\303\251ken" }}{PARA 0 "" 0 "" {TEXT 
201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT 206 16 "9.1. Elliptikus " }{TEXT 206 21 
"g\303\266rb\303\251k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 64 "# This routin
e randomly choose an elliptic \"curve\" modulo n,\n" }{MPLTEXT 1 0 53 
"# where gcd(n,6)=1. The coordinates x,y are choosen\n" }{MPLTEXT 1 0 
55 "# randomly, the parameter a too, and b is calculated.\n" }{MPLTEXT
 1 0 57 "# The list [x,y,a,b] is given back or a divisor d of n.\n" }
{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 39 "ellrand:=p
roc(n) local x,y,a,b,r,d,f;\n" }{MPLTEXT 1 0 19 "r:=rand(n); d:=0;\n" 
}{MPLTEXT 1 0 14 "while d=0 do\n" }{MPLTEXT 1 0 52 "  x:=r(n); y:=r(n)
; a:=r(n); b:=y^2-x^3-a*x mod n;\n" }{MPLTEXT 1 0 36 "  d:=4*a^3+27*b^
2 mod n; gcd(d,n);\n" }{MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 1 0 34 "if %<n
 and %>1 then return % fi;\n" }{MPLTEXT 1 0 15 "[x,y,a,b]; end;" }
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"6)I\"xGF
%I\"yGF%I\"aGF%I\"bGF%I\"rGF%I\"dGF%I\"fGF%F%F%C'>F+-I%randGF%F#>F,\"
\"!?(F%\"\"\"F5F%/F,F3C(>F'-F+F#>F(F9>F)F9>F*-I$modGF%6$,(*$)F(\"\"#F5
F5*$)F'\"\"$F5!\"\"*&F)F5F'F5FGF$>F,-F>6$,&*&\"\"%F5)F)FFF5F5*&\"#FF5)
F*FCF5F5F$-I$gcdGF%6$F,F$@$32I\"%GF%F$2F5FYOFY7&F'F(F)F*F%F%F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ellrand(97); " }{MPLTEXT 1 
0 13 "ellrand(97); " }{MPLTEXT 1 0 13 "ellrand(97); " }{MPLTEXT 1 0 
13 "ellrand(97); " }{MPLTEXT 1 0 12 "ellrand(97);" }{MPLTEXT 1 0 0 "" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"&\"#e\"#V\"#<" }}{PARA 11 "" 1 "
" {XPPMATH 20 "7&\"#P\"#o\"#E\"#a" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&
\"#&*\"#;\"#*)\"#a" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"#L\"#<\"#^\"#9
" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"#b\"#U\"##)\"#Z" }}}}{SECT 0 
{PARA 4 "" 0 "" {TEXT 206 12 "9.2. Hasse t" }{TEXT 206 8 "\303\251" }
{TEXT 206 5 "tele." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 61 "# This brute force pr
ocedure calculate the number of points\n" }{MPLTEXT 1 0 55 "# on an el
liptic curve modulo p>3, a prime. The curve\n" }{MPLTEXT 1 0 25 "# par
ameters are a, b. \n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 34 "ellcount:=proc(a,b,p) local x,c;\n" }{MPLTEXT 1 0 7 "
c:=1;\n" }{MPLTEXT 1 0 67 "for x from 0 to p-1 do c:=c+numtheory[jacob
i](x^3+a*x+b,p)+1; od;\n" }{MPLTEXT 1 0 7 "c; end;" }{MPLTEXT 1 0 0 ""
 }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"aG6\"I\"bGF%I\"pGF%6$I\"xGF%I
\"cGF%F%F%C%>F*\"\"\"?(F)\"\"!F-,&F'F-F-!\"\"I%trueG%*protectedG>F*,(F
*F--&I*numtheoryGF%6#I'jacobiGF%6$,(*$)F)\"\"$F-F-*&F$F-F)F-F-F&F-F'F-
F-F-F*F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ellrand(97)
; ellcount(%[3],%[4],97);" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 38 "ellr
and(97); ellcount(%[3],%[4],97);\n" }{MPLTEXT 1 0 38 "ellrand(97); ell
count(%[3],%[4],97);\n" }{MPLTEXT 1 0 38 "ellrand(97); ellcount(%[3],%
[4],97);\n" }{MPLTEXT 1 0 36 "ellrand(97); ellcount(%[3],%[4],97);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "7&\"#f\"##*\"#8\"\"%" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "\"#))" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"#\\\"#Y\"\"(
\"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$:\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "7&\"#X\"#V\"\")\"#*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$
7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"#w\"#e\"#:\"#R" }}{PARA 11 ""
 1 "" {XPPMATH 20 "\"$0\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"!\"#p
\"#w\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#(*" }}}}{SECT 0 {PARA 4 
"" 0 "" {TEXT 206 15 "9.3. Gyakorlat." }}{PARA 0 "" 0 "" {TEXT 201 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}{PARA 11
 "" 1 "" {XPPMATH 20 "7gnI,InteractiveG6\"I(animateGF$I*animate3dGF$I-
animatecurveGF$I&arrowGF$I-changecoordsGI(_syslibGF$I,complexplotGF$I.
complexplot3dGF$I*conformalGF$I,conformal3dGF$I,contourplotGF$I.contou
rplot3dGF$I*coordplotGF$I,coordplot3dGF$I-cylinderplotGF$I,densityplot
GF$I(displayGF$I*display3dGF$I*fieldplotGF$I,fieldplot3dGF$I)gradplotG
F$I+gradplot3dGF$I,graphplot3dGF$I-implicitplotGF$I/implicitplot3dGF$I
(inequalGF$I,interactiveGF$I2interactiveparamsGF$I-listcontplotGF$I/li
stcontplot3dGF$I0listdensityplotGF$I)listplotGF$I+listplot3dGF$I+loglo
gplotGF$I(logplotGF$I+matrixplotGF$I)multipleGF$I(odeplotGF$I'paretoGF
$I,plotcompareGF$I*pointplotGF$I,pointplot3dGF$I*polarplotGF$I,polygon
plotGF$I.polygonplot3dGF$I4polyhedra_supportedGF$I.polyhedraplotGF$I'r
eplotGF$I*rootlocusGF$I,semilogplotGF$I+setoptionsGF$I-setoptions3dGF$
I+spacecurveGF$I1sparsematrixplotGF$I+sphereplotGF$I)surfdataGF$I)text
plotGF$I+textplot3dGF$I)tubeplotGF$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "implicitplot(y^2=x^3-10*x+10,x=-5..10,y=-20..20,numpo
ints=10000);" }}{PARA 13 "" 1 "" {TEXT 207 0 "" }{GLPLOT2D 400 400 400
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Fher7$FhasFi]s7$F_as7$$\"3R`4w1<\">%eF0$\"3_YP'[aB)G7Fher7$7$$\"3*\\m@
Lqh5(eF0$\"3@++++++S7FherF_bs7$Febs7$$\"3S\"[P+fgS!fF0$\"3KRLK40D_7Fhe
r7$7$Fdas$\"3a5Q_f]Ip7FherF[cs7$7$$\"3?.++++++hF0$!3`<nloi\\E8Fher7$$
\"3Mne\"[p(R#3'F0$!3N,+++++?8Fher7$7$F[ds$!3<,+++++?8Fher7$$\"3r0Ipa58
TgF0$!39-[e9;I/8Fher7$7$$\"3E\"3@Z%*[s(fF0$!39,+++++!G\"FherFcds7$Fids
Fcas7$Facs7$$\"3;Xt\\&=Di'fF0$\"3dvt'QGtcF\"Fher7$7$$\"3rx5sW*[s(fF0$
\"3D++++++!G\"FherF`es7$7$Fges$\"32++++++!G\"Fher7$$\"3Jid#GfNy-'F0$\"
33JJ\">%QC*H\"Fher7$7$$\"3zje\"[p(R#3'F0$\"36++++++?8FherF_fs7$7$Fffs$
\"3G++++++?8Fher7$$\"3W!\\bYJ\"y*3'F0$\"3'p`eF)\\sA8Fher7$7$Ffcs$\"3q<
nloi\\E8FherF^gs7$7$$\"3c.+++++]iF0$!3uzM/j^?%Q\"Fher7$$\"3aa.q1y+'='F
0$!3@,+++++g8Fher7$F]hs7$Ffcs$!3q<nloi\\E8Fher7$Fdgs7$$\"31eo\")\\k0^h
F0$\"3d0brY\\QY8Fher7$7$$\"3x_.q1y+'='F0$\"3]++++++g8FherFghs7$7$F^is$
\"3K++++++g8Fher7$$\"3*HxR<(yY7iF0$\"3^hg`n&3+P\"Fher7$7$Figs$\"3czM/j
^?%Q\"FherFfis7$7$$\"3#R++++++S'F0$!3*3&o]J\\oU9Fher7$$\"3oLe#[=uGR'F0
$!35,+++++S9Fher7$7$Ffjs$!3G,+++++S9Fher7$$\"31,ykNc8xjF0$!3>ZF<OG!RV
\"Fher7$7$$\"3(>7+sdA'*G'F0$!3C,++++++9FherF^[t7$Fd[tFhgs7$7$Figs$\"3u
zM/j^?%Q\"Fher7$$\"3#pE<J'4utiF0$\"3Sj?]w!pOR\"Fher7$7$$\"3I>,?xDi*G'F
0$\"3O+++++++9FherF^\\t7$Fd\\t7$$\"3TLfecQiMjF0$\"3s_PCQOV<9Fher7$7$$
\"3#>$e#[=uGR'F0$\"3d++++++S9FherFj\\t7$7$Fa]t$\"3R++++++S9Fher7$$\"3Y
72GFt#eR'F0$\"3sW^s7F6T9Fher7$7$Fajs$\"3*3&o]J\\oU9FherFi]t7$7$$\"3E/+
++++]lF0$!3r,++]7d,:Fher7$$\"3)o/>g[2T\\'F0$!39,+++++![\"Fher7$7$Fi^t$
!3J,+++++![\"FherF`js7$7$Fajs$\"3r]o]J\\oU9Fher7$$\"3%ytSAqmiX'F0$\"3/
Q\"pga&*\\Y\"Fher7$7$$\"33X!>g[2T\\'F0$\"3V++++++![\"FherFe_t7$F[`t7$$
\"3.!z-45Yp^'F0$\"3&4f#4tV\"))[\"Fher7$7$Fd^t$\"3r,++]7d,:FherFa`t7$7$
$\"3g/++++++nF0$!35)pbkJv7c\"Fher7$$\"3G/+9mzo'p'F0$!3@,+++++g:Fher7$7
$Faat$!3R,+++++g:Fher7$$\"31]q]j)3)*o'F0$!3<!)=gjBGd:Fher7$7$$\"379/p2
nV&f'F0$!3N,+++++?:FherFiat7$F_btFc^t7$Fg`t7$$\"3vV<\"o>8ud'F0$\"3aON=
9)*o7:Fher7$7$$\"3O7/p2nV&f'F0$\"3Y++++++?:FherFfbt7$F\\ct7$$\"3\"4?>l
Yuvj'F0$\"3s9#G*3okO:Fher7$7$$\"3]-+9mzo'p'F0$\"3]++++++g:FherFbct7$Fh
ct7$$\"3O/o%*eJ/)p'F0$\"3%Q&3GC=_g:Fher7$7$F\\at$\"3$zpbkJv7c\"FherF^d
t7$7$$\"3(\\++++++&oF0$!3\"e2uSKb8i\"Fher7$$\"3g<qPF0#ez'F0$!3U,++++++
;Fher7$F]et7$F\\at$!3$zpbkJv7c\"Fher7$Fddt7$$\"34)*ymK(Gxv'F0$\"3)>c)G
rcg%e\"Fher7$7$$\"3#\\,xt_?ez'F0$\"3O+++++++;FherFget7$F]ft7$$\"3^3]vt
Qo<oF0$\"3vK`mLwh3;Fher7$7$Fidt$\"3\"e2uSKb8i\"FherFcft7$7$$\"3K0+++++
+qF0$!3+\\w6%HNAo\"Fher7$$\"3+wWacHK%*pF0$!39,+++++!o\"Fher7$Fbgt7$$\"
3(3Gj5XPK)pF0$!3=3-&p)*Hbn\"Fher7$7$$\"3wnJ&>4C^*oF0$!3G,+++++S;FherFh
gt7$F^htFhdt7$Fift7$$\"3*G$yn`XUxoF0$\"3x7fo&y'oK;Fher7$7$$\"3+mJ&>4C^
*oF0$\"3@++++++S;FherFeht7$F[it7$$\"3.;$y$f@(o$pF0$\"3Y]C$34Mol\"Fher7
$7$$\"3MtWacHK%*pF0$\"32++++++!o\"FherFait7$Fgit7$$\"3amV$)*G:m*pF0$\"
3*R]x$*e-4o\"Fher7$7$F^gt$\"3O\\w6%HNAo\"FherF]jt7$7$$\"3p0+++++]rF0$!
3eZ.JHE_V<Fher7$$\"3oE$z'>9g\"4(F0$!3*4++++++s\"Fher7$7$$\"3yD$z'>9g\"
4(F0F_[uF]gt7$Fcjt7$$\"3uou4VJibqF0$\"3Zw1%=;n^q\"Fher7$7$$\"37B$z'>9g
\"4(F0$\"3$*************><FherFf[u7$F\\\\u7$$\"3'e'*RAMj[6(F0$\"3LxESv
(p$H<Fher7$7$Fhjt$\"3BZ.JHE_V<FherFb\\u7$7$$\"3/1++++++tF0$!3O67z37a0=
Fher7$$\"3yc>Cx@C'G(F0$!3s+++++++=Fher7$Fa]u7$$\"3$pa4k&p'4E(F0$!3_yeP
)=\"f*y\"Fher7$7$$\"3eK)))\\37\"*=(F0$!3&3++++++w\"FherFg]u7$7$$\"3pJ)
))\\37\"*=(F0F`^u7$$\"3![++++++:(F0$!3BZ.JHE_V<Fher7$Fh\\u7$$\"3KAYJ<8
&R<(F0$\"3BN%\\?)Hh`<Fher7$7$$\"3\"*H)))\\37\"*=(F0$\"3z************f<
FherF\\_u7$Fb_u7$$\"3O!3_*4YpKsF0$\"3&)zF,/\"[zx\"Fher7$7$$\"3*e&>Cx@C
'G(F0$\"#=F]hrFh_u7$7$$\"3+b>Cx@C'G(F0$\"3k*************z\"Fher7$$\"3a
'R**HbA<H(F0$\"31i,?>t?-=Fher7$7$F]]u$\"3O67z37a0=FherFi`u7$7$$\"3R1++
+++]uF0$!3%3EK!zt/o=Fher7$$\"3[\"R&R%R&)=Q(F0$!3d++++++S=Fher7$FhauF
\\]u7$F_au7$$\"3Ni7\\YI7]tF0$\"3%z**oU&QjE=Fher7$7$$\"3r*Q&R%R&)=Q(F0$
\"3]************R=FherF_bu7$Febu7$$\"3uK(zhle'3uF0$\"3gs?NeV-^=Fher7$7
$Fdau$\"3%3EK!zt/o=FherF[cu7$7$$\"3w1++++++wF0$!3kh>td_<J>Fher7$$\"3J
\\:\"Q3!)Gd(F0$!3G++++++?>Fher7$Fjcu7$$\"3W\\#oQeUf_(F0$!3X)>)pb8D+>Fh
er7$7$$\"3emZ\"pE5yZ(F0$!3V++++++!)=FherF`du7$FfduFcau7$Facu7$$\"35X3Q
/?;nuF0$\"3DU%)\\lMUv=Fher7$7$$\"3#Rw9pE5yZ(F0$\"3O************z=FherF
]eu7$7$$\"3\"[w9pE5yZ(F0Ffeu7$$\"3%Hc\\;F6__(F0$\"3c]M*3mV***=Fher7$7$
$\"3lY:\"Q3!)Gd(F0$\"3A************>>FherF\\fu7$Fbfu7$$\"33;$*=n^`$e(F
0$\"3I]G3A1RC>Fher7$7$Ffcu$\"3kh>td_<J>FherFhfu7$7$$\"362+++++]xF0$!3s
!zyy.5\\*>Fher7$$\"3:D#)oF:3nwF0$!39++++++g>Fher7$FgguFecu7$F^gu7$$\"3
`heT@KRTwF0$\"3;QCi2='*[>Fher7$7$$\"3f@#)oF:3nwF0$\"33************f>Fh
erF^hu7$7$Fehu$\"3s)***********f>Fher7$$\"3!QD(p\"GN#*p(F0$\"3jmS@es`t
>Fher7$7$Fcgu$\"33\"zyy.5\\*>FherF]iu7$7$$\"3gsstDUghxF0$Fd`oF]hr7$Fcg
u$!33\"zyy.5\\*>Fher7$Fciu7$$\"3iut1nn@dxF0$\"3/M?yGb2)*>Fher7$7$$\"3%
*pstDUghxF0$\"3#*)*************>FherF_ju-%&COLORG6&%$RGBG$F\\hr!\"\"$F
]hrF_[vF`[v" 1 2 2 1 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}
{TEXT 207 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 15 "9.4. Gyakorla
t." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 65 "# Doubling on an elliptic \"cur
ve\" modulo n, where gcd(n,6)=1.\n" }{MPLTEXT 1 0 57 "# P is the point
 to double and a, b are the parameters.\n" }{MPLTEXT 1 0 52 "# The ret
urn value is the double of the point P or\n" }{MPLTEXT 1 0 21 "# a div
isor d of n.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 
1 0 34 "elldou:=proc(P,a,b,n) local l,d;\n" }{MPLTEXT 1 0 29 "if P[3]=
0 then return P fi;\n" }{MPLTEXT 1 0 35 "if P[2]=0 then return [0,1,0]
 fi;\n" }{MPLTEXT 1 0 26 "d:=igcdex(2*P[2],n,'l');\n" }{MPLTEXT 1 0 
34 "if 1<d and d<n then return d fi;\n" }{MPLTEXT 1 0 26 "l:=(3*P[1]^2
+a)*l mod n;\n" }{MPLTEXT 1 0 19 "l^2-2*P[1] mod n;\n" }{MPLTEXT 1 0 
30 "[%,l*(P[1]-%)-P[2] mod n,1];\n" }{MPLTEXT 1 0 4 "end;" }{MPLTEXT 
1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I\"PG6\"I\"aGF%I\"bGF%I\"
nGF%6$I\"lGF%I\"dGF%F%F%C)@$/&F$6#\"\"$\"\"!OF$@$/&F$6#\"\"#F2O7%F2\"
\"\"F2>F+-I'igcdexGF%6%,$*&F8F;F6F;F;F(.F*@$32F;F+2F+F(OF+>F*-I$modGF%
6$*&,&*&F1F;)&F$6#F;F8F;F;F&F;F;F*F;F(-FJ6$,&*$)F*F8F;F;*&F8F;FPF;!\"
\"F(7%I\"%GF%-FJ6$,&*&F*F;,&FPF;FZFXF;F;F6FXF(F;F%F%F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 65 "# Addition on
 an elliptic \"curve\" modulo n, where gcd(n,6)=1.\n" }{MPLTEXT 1 0 
61 "# P and Q are the points to add and a,b are the parameters.\n" }
{MPLTEXT 1 0 66 "# The return value is the sum of the points or a divi
sor d of n.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 
0 36 "elladd:=proc(P,Q,a,b,n) local l,d;\n" }{MPLTEXT 1 0 29 "if P[3]=
0 then return Q fi;\n" }{MPLTEXT 1 0 29 "if Q[3]=0 then return P fi;\n
" }{MPLTEXT 1 0 40 "if P=Q then return elldou(P,a,b,n) fi;\n" }
{MPLTEXT 1 0 38 "if P[1]=Q[1] then return [0,1,0] fi;\n" }{MPLTEXT 1 
0 29 "d:=igcdex(P[1]-Q[1],n,'l');\n" }{MPLTEXT 1 0 34 "if 1<d and d<n \+
then return d fi;\n" }{MPLTEXT 1 0 25 "l:=(P[2]-Q[2])*l mod n;\n" }
{MPLTEXT 1 0 22 "l^2-P[1]-Q[1] mod n;\n" }{MPLTEXT 1 0 30 "[%,l*(P[1]-
%)-P[2] mod n,1];\n" }{MPLTEXT 1 0 4 "end;" }{MPLTEXT 1 0 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "f*6'I\"PG6\"I\"QGF%I\"aGF%I\"bGF%I\"nGF%
6$I\"lGF%I\"dGF%F%F%C+@$/&F$6#\"\"$\"\"!OF&@$/&F&F1F3OF$@$/F$F&O-I'ell
douGF%6&F$F'F(F)@$/&F$6#\"\"\"&F&FBO7%F3FCF3>F,-I'igcdexGF%6%,&FAFCFD!
\"\"F).F+@$32FCF,2F,F)OF,>F+-I$modGF%6$*&,&&F$6#\"\"#FC&F&FZFLFCF+FCF)
-FU6$,(*$)F+FenFCFCFAFLFDFLF)7%I\"%GF%-FU6$,&*&F+FC,&FAFCF]oFLFCFCFYFL
F)FCF%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "n:= 97; P:=[5
9,92,1]; a:=13; b:=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#(*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "7%\"#f\"##*\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "elldou(P,a,b,n); elldou(%,a,b,n); e
lldou(%,a,b,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"#!)\"#g\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "7%\"#x\"#f\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "7%\"#n\"#H\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "elladd(P,P,a,b,n); " }{MPLTEXT 1 0 19 "elladd(%,P,a,b,n); " }
{MPLTEXT 1 0 19 "elladd(%,P,a,b,n); " }{MPLTEXT 1 0 18 "elladd(%,P,a,b
,n);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"#!)\"#g\"
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"#n\"#o\"\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "7%\"#x\"#f\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%
\"#\"*\"#'*\"\"\"" }}}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT 205 65 "10. Faktoriz\303\241l\303\241s elliptiku
s 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\2
66rb\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. A
z AKS-teszt" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 36 "14. A szita m\30
3\263dszerek alapjai" }}{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" }}}}
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33 1 1 }