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{SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Codage RSA :" }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 19 "Choix des nombres :" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 73 "Effectuons une application num\351rique avec des nom
bres de taille modeste :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 
"x:=rand(1..100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p:=ith
prime(x());q:=ithprime(x());k:=17;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"pG\"$*Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"$n\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"#<" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "n:=p*q;phi:=(p-1)*(q-1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"nG\"&j\\'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phi
G\"&3W'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "On r\351soud : (phi mo
d k)*a+1=0 mod k ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "phi m
od k;(phi mod k)*[1,2,3,4,5,6,7,8,9,10] mod k;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"#7\"\"(\"\"
#\"#9\"\"*\"\"%\"#;\"#6\"\"'\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 35 "Donc a vaut 7 et alpha=(a*phi+1)/k." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "alpha:=(7*phi+1)/k;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&alphaG\"&@l#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res
tart;" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 24 "Proc\351dures n\351c
\351ssaires :" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "On utilise des pr
oc\351dures de chiffrement classiques qui seront d'ailleurs utilis\351
es plusieurs fois." }}{PARA 0 "" 0 "" {TEXT -1 68 "On prendra ici cell
es qui sont bijectives pour \351viter tout probl\350me." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1241 "alphabet:=abcdefghijklmnopqrstuvw
xyzABCDEFGHIJKLMNOPQRSTUVWXYZ:\nnb_lettres:=length(alphabet):\nfor ni \+
to nb_lettres do\n     lettre[ni]:=substring(alphabet,ni..ni) od:\nlet
tre[0]:=` `:\nlettre[53]:=`\340`:lettre[54]:=`\351`:lettre[55]:=`\350`
:lettre[56]:=`\371`:lettre[57]:=`.`:lettre[58]:=`,`:lettre[59]:=`;`:le
ttre[60]:=`:`:lettre[61]:=`?`:lettre[62]:=`/`:lettre[63]:=`-`:lettre[6
4]:=`(`:lettre[65]:=`)`:lettre[66]:=`+`:lettre[67]:=`*`:lettre[68]:=`=
`:lettre[69]:=`0`:lettre[70]:=`1`:lettre[71]:=`2`:lettre[72]:=`3`:lett
re[73]:=`4`:lettre[74]:=`5`:lettre[75]:=`6`:lettre[76]:=`7`:lettre[77]
:=`8`:lettre[78]:=`9`:lettre[79]:=`&`:lettre[80]:=`~`:lettre[81]:=`
\343`:lettre[82]:=`\365`:lettre[83]:=`\361`:lettre[84]:=`\"`:lettre[85
]:=`#`:lettre[86]:=`'`:lettre[87]:=`\{`:lettre[88]:=`\}`:lettre[89]:=`
[`:lettre[90]:=`]`:lettre[91]:=`!`:lettre[92]:=`\247`:lettre[93]:=`@`:
lettre[94]:=`_`:lettre[95]:=`>`:lettre[96]:=`^`:lettre[97]:=`\342`:let
tre[98]:=`\352`:lettre[99]:=`\356`:lettre[100]:=`\364`:lettre[101]:=`
\373`:lettre[102]:=`\250`:lettre[103]:=`\344`:lettre[104]:=`\353`:lett
re[105]:=`\357`:lettre[106]:=`\366`:lettre[107]:=`\374`:lettre[108]:=`
\243`:lettre[109]:=`$`:lettre[110]:=`\244`:lettre[111]:=`\265`:lettre[
112]:=`<`:lettre[113]:=`\347`:nn:=114:\nfor ni from 0 to nn do\n     i
mage_lettre[lettre[ni]]:=ni od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 161 "code:=proc(exp)\nlocal ni,res,element;\nres:=0;\nfor ni to le
ngth(exp) do\n   element:=image_lettre[substring(exp,ni..ni)];\n   res
:=res+element*(nn)^(ni-1);\nod;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 122 "decode:=proc(nb)\nlocal ni,liste;\nliste:=convert(nb
,base,nn);\nseq(lettre[ni],ni=liste);\ncat(seq(lettre[ni],ni=liste));
\nend:" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 256 16 "Proc
\351dures RSA :" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Voici donc 2 pr
oc\351dures qui utilisent le syst\350me de codage RSA." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "codersa:=proc(mot,k,n)\nlocal m,i,
c;\nc:=0;\nm:=code(mot);\nm:=convert(m,base,n);\nfor i from 1 to nops(
m) do\n   m[i]:=m[i]^k mod n;\nod;\nfor i from 1 to nops(m) do\n   c:=
c+m[i]*n^(i-1);\nod;\ndecode(c);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 213 "decodersa:=proc(mot,k,n)\nlocal m,p,i,d;\np:=array;d
:=0;\nm:=code(mot);\nm:=convert(m,base,n);\nfor i from 1 to nops(m) do
\n   p[i]:=m[i]^k mod n;\nod;\nfor i from 1 to nops(m) do\n   d:=d+p[i
]*n^(i-1);\nod;\ndecode(d);\nend:" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 10 "Exemples :" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 183 "On peut util
iser majuscules, ponctuation et tous les caract\350res particuliers qu
i ont \351t\351 pr\351vus par les proc\351dures code et decode.(On cod
e avec k et n puis on decode avec alpha et n)." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 91 "codersa(`Bonjour tout le monde! Voici une appl
ication du syst\350me de codage RSA.`,17,64963);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%bo|dyR8j|`z-33/cCZXk=vD|_yc@|_zFgq|cyPk|_u#i^|dz&:9ku
\"XP<n'|cu\"pAxXM|byO+t|fywJFKD[z;J*v|^y|[y|fyMaG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 25 "decodersa(%,26521,64963);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%aoBonjour~tout~le~monde!~Voici~une~application~du~sy
st|cyme~de~codage~RSA.G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Le pro
bl\351me de ces proc\351dures , c'est qu'elles ont besoin d'\351lever \+
\340 des puissances gigantesques..." }}{PARA 0 "" 0 "" {TEXT -1 99 "Et
 , de ce fait , les calculs sont tr\350s long : le calcul de d\351coda
ge ci dessus aura pris 2 minutes." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
79 "Et cela fonctionne \351galement dans l'autre sens , dans le cadre \+
des signatures :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "coders
a(`Plus les nombres sont grands, plus il sera difficile de \"pyrater\"
 le message.`,26521,64963);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%hokY|^
uujVIm|jy|`v?B\"Dmnh|eyp^c|cy^pQ~OzP|azQfR|`zsGlH|hrJ?|`zj/^7ZF_q8k~|_
u]|hr6o<s_]KDpL1|iym|]yD9:e\"J.G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "decodersa(%,17,64963);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%hoPlus~les~nombres~sont~grands,~plus~il~sera~difficile~de~\"pyr
ater\"~le~message.G" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Exponent
ielle modulaire :" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Pour \351vite
r ces probl\350mes de temps de calcul on utilise une \"exponentielle m
odulaire\":" }}{PARA 0 "" 0 "" {TEXT -1 204 "Elle utilise une methode \+
d'exponentiation tr\350s classique qui permet de faire beaucoup moins \+
de calculs et , de plus , en passant au modulo n a chaque fois , elle \+
se ram\350ne \340 des nombres plus petits que n." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 255 "expomodulaire:=proc(m,e,n)\nlocal L,i,facteu
r,res;\nL:=convert(e,base,2);\nfacteur:=m mod n;\nif L[1]=1 then\n   r
es:=m;\n   else res:=1;fi;\nfor i in subsop(1=NULL,L) do\n   facteur:=
facteur^2 mod n;\n   if i=1 then\n      res:=(res*facteur) mod n;fi;\n
od;res;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "expomodula
ire(5247,17,11659);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$l#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "5247^17 mod 11659;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"$l#" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
26 "Nouvelles proc\351dures RSA :" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
84 "Reprennons les proc\351dures pr\351c\351dentes en y inserant notre
 \"exponentielle modulaire\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 212 "codersa2:=proc(mot,k,n)\nlocal m,i,c;\nc:=0;\nm:=code(mot);\nm:
=convert(m,base,n);\nfor i from 1 to nops(m) do\n   m[i]:=expomodulair
e(m[i],k,n);\nod;\nfor i from 1 to nops(m) do\n   c:=c+m[i]*n^(i-1);\n
od;\ndecode(c);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "d
ecodersa2:=proc(mot,k,n)\nlocal m,p,i,d;\np:=array;d:=0;\nm:=code(mot)
;\nm:=convert(m,base,n);\nfor i from 1 to nops(m) do\n   p[i]:=expomod
ulaire(m[i],k,n);\nod;\nfor i from 1 to nops(m) do\n   d:=d+p[i]*n^(i-
1);\nod;\ndecode(d);\nend:" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "E
xemples plus grands :" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Utilisons
 de plus grandes applications num\351riques :" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "p:=nextprime(9853417379137);k:=37;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"pG\".f\"ztT`)*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"kG\"#P" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "q:=next
prime(9849789961773);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\".$yh*
*y\\)*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n:=p*q;phi:=(p-1)
*(q-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\";(\\!oIYZ)\\!f\"4aq
*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG\";c&RLUr(y/f\"4aq*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "On r\351soud : (phi mod k)*a+1=0 m
od k ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "phi mod k;(phi mo
d k)*[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] mod k;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#72\"#K\"#F\"
#A\"#<\"#7\"\"(\"\"#\"#M\"#H\"#C\"#>\"#9\"\"*\"\"%\"#O\"#J" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 36 "Donc a vaut 15 et alpha=(a*phi+1)/k." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "alpha:=(15*phi+1)/k;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\";$zs-#QR\"\\ZLDY$R" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 45 "Il n'y a plus qu'a relancer les proc\351dures..." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "codersa2(`Ainsi, on peut
 crypter avec l'algorithme RSA en utilisant des nombres relativement g
rands sans aucun probl\350me de temps de calcul.`,37,97054091590498474
630680497);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%fs|_u=|dy|_zQ|\\z1oy|j
y;d|`z7O?:VLp|bu^o2V:i|[ywcXX|eyBz|by#:|\\z|[yS|cy5B(gUvz4:6cI&ft2]d?|
by|jy,C|iyV|]y!i$)qD|azh|fr?(S,]ZT|[yIg:[#i|_zr|`z6A'P|fr\"9)/h(MvU02X
*HvK'58UgKhg|frV$lfG|[ygE|by@|hr|iyE):G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "decodersa2(%,39346253347491393820272793,9705409159049
8474630680497);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%\\sAinsi,~on~peut~
crypter~avec~l'algorithme~RSA~en~utilisant~des~nombres~relativement~gr
ands~sans~aucun~probl|cyme~de~temps~de~calcul.G" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 40 "Et pour finir , un exemple de signature." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "codersa2(`Je t'envoies ce message \+
en le codant avec ma clef priv\351e, ce qui te permet, en le d\351coda
nt, avec ma clef publique, d'\352tre s\373r que c'est bien moi qui te \+
l'envoies.`,39346253347491393820272793,97054091590498474630680497);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#%`uH)hNzTZ2-f**|_u+SY)7^|bucYzUu|gz|d
yKcY3K|^y:Pdp!|fr|irqu|hrxy$P|^yj|\\zAEq|jyMYcrZC?mT-|^u5|\\z\"|\\zf0x
k8e|frw|bu|fzE~!&<Lu|azSW/~]O|bys?BOl.Am:yj|_uD|^uWZ,5fdj|by=:u&]|\\z8
:|`zXae6-T.4|\\z|_u5ok|ey6|frR|`vtalBuwb|fr-|`z7?|\\zl|hrhD|_uEL-aG" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "decodersa2(%,37,9705409159
0498474630680497);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%]uJe~t'envoies~
ce~message~en~le~codant~avec~ma~clef~priv|dye,~ce~qui~te~permet,~en~le
~d|dycodant,~avec~ma~clef~publique,~d'|eytre~s|fzr~que~c'est~bien~moi~
qui~te~l'envoies.G" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Piratage
 :" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Petits piratages :" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Pour de petits nombres, il est tr
\350s facile de pirater un code RSA ..." }}{PARA 0 "" 0 "" {TEXT -1 
37 "Il suffit de factoriser le nombre n :" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "ifactor(64963);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
*&-%!G6#\"$n\"\"\"\"-F%6#\"$*QF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
113 "On trouve ainsi p et q sans difficult\351s et on en deduit phi pu
is a et enfin alpha et on peut finallement pirater." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 86 "Ici, il est facile de pirater de telles applica
tions num\351riques parce que n est petit." }}{PARA 0 "" 0 "" {TEXT 
-1 113 "Par contre la proc\351dure de factorisation de maple ne foncti
onne ais\351ment que pour des nombres inf\351rieurs \340 10^30." }}
{PARA 0 "" 0 "" {TEXT -1 30 "Au dela, il bug r\351guli\350rement." }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Plus grands :" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 112 "Pour trouver des nombres premiers de cette taill
e dans maple, il vaut mieux utiliser \"nextprime\" que \"ithprime\"." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "p:=9853417379159;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\".f\"ztT`)*" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 17 "q:=9849789961783;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"qG\".$yh**y\\)*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "n:=p*q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\";(
\\!oIYZ)\\!f\"4aq*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ifact
or(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%!G6#\".$yh**y\\)*\"\"\"-
F%6#\".f\"ztT`)*F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Ce calcul, \+
pour un nombre n de 26 chiffres a \351t\351 effectu\351 en une dizaine
 de secondes par maple." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "
p:=nextprime(52684*10^12+12345);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"pG\"2hB,+++%o_" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "q:=next
prime(63485*10^12+56789);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"2
Bo0+++&[j" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:=p*q;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"C.\"*Q-2+<5SyP+SPkWL" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ifactor(n);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*&-%!G6#\"2hB,+++%o_\"\"\"-F%6#\"2Bo0+++&[jF(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Cette factorisation de 34 chiffres
 aura \351t\351 effectu\351e en 50 secondes ..." }}{PARA 0 "" 0 "" 
{TEXT -1 46 "Mais au del\340 cela devient vite tr\350s difficile." }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Nombres premiers :" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 64 "Maple permet tout de m\352me de trouver d
e grands nombres premiers." }}{PARA 0 "" 0 "" {TEXT -1 83 "(Mais il lu
i sera impossible de calculer \"alpha\" avec des nombres de cette tail
le)." }}{PARA 0 "" 0 "" {TEXT -1 53 "Par exemple, voici un nombre prem
ier de 200 chiffres." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "nex
tprime(12548595956858748595*10^180+12545895784858698754685468);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#\"cwjeoa()pe[y&*ea7+++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++&f[(eo&ff[D
\"" }}}}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }