Multiplication-addition modulaire: algorithmes itératifs et implantations sur FPGA
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Multiplication-addition modulaire: algorithmes it´eratifs et implantations sur FPGA Jean-Luc Beuchat et Jean-Michel Muller. No 4840 Juin 2003. ISSN 0249-6399. ISRN INRIA/RR--4840--FR+ENG. ` THEME 2. apport de recherche.
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(210) %p
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(245) !f (+*-, (+*-, % m E #&%
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(263) hMyuy{\-l
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(266) ®Ê\-l
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(270) '( %L $&L
(271)
(272)
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(276) b
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(292) 570. Avec le théorème 1 (n=16) Sans le théorème 1 (n=16) Avec le théorème 1 (n=32) Sans le théorème 1 (n=32). Surface [slices]. 310 165 85 50 35 25 17 1. 2. 4. 8 Nombre de moduli. 16. 32. 64. Î Á=- kkÂI=¾ãqZÀ yunk[Fl\h qqZgfedjkb
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(294) JmY-ruq=\khk«v;Z\gpnkgi[l\no[]d\\dGd]m%\
(295) hMmdy{l hkm{efqZhMm{dnojM\ªq\dZ®;hk\[l m{vZqZy{b-\r%n²\M\-[]l3hDCq=qZÂigieK$ru¥iqZkyz¥ kqZkde
(296) ¥lefyl
(297) qZefmðÁªefyumu\¹ lnkhkgfhMdZy{jMr¤yuq \ 8N : OQM4«Mh,Ä][]Lh\-rm¤qZgpginh¯&y{b-wG\-dqZ\-dl
(298) l
(299) \ctlgf\\%rzhkdK XZl hkmueiy{hkgfhMdjkdZ\M\¥[]s\\%dGdZm(hM[/q0vZl
(300) y{ei\yl
(301) qZ\efm-[¥q=qZqZgm{yefqZgfd0eplnolmuefyeihkl
(302) qZdefrmð¾ÃnMÁªZefyumuefmu\ei¹Ghk¾ÃdÂrI«GZdZb
(303) hkmuqK\-yurð[]hMeivdZmub\-\dZrIhkdnkrðyðrno\y¤l
(304) hM\
(305) ¹d\[]\(Z\-rgi\m -¼ ^,^ H/;hkqZyc\-rShk;bynodZ\-r\ R ¼-½vemr¥Kx nkyªlhkdrub-wGqZ\-dMm«ZdZhMqr\®;\l muqhkdrS\d==efy{hkdrue¹ []efgi efhMdrS hk;byn+m{efhMdr ]gpn]r\l
(306) hkdK) \\dqmueigfepr{nodGmqZdZepwGqZ\[]\dGm+Åm{y{nkdlXZ\-r-¥. ¤
(307) $@ dZhMei[/[ GcvZno\y{m2d\rªq=l
(308) l
(309) hMhM\vKdZmªjk\-gfy{no[q yumunoep d@l
(310) gi«@7\kN dZ«ZO5hMdqhk4 rqr;n²kn²==hMM[]dhkhdr%r(Z¯¡bq muq¥sy{Zb-ÂzefdbFqefly{nk\-\rgilnonkqZZgfgpjMnohkdGyuh²y{me¹m{gieiXZn[][]v;no\-muhkr%eiy{kdZef\-mu\[Fb-y{ru\-noqZdGmuKefmb-ryuei \qZ\qd¦yu[\]nMqZ\
(311) l gf¹ZmumueinM\-ZqZl gfmuyepl\noZZmu\ei\-qhkr¹¡dm{¾ãgink\ny{[FruqZe\m{yur efhMnkd©lg\][]!Z1Ih=\-ZrkqZh¤hMginkKef\y{b-\)y{ g no y{nkmu\
(312) gf\-jMmuqZhMhky{qZy{rcey{m{dm{XZy{no[]nkdGem%\mnoqZdGd \m qZy{b-d¶qK\-l dmurei\-hk[/d vZ[]gi\©hqZ\©gpno[]eiyuh\cdZqZbgie%l
(313) \-¯©r{r{°+no±eiyu¯ \ ³kF±-´g ´ hk´v±¯©m{\dGµ+mu¥ eihkd©hMqr5\ n²7 NkhMOd4 rF;bjG=no[Fgi\h[]%\¯ dGm][Fb;hM\dGdmuy{ b wGq©qZ\ lXZ ghM nke¹0y{lZX\)em{¯\-l m{¥ qZ y{\qÄlefyl
(314) qZefm\
(315) ®Ê\-l
(316) muqnkdMmgpn hMKb-y{nomu\qySeidrefy{b\Fg nkgfjMhky{em{XZ[]\/\ ef[ \
(317) mq=hMvK\-gf[noddZb-l\-r{refmu\/muhMqmu\hkeprªwGqZ\Fg hk;bynodK\Oruhkefmªeidby{ef\-qZy ¯¡¥ Dãg\-rmShkdldZb-l\-r{runkefy{\c \®;\l muq\ySqdZ\y{b-ql
(318) mueihkd []hqZgpnoeiyu\q y{b-ruqZgfm{n+mSruel\giqZef¾Ãles\-rmqmueig eirub/l
(319) hM[][F\)hMKb-y{nkdj\ O qdZ\ \-nk dGq no=mugiy{epjkru\hMnkyujk[/ef\-muqZhkXgfd[FmureiZ\gprgine l-Zn+l
(320) nkm{hMdefdhMr/l
(321) ds\-¥@\0~%mueiZ§Khkgfdd qKr\jk qZyyud nobdKrhMhMK\-qb-r)y{y{novK\munk\-rul
(322) qZ\-\yrZnoy{§ hk\
(323) dvZ¹giKY hM[] dZb
(324) \Mmu\qK«sdGm{dZeiefhkno\-qKmuyeir)hkgi\-d n²rk[]hkl
(325) dKhMh[FrqZZgpyuyunohMhMeiK[Fyu\hGeprr\b]¹=\qZdGdgfm{hMyu\e\mdZnogihkdG\qZmdZMdZhM\ho[/gifm{\vZyu\0y{y{\0\hkgp;n+ m{b efefymuhMn+bd¡muy\-n+ZqZm{\efy/hMyudnkbr-efl
(326) d«sqZrugieðy{nEyuwG\-rqZdqZ\l
(327) yu\Mg nk¥ lb
(328) mu\qKhMqq \r lefyl
(329) qZefm\
(330) mgi\)bvemS\rlnkgilqZgpr¥ 7 3 % Ï "#L'%)
(331) %\-[noywMqhkd¯rªmuhMRqm.0/ 1¡ nov;¯hky« wGqZ\]gpn0hMdl m{efhMAd Pb;\dK¡Zq[]hqZgih ¯ lXZhkeprueQ[¥ DãgI\-rm)nklefgi\\]kb-yuef§\y)wGqZ5\ J7^w=3R^« J7} ¼ =QRi.,0[Fh(F ° 4 j¼ _£¯ _*.° 0/1 ¼ ° }7.,=>R. 0 ° []h=(¯ R .,. 0 0 ° 11Kuo.+¯¯ ruruete . .,00 T ° 4 :&uj¼ ! _4 ¯ _" . 0 :u! .
(332) ( q #. .
(333) \
(334) m. k ¯ r t.,0UT ° 4 .0T ³ 4 ¼!_4¯_.021 ¼ e J7uL=>RS7u!V . 0 ==[]h(¯ M¯o¯ rreet., C7m0U.,T 0 ° 4 ° 4 ¼j_.0¯= :4 _#!}4 C7m.¼j0 _£ ° ¯4 _*.0.= 0:T4 !° 4 .0UT ³ jMefgðnk\-y{rnksmdM\ðm{b-muei=r{XZrueib-\hkdG\-y{dGmðYm)[]wGqZwG\¤\;qZru\c^\¤^ _ b-_ []N hkT[dGo g mub`O+y{\ð4 1T ¼nkh}y@T y{_ o b-l_4¯¡qZyu¯¥ty{\q=¥GdKqZxIl
(335) no\ðKy(ruhGqZl
(336) rhky"hMdKdginSrrbvK[wMhMqyuno\dZeidGdM\m-m{r«U\qd.Kf5nkb-dGg yum)>ei\1wGqZqZy{k\h&\¤4 Z\ Ng 3 T 1Ig oOJk13hð4 kr{hÃn+¥ muS epT[rhoomunkhM_rr{dr\Fr".,wMf5gqeig dZQ\ðb1gijG\-norkmgihXGe4Em{t=b;¯ho$,m{¥ ª\XZÎ Y-
(337) hMrum[F\-r"[]yuhMq\3qZf5m{kXZhMg bb[dhMh[rªyuRY-wG[]qZ^Z\\ « l
(338) ef¾Ãg !ZM\-¥+1 r{rqkqeQr-1T^A¥ ¼ _ h¤hMgiqn©r%g 3Mbep1 r}y{m{e§KefkdhP\jk_|bqZ jMhM.nkd0gfr%\-[]1TwGq\¼Mn+dG«sm{myudZ\)\hkd qKlnMr)qr(m{ef\b-gidLeprunkqhkdMeirudmhkl
(339) dgimun©r/eihky{d©\\gpn+gi\)m{nefhMgpnFy{d¡\²gpZn+nkm{\gfef\-hMqZyud¡byl
(340) qZ\0\y{y{\yu bdg l
(341) +lqZ\ 1Ay{yu\-kLh>d)\l
(342) '
(343) \ m/gpw nwGbqZ§c\ dZ^efmuei_|hkd f5g\jgp1 nkh+hkdK_|l mu. eihk0 dI1T ¼]\
(344) hMm/dZwGdZqZb-\\ ^2_ g 1<k1¡¼ h[_.,0 ° 4¯e1.Z¥ Î hk[][]\S¯ nonoyumuei\dGm &.0UT ° 4 ¼M±´´-´
(345) ±@.,0P1¡¼ G«Lf5g 1<kmh;\
(346) m g >1#kE1¼hrunomuepr}hMdGm ^Z¥Jy{@.,\-0 hkru y;ru³ \- wGl 1¡qZm{\Pef¯sM.,\0p[]1 _\.dG¥m%@gig \-\-Wr(r(1ceieidZdZkh8bbjGjG_rnonogigi.ee0m{m{b-b- rjjr ° 1EUU ¤\\¼k«k
(347)
(348) mmdhkmmq$L$L(r ¥rIhMhMvdGmu\-m dZFhkdKdZrhMqZ.k0P\1noqL¯ kb_ y{eff5§bg \Wr1c¥ kht_*.0 ° 1¯ 1¼%\Jm .,0 ° 1 ¯ _ g W1ck1E ¼ h8_ ¥Âðd nkZZgie wGqnkdGm( gi\)[ []\Zy{efdlef;\wGqZ\l
(349) ef¾ã\-r{rqKr«=dZhMqr%b-ZqZeiruhkdKrwGqZ\ . 0 ° 1 .+¯ _ f5g P1Akh8_. 0 ° 4 . 0 1K.o¯s14¼rue . 0 ° :&u! 4 ¼j_4¯ _r. 0 1 ¼ .0 ° 1Kuo¯ _ f5g P1Akh8_.,0 ° 4 .0/1Auks ¯ 14¼rute .0T ° 4 j¼ _4¯ _" .,0 ° :u! \m . 0 ³ 1Kuo¯_ g +1Akt1 ¼ ht_. 0 ³ 4 . 0 ° 1Kuo¯{1 . re m. 0 ° :&u !Q4T¼!_ ¯_r. 0 1 ¼ .,0 ³ 1 4+ ¯ _ g +1Akt1 ¼ ht_.0 ³ 4 .0 ° 1 4+{ ¯ 1 . rte .,0UT ° 4 !¼ _4¯ _# .,0 ° :&u! gihkyr{wMq!\ .,0 ° _ g +1lkht_.0 ° 4 .,01 ¼k³ ¥ q=We .0UT ° ° 4Äj¼ _4¯ _" m.0 ³ ° :&u !M«eigs\-rm(nkl
(350) eig \\)kb-yuef§\y%wGqZ\ ' . 0 4 . 0 154os ¯ 1 .2_r. 0 1¯ 1K. KJ¥ ru@qZ\ei+no\-dGyumud\efr!\-yL' lnMr bmuqZef\-yb-K\-dTZ\Egpn +nogi\qZy\E¯¡¥ hkqKrL\-khMdr\dT\
(351) ®Ê\
(352) m©l
(353) hkdKrepby{\y0gf\rL\-q¹ nogfmu\-yudnomueik\r ³ . 0 ° 1Auks ¯ 1 . re m. 0 ° :&u !Q4 j¼ _£¯ _. 0 1 ¼ . 0 ° 4 . 0 _ g +1Akh8_ ..0 0 ³ 41¡s ¯ 1 . r te .,0UT ° 4 j¼ _£ ¯ _ .,0 ° :&u! nFyu\-ginomueihkdLZ\)yubl
(354) qZy{yu\-dl
(355) \ w (n oeidrueswMq\cgpnb§dZefmueihkd \)ginhMdl m{efhMd £dZhkqKr(K\-yu[]\
(356) mumu\-dMmSZ\b-qefy{\)wMq2\ ' .0 ° 4 .,021Auk¯ _rf5g +1Akht_r.,0 ³ 4r.0 ° 1¡½ks ¯ 1K. rue .0T ° 4 .0UT ³ 4Ä!¼ _ ¯ _.,01 ¼ ° ³ ° . 0 4 . 0 1 M¯ _rf5g +1Akht_r. 0 4r. 0 1 "+¯s1K. rue . 0 ° :u!}4 ¼!_4¯_. 0UT ° 4 . 0UT ³ .0 ° 4 .,021 M¯ _rf5g +1Akht_r.,0 ³ 1 4+{ ¯ 1 . u r e 7.0 ° 4 .,0X= : 4! 4Äj¼ _ ¯ _# m.0 ° :&u ! ° ³ ° ° . 0 4 . 0 154o¯ _rf5g +1Akht_r. 0 1 ½o{ ¯1. rue . 0T 4 3¼ _ ¯ _ 7. 0 4 . 0 =@: 4! Shkqrðginkeir{ruhkdrInkqgf\l mu\-qZyðgi\Sruhkeid\Mby{e§\-yzwGqZ\ f5g W1ckhÊr{n+m{eirnoefmzg eidZbjGnogiem{b U ¥ g W1ck1E ¼ hKMby{e§K\%noyðlhkdrub-wGqZ\-dGm g efdZb-jMnkgfe mu5b m$L\
(357) m%gi\muXZb-hky{Y[]\\-rmSb[]hkdGm{yubM¥ s\[]hqZgih¯8nonoyumu\-dnodGm &.0UT ° 4 ¼M±´´-´
(358) ±@.,0j1 ¼M«X.0 ° 4 ° .,0j1 .+¯1 ¼c\-rmr}m{yuepl m{\[] \dGm%rqKb-yuei\qZy .0 ° 1¯1 ¼c\
(359) m ginFv;hky{dZ\ruqZKb-yuei\qyu\\ g +1Akhs\-rm%nomm{\eidMm{\)KhMqZy%F¯ R .,0 :u! 4 ¼k¥ Î hk[][]\ ° .L= :&u ru8e bE\r}mKnoeiy m. 0 ° :&u !R 77.. 0 0 ° 1K 14¼ = :&u rue8bE\r}mei[]noeiy hkd hkvZmuei\dGm(§dnogi\[]\-dMm ³ ³ 4L= :&u ru8e bE\r}mKnoeiy f R 7m7m.,.,0 0 ³ 44 ..0/0/15 \ m !R .0 ³ 1Aurune bE\r}mKnoeiy 1 ",= :&u ru8e bE\r}mei[]noeiy .0 1 rune bE\r}mei[]noeiy u!V-.0/1Au FR u!V-.0/1 u!V-.0/154.
(360) 3 % "#L'%)
(361) M jMy{b-y{runkhkd@q]\)Z\yumudG\cXZmub-geihk \eiy{dZyJYb-F[]wGqmu\\-nog;/mueiwGhkqZhkd©\2qZry{7 qdZN efefO++mSno4qZdGdZm{;\\2==vK' []hMhyudZ(\F¯ rq4lK=b-¯¿yuei\qZruhky{ef\m¤;eidhkqZby3y{ei\f5qZg y ;1Kf5khg \-d©1<kmhkhÊdwGl
(362) qZmu\eigphkrðdEwGZqZ\/\cr¯¡hMef¥\-DãdMg m rN q«U ]Om\mhkdl¥UDãgK\/noqbmmu\-yuno[]yeil
(363) dZhM\dycrbgf\/wGqZZ\gfqKdGmr 7C7 O4 3==[]h= ¯ = 4!¯ _rf . N \. Shkqr%hMvmu\-dZhkdKr%noeidre ' ° .0/1K.os¯ 1 -¼ = :+¯ !rue .,0 ° ° :u! 4 j¼ _4¯ _.° 0/14¼ 5R C C7m7m.,.,0 0 ° 41¡s ¯ 1 -¼ = :+¯ ! rute .0UT 4T3¼ _ ¯ _" .0 :u! Âzd©\®;\l m{qnodGm%ginZef=epreihkd\dGmueiYy{\k«dhkqr(bm{novgfepruruhkdKrð§Kdnogi\[]\dGmSwGqZ\ ' ° ° ° ° ³ 5R .¼rrette .,.,0U0UTT ° 44 .¼j0U_£T ³ ¯_4_ ¯ .,_r0 .0/:&1 u!¼ hkq .,0 :&u!}4 ¼!_4¯ _r.0T 4 .0UT 14¼ @\muXZb-hky{Y[]\\-rm%nkefdKreºZy{hkqkbk¥
(364) 2 %*+ ( " % %$ "i %/~Sdy{\-nkrÁ¥ Î qZy{efjM\y ¥ W;/U I%V%G*[ m % X 5 Q E 2; > # & «"khMgfq[F\^kÀLhk; n 5 ` /%B# m/% . Q¥ HSnkym{qZdZjo¾}`chky{y{\SÁI\y{gpnoj«º MZ¥ "ß ^ % zhMdZjo¾ãkeidM\hkdjnod ¦n²t=dZ\ðx ¥+¢qZy{gf\rhMds¥o;Á WqUDsnoy{yn²tªnogijkhMyuefmuX[]r@nodnky{lXem{\-l m{qZy{\-rºhMy qZ~£[]hqZgpnoy[/qZgfmueiZgie l-n+mueihkd@¥ "!X/ m % X %# E ?&L # $ ,/ m % `E / « $XQL^ 'ß^M ( ^Z²Å«MMqZdZ\ GÅ¥ " & %/q=qdZjk¨h=hk¸ ei[6nkd`c\ynogpFÂ¥ q=hkv;\gi[nodsn¥ GªefjMemu¾Ãru\y{einkg[]h=ZqZginkyð[qZgm{efgfeplnomueihkdqruefdjr¸M\¨(¾Umuhkgi\ynodGmzhM[FeidZh Î )&% qÊ¥ Dã( d nkjk'}\rª/%'M
(365) ²@
(366) Åo ( M² ÅoÀZ%-<¥ Dã )ÂzÂð*Â + ,Î hk -[]Z&qEm{ \&y q=h ml
(367) % ei\
(368) m} tM# «K^oVM% Z.M ¥ /
(369) % .$%U
(370) %
(371) *@
(372) KE #/'}/%'@
(373) k«¤MhkgiqZ[]\©^Z« " % zqmnoe ) nZ¥ ~8ruef[]Zgie§K\-ÄnoylXZefmu\l muqyu\Lhky[]hqZgi* h 7m.,0#4 -¼ =[/qgm{efZgiepln+m{efhMds#¥ 0!X/ m % X %1;%G*[
(374) « Å w ' kM ( MGÅ« ) noylXE MZ¥ " $ %/~Sgy{\-]K¥ ) \dZ\ \r«Mx nkqZg Î ¥o+nod % hMy{r{lXZhom«+nod5 qlhommð~/¥oÁznodKr}m{hkdZ\M3¥ 2 L %'%)(%-4* *#
(375) 5Y * +% ,/ *6U+¥ Î Î xðyu\rur-« GÅ¥ " &À %x \
(376) m{\}y z¥ ) hkdGmujMhk[]\y{tk¥ ) hqginky¤[qZgm{efgfeplnomueihkd¨%efmuXZhMqmðm{yuepnogÊei=eiruefhMds7¥ & 8X m % 9V%G*[ m % «=k }+Å+L ' $Z ( $M^k«Ê kL $Z¥ "è-Å %/q=epZei¸+n¢¤\-yudK n :ªyr-« s\}|}gpn)¢(n+m{efdn«k¢(noyum"xðyu\-dZ\\-gQ«GnodFkh=hGr Áznod\-¨nkgfi\kE¥ HSnoy¨(noy{\ðei[]Zgf\-[]\dGm{nomueihkd]hoÊn ) hkdGmujMhk[]\y{t [] &hG*=qZ% gpno y [/FqZ¥ gfDãmuÂzeiZÂzgief \-y%Î efd hM[FnFrqtmur\-muhky giq=eplShnol
(377) eiy{\
(378) ym}n²tktk«8¥ ^kDãk;d MZ'}¥ /%'
(379) @
(380) U%-! )<=B )>& && % #?' /# # I#A@ $
(381)
(382) '} %)
(383) @ " & %Gcno[q0nkZXno¸=y{eiruXZdnkdnod zhMdZ9j zqnods¥ h+k\-g;nkZZy{hMnMlXZ\-r m{h/m{XZ\ªZ\-ruefjMd0hos3Á qUD¤ ;q[/qZgfmueiZgief\-y{r-B¥ + , -! X I % U% CY / [
(384) ED+/ F #&%% $ , #nE #G'} %)
(385) @ o« - ' $k^ ( $GÅ=«MGnod=qnkyut© G^¥ " &%/%\muIh H;ei[F[]\-yu[nodds¥ZÂ l
(386) ei\dGmz3Á qUDef[]Zgi\[]\dGmn+mueihkdFho;[]h=ZqZgf5 h 7.0AJ© ¼ =@nMZefmueihkd]nod[/qZgfmueiZgie l-n+m{efhMds¥ Dã*d '}/%'@
(387)
(388) & & %-! )9<LKX 8>+ , &G*=% %C;%G*[ $; )
(389) «ZKnojk\rc- $o ( ÀGÅ=w¥ DãÂzÂz ΠhM[]Zqmu\-y q=hl
(390) ei\
(391) m}tM«º Z¥ -.
(392) Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN Unit´e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unit´e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. ´ Editeur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)
(393) ISSN 0249-6399.
(394)
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