Opérateurs itératifs de multiplication-addition modulaire pour FPGA
22
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Op´erateurs it´eratifs de multiplication-addition modulaire pour FPGA Jean-Luc Beuchat et Jean-Michel Muller. No 4937 Septembre 2003. ISSN 0249-6399. ISRN INRIA/RR--4937--FR. ` THEME 2. apport de recherche.
(3)
(4)
(5)
(6) !#"%$&' !'"()!*
(7) #+ ,-!#
(8) .0/2143 576987:<;= >?A@B6C>?DE8GF6HF57698G:; IKJL?D6CMNIO>MPML6CQ RS&T UWV#X#Y[Z]\ ^`_aVcbedgfH_eh _aVib VjNhiklbmhn&bporq9U's7dgbe_etCn&V uwvxdlyzV j {Nvr\i^`kl_evxV |}kg~&~GdHvj}VcvxVihS`V vhS&V^'lC g9Y[9V ~&jrV UWs&vxV'XlHgcYiB~`kgfgVio. % i&lH RS&_moE~`kg~GVivwVioxhvx_esGVoEoVigV vklb9_eUW~&vrdHV UWV ^Cjodl7kg^W_jxV vkjr_egV kgbafHdgvx_jxS&UdgvEUWdn&bmklvEUn&bajr_e~&ba_mh kljr_edg^Wdgvx_efg_e^`klbebaq ~&vxdg~7dHorVis9qgVidg^&fkl^` n`vrbeViordg^p ¡{£¢`vxojUWd_a¢Gh kljr_edg^¤dl jrS`V vxVih n&vxvrVi^`hV*vxV bmkjx_adH^klbebad¥}o'n`ojrd_aUW~&beV UWVi^Hj*k¦nGoV U'n&bajr_e~&beq¦kl^`¦kH&¦n&^&_aji §RS&V ^p¨7¥©V'oS`d¥KS&d¥ªjxd%vrVn`h VcjrS&VBh_evxh n&_aj§klvxVikWs9q«kW¬kghjrdgvNjz¥ d*¥ S`V ^¦jxS&V'dH~GVivxkljrdgv dg7Vivxo}jrS&V ~GdCoror_es&_abe_ajzq'jxdhS&d9dHorVjrS`VNUWdn&bed'klUWdg^`f#k#oV j ®¯±°r® ²g° ³ ³i³ °r®«´± E{O^&V ¥µ_ajrV vkjx_aHVNklbefgdHvr_ajrS`U¶U*kl·9_e^&fcjrS`V}_aUW~&beV UWVi^Hjkjx_adH^ dl UWdn&bmklv'V ¸~GdH^&V ^Cjr_mkjx_adH^¡Vikgor_eV v#_mo#VigV ^Cjxn`klbebaq_morh n`ororVip ¹&dgvºi»¼½s&_aj'^9n&U's7V voi¨dHn&v#dg~7V vkjxdgvo ~GVivdHvrU¾dgv_a^Gozjkl^`h V ¿ UW_ebabe_edg^`oBdl§dg~7V vkjr_edg^Go'~7V v*orVihdH^`ÀdH^¤kAÁN_evjxV¸9¼PÂÃViC_mhV ¥ S&_ebaV dg^`baqvxVitCn&_evr_e^&fÀorbe_eh Vioi ÀRS&ViorVdH~GVivxkljrdHvxo'_a^9gdHbaHV oU*kgbabÄjxkls`baVo#V ~7V ^G_a^`fdg^jrS`V%vrVtCn&_avxVi¡orVj'dg UBdn`ba_P {£ozjxvxkg_afHSHjrdgvx¥©kgvx-kl~&~`vrdCkghS_eo jxdAkln&jrdgU*kljr_mh klbebeqfgVi^&V vkjxV%k Á§Å§Æ§ÇµVorh vr_e~jr_edg^AdlEjrS`VUWdn&bmklv U'n`bjx_a~&be_eV v kgh h dgv_e^&f*jrd ® ¯ °® ² °i³ ³i³°® ´ cRS`klj Vo_efg^AUBV jrS&ddHbadHfgq«kgoxon`UBVo}jxS`kj jrS&VNUBdn`ba_`kgvrV·9^&d¥ ^Wk]~&vr_edgvx_&kl^G¥©dgvx·odg^&beq#dHvwoU*klbeb`orVjodlÈUBdn`ba_P ÉÊ^BdHvx&V vjrd#k±Hdg_m#jxS&ViorV}vk±¥ s`kgh·oi¨¥©V ~&vxdg~7dHorV S&V vxVckl^¦klbefgdHvr_ajrS`UÃjrS`kljNklbebad¥}o jrdWs&n`_abm jrS&V]jkls&beVc¥ S&_ebaVc¥©V ~GVivdHvrUjxS&V ¢`vozj}ojrVi~`o dljrS&VcUWdn&bmklv Un&bajr_e~&ba_mh kljr_edg^p ˦ÌCÍÈÎrÏ'Ð ÑH,Ò dgUW~&n&jrV v}kgvr_ajrS`UBV jr_mhl¨UWdn&bmklv Un&bjx_a~`ba_mh kljr_edg^p¨`¹u Z {. ßpà±áãâÖäPåÊæ äáãâ<çèãâLéçêçáãèëçì±èãåwçâç©íLåzâLårçí¬î½àcí¬åzïgé+íäéð`äLà±åEñçìlé+íLçxäPé+áãí¬åÄò±åwèmó ôöõðeé+í¬÷}çxäLáãøiù±åò±ù ú&çíLçèãèãûèãáãâL÷ åüý+ýþ`ÿ. ÓmÔÖÕr×½ØiÙÕ ÚÛe×½ÜPÝÞ
(9) . Unit´e de recherche INRIA Rhˆone-Alpes 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN (France) T´el´ephone : 04 76 61 52 00 - International : +33 4 76 61 52 00 T´el´ecopie : 04 76 61 52 52 - International : +33 4 76 61 52 52.
(10) -
(11) £
(12) + +« © !#"%$`# !#"()!*
(13) '+ ,!#
(14) .¤/2143 #Ò Vj©kgvjx_eh baV§~&vx\iorV ^CjrV jxdgnj kls7dgv*_egV vorVioÄkgUW\ be_adHvxkljr_edg^`o n`^klbefgdHvr_ajrS&UWV}_ajr\ivxkljr_aV§U'n&bajr_e~&be_ehikjr_edg^UWd9&n¼ bmkl_evrVc~`vrdH~GdCo\ V ^¡ g C#~`kgvNgVidg^&fWV jN n`vrbeViordg^p }^&V'or_eUB~`baVcUWd_a¢Gh kljr_edg^¦VcbmkWvr\hn&vxvrVi^`hV ~7V vxUWVj} b _eUW~&bekg^Cjxkjx_adH^¦ n`^&V Un&bjx_a~`ba_mh kljr_edg^¼Êkg&&_jx_adH^UBdn`bekg_avxVg
(15) }dHn`o#UWdH^HjxvrdH^`ocVi^`on`_jxVhdHUWUBVi^Cj'vr\n&_evrV% n&^À¬kgh+jxV n&vBV n&¸bmk«jxkg_abebeVnh _avhn&_aj bedgvortCn&V* b dg~7\ vkjrVin&vcdlÈvxV*baV%hS&dH_¸¡nUWdn&bed¦~`klvxUW_wn&^V ^GoViU's&beV*®¦¯l°®«²l°i³ ³i³°®¦´
(16) }dgnGo#~&vxdg~7dHordg^`o]n&^^`dgn&HV bÄkgbafHdl¼ vx_jxS&UWV'¬kHh_eba_ajxkg^Cj§bmk%vx\iklbe_morkljr_edg^Vb V¸~7dg^&Vi^Hjx_ekljr_edg^UWdn&bmkl_evxV'Vj ~&vx\iorV ^CjrdH^`oNtCn&VibetCn&VoNvx\iorn&bajxkljxoN&VBorq9^HjxS&TiorV'~7dgn`v]Vo h _avhn&_ajxoc¹u Z {£VWbek¬klUW_ebabeVWÁN_evjxV¸-&V N_eba_e^¸Ö WudHn&vcVio ^`dgU's`vrVocV«ºi»s&_jo ¨baVo dg~7\ vkjxV n&vo \ HV bedg~&~7\io]~7V vxUBV jjxV ^Cj ~`kgvWV¸V UW~&beV¦ V 7Vh+jxn&V v ¿ UW_abebe_adH^`o* dg~7\ vkjr_edg^Go bmk-orVih dg^`V V ^0njx_abe_eoxkl^Cj*n&^&_mtCn&V UWVi^Hj¦«jxvxkg^`hS&Vioi Ò Vo*_a7\ivrVi^Hjo dH~G\ivxkljrV n`vxo]^&\hVioxor_jxV ^Cj jrdgn&jrVdH_eo'VW~GV jr_ajrVio#jxkgs&baVoc\ ~7V ^G&kl^Cj#n¡UWd9&n&bad¦® hS`dg_mo_wkgn-UWdgUWVi^Hj#V* b \ihvx_ajrn&vxV*nhdV Á]ÅNÆ]Çw `{}¢`^¦V vrViUB\_eV v WhVj _e^`hdH^9g\ ^`_aVi^Hj¨^&dgnGo ~`vrdH~GdCodH^`o©n&^&V UW\jrS`d9&V]vx\ih n&vo_egV n«h klbmhn`bpVio©jkls&beVio o V 7Vh+jxn`kl^Cj ~`kgvxkgbabeT beV UWVi^Hj}kgn¸~&vxV UW_eT vxVio©_jx\ vkjx_adH^`o VcbmkUn&bajr_e~&ba_mh kljr_edg^ UWd9&n&bekg_avxVg Ð i Î {Nvr_ajrS`UB\ jr_mtCn&V#ViodHvx&_a^`kljrVin&vxoi¨9U'n&bajr_e~&be_ehikjx_adH^ UWdn&bmkl_evxVg¨&¹u Z {.
(17)
(18)
(19) "!#$ % & ('))* +, -+. (+.0/21-34576*88 9:9
(20) ;<=7
(21) 2)$ -+)*
(22) ,! > -'? @.!- -A 79 ,!- @ B *CD
(23) E'FE G @., % )F)$ @ >'EH'IG,!! -' J,BK'LMENI,!#)DF'O > *EH%PQEH +%7EH'%G,!! (' ?+JRSTB/<. " <!*
(24) w !#" Ç<kNU'n&bajr_e~&be_ehikjr_edg^'UBdn`bekg_avxVÄ_e^CjrV vx9_aVi^Cji¨V ^CjrvxV kgnjrvxVioi¨`kl^`obaVokl~&~&be_mh kjx_adH^`o<&VÄjxvxkg_jxV UWV ^Cj&nor_afH^`klbHjr_evxkg^Cj~`kgvjx_CV©bmk vrVi~&vx\iorV ^Cjxkljr_edg^%UWdn&bmkl_evrVcVo©^&dgUs&vrV&o W#XZY\[]K^,_`Yba>_Dc?d Y egfih[j-Y\c>k Vj}&kl^Go ~`ban`or_eV n&voklbefgdgvx_ajrS&UWVioVch vrq9~jxdgfgvkl~`S&_aVCl m,nP Ò VjrjrVcdg~7\ vkjx_adH^ kgvr_ajrS`UB\ jr_mtCn&Vck~Gklv}hdH^`or\itCn&V ^Cj¬kl_aj b dgs&yzVjNVc^&dHU's&vxV nGoVo \ jrn`&Vio dH^Hj}Hdg_mh_ptCn&VibetCn&VoV¸V UW~&beVio&o V vrViUB~`bekHhV v§baVoNdH~G\ivxkljr_edg^`oNUWd9&n&bad® ~`klv Vio prq dg^CjrfHdgUWV vxqk~&vrdH~GdCo\n&^AkgbafHdgvx_jxS&UWVdH^Hj]baV~&vx_a^`h _a~7VWhdg^Go_mozjx hiklbmhn&bmoEUWdn&bedcn&^&V}~`n&_eoxoxkl^`h V&V}bek sGkgorV}&kl^`owbekHtHn`V bebaV ordg^CjwvrVi~&vx\iorV ^Cjr\obaVoEdg~7\ vkl^`&Vi7o l ¿ n½ gÇ<k vx\\ivrVi^`h&V lëF n&dHn&vx^&_j n`^&VcsGdH^&^&V s&_es&be_adHfgvkl~&S&_eV vrVibekljr_egV kln¸_aUW~&bmkl^Cjxkljr_edg^`oU*kljr\ vx_eV bebaVo©&V#hVjNkgbafHdgvx_jxS&UWVg V s
(25) tvu0ºcVgj s*t w ºH¨¬kgh _abe_ajxkl^CjNbmkWvx\iklbe_morkljr_edg^«V#bmk p uÄban`or_eV n&vo}hS&VivxhS&Vin&vodg^Cj}\jxn`_e\'VioNUWd9&n&ba_~`klvrjr_mhn`ba_eV vo ¨&h dgUWUWC Un&bajr_e~&ba_mh kljr_edg^ UWd9&n&bekg_avx?V WHdg_ev ~`klv V ¸9ViUW~&ba"V l `¨Ê, n k+ ® y| }7Vioj<_modHUWdgvx~&S&VEkln fgvxdgn`~GVwkH&_ajr_aD ygzDx~® ucº |+yleg¨ p Ç<dgvortCn&VbeVEUWd9&n&badN® Vioj<~&vxV UW_eV v¨+baVwfHvrdHn&~7VEU'n&bajr_e~&be_ehikjr_a xKybz{
(26) nP GÉÊbVozj]kl_e^`o_~GdCoror_es&baVV'vxV UW~&bmkgh V v}bmk*U'n`bjx_a~&be_mh kjx_adH^UWdn&bed%®¾~Gklv} b kH&_ajr_edg^UWdn&bed% ® uµºg GÇ _e^`h dg^9g\i^&_aVi^Cj &V h VjrjrVcUW\jxS&dVcvr\o_mVc&kg^`obaVo s&bedh o VcUW\ UWdH_avxV§_e^`_mor~GVi^`orkgs&beVio}kgn hiklbmhn&bpVc b _eordgUWdgvx~&S&_morUBVH Ƨkg^`oh Vj«klvrjr_mhbeVg¨Ä^`dgn`o\jrnG_adH^`o Vioklbefgdgvx_ajrS&UWVio%s`kgor\ioon&vbaVorhS`\ U*kÀVAÅNdgvx^&V vVj«h klbmhn`bekg^Hj%_ajr\ivxkljr_egViUBVi^CjbaV ~&vxd9&n&_j}UWdn&bmkl_evxV]V#&V n¸dH~G\ivxkg^`Vo
(27) Ndgn`o~&vxdg~7dHordg^GowjxdgnjN kls7dgv _aHV voVoklUW\ be_adHvxkljr_edg^`o© n&^&Vc_ajr\ivxkljr_edg^ ~&vrdH~GdCo\iV ~`klv§gVidg^&f*V j§©n&vrbeViordg~^ lãX n~7V vxUWVjrjxkl^Cj§ V 7Vh+jrn`V vNn`^&V#U'n`bjx_a~&be_mh kjx_adH^¼Êkg&_ajr_edg^«UWdn&bmkl_evrV jxdgnj§V ^¦vx\in&_moxkl^Cj}bek*orn&vr¬kgh V Vb dg~7\ vkjxV n&v_e^&_jx_ekgb W¬orVihjr_edg^BX& a k+
(28) Ndgn`o\hvx_aHdg^`oV ^`orn&_ajrVn&^&V©^&dHn&gVibabeV _ajr\ivxkljr_edg^'¬kgh_ebe_jkl^Cj b _aUW~&bmkl^Cjxkljr_edg^W dg~7\ vkjr_edg^Go hdHUBUWV]b V¸~7dg^&Vi^Cjr_mkjr_edg^UBdn`bekg_avxCV WLoVh+jr_edg^«X ã*X kwVj~&vr\oVi^Cjrdg^GoÄtCn&V bmtCn&Vio©vx\iorn&bajxkjoÄdHsjrVi^9n`oÄ~7dgn`v©Voh_evxh n&_jo ¹u Z { V bek¬kgUB_ebebaVcÁ§_avrjrV ¸C¼½Â V N_ebe_a^¸ W¬orVihjr_edg^« k+ Ò Vio~&vxV UW_eV vo<dH~G\ivxkljrV n`vxo<^&\ih Vioxo_ajrVi^HjjrdHnjrV dg_moV ~7Vjx_jxViojkls&beViIo W¬&V ¿ UWdljo&V bmk}jkl_ebabeVViodg~7\ vkl^GViMo kp\i~GVi^`&kg^Hj nWUWd9&n&bad ®£hS`dg_mo_P Ò V ~`klvklUWTjxvrVdH_jE~`kgvwhdg^Go\tHn`V ^COj jrvxV hdg^`^CnWklnBUWdgUWV ^CjÄVbmk Vioxhvx_a~&jr_edg^BÁ§ÅNÆ]ǦnWU'n`bjx_a~&be_eV n&v tCn&_g^&Vwdg^`hjr_edg^&^&Vivxk tCn&VÄ~GdHn&v<n&^`Vwn&^&_mtCn&VwklbeV n&vVw® Ò V jjrV h klvkghjr\ivr_mozjx_etCn&V~7V njo k±g\ivrVivÈjrvxdg~ vxViojrvx_ehjr_egVw&kg^`ophVivjkl_e^&Vio kl~&~`ba_mh kljr_edg^`oi i _abHVioj<~7dHoxo_es&beVwVEfg\i^&\ vxV vÖbeViopjxkls`baVophdHvrvxVior~GdH^`&kg^Hj n&^cVi^`orV U's`baIV ® ¯ ° ³ ³i³°® ´ &VwUWd9&n&ba_P¨ _ebg^`dgn`oporV Us&beV ~&ben`o#_e^Cjr\ vxVioxoxkl^Cj#V%\ HV bedg~&~7V vcn`^^&dgn&HV bwdH~G\ivxkljrVin&v#dg^Cj#beV*UWd9&n&bad¦® Vioj#b n&^&VViocVi^Cjrvx\ Vioi
(29) }dHn`oc~&vxdg~7dHordg^Goc&kl^`o bek*orVihjr_edg^«*n&^&VcUW\ jrS&dV#vx\ih n&vo_egV V'h dg^`ojrvxn`h+jx_adH^ &Viojkls&beVio}o VÈVih+jxn`kl^Cj}~`kgvxkgbabeT beV UWVi^Hj}kgn¸~&vxV UW_eT vxVio_ajr\ivxkljr_edg^`o V b klbefgdHvr_ajrS`UBVcV Un&bjx_a~`ba_mh kljr_edg^«UWd9&n&bekg_avxVg |}kl~`~GVibadH^`octCn&V%beVio#¹u Z { h dg^`or_m\ vx\ioc&kg^`ochVWjxvxk±kg_abwordg^Cj#VioxorV ^Cjr_eV bebaViUWV ^Cj'hdH^`ozjx_jxn&\io'V*s&bed9hiocbedgfg_mtCn&Vio ~`vrdHfgvklUB¼ U*kls&beViCo W Ò Ç<Odgrn ,F*_De*d Y>*]K 7@M k¨7 n&^`V#U*kjrvx_mhV#V#vxdgn&jxklfHVchdg^&¢`fgn&vkls`baVH¨`Vcs&bedh oN&V#UW\ UWdg_evrVcV j§V#s`badh o V ^Cjrvx\ ,V ordgvrjr_eVg Ò SGkgtCn&V Ò Ç¤Vozj_eC_mor\NV ^ V n&¸Bjxvxkg^`hS&VoÄhdH^CjrV ^Gkl^Cj©hS`kHhn&^`VNV n&¸WfH\ ^&\ivxkljrVin&vxo V}dH^`h+jx_adH^`o©V§tCn`kljrvxV klvx_ekgs&baV>o W¬Ç NRªdgn F
(30) -_Fj(d\Y k+¨7V n&¸ \iba\iUBVi^Cjxo}V'UB\iUWdgvx_eoxkjx_adH^V jNV#bmkWbadHfg_mtHn`V#kg&_ajr_edg^`^&V bebaVc~7V vxUWVjrjxkl^CjNn`^&Vc_aUB¼ ~&bmkl^Cjxkljr_edg^%V *hikgh V§ kg&&_jx_adH^&^&V n`vxo 'vxVjxV ^9n&V§~&vrdH~`klfH\ VH Ò V V vx^&_eV v©~GdH_a^Cj Vozj_eUB~7dgvrjxkg^Cgj oCV ^~&vxkljr_mtCn&Vg¨C_abÈ^ Vioj kgn`hn`^ _a^Cjx\ v jN njr_ebe_eorV v}orn&v}¹u Z { Vo©jrVhS&^&_mtHn`Vio} kg`_jx_adH^vxVi&dg^`&kg^CjrVchdHUBUWV b kH&_ajr_edg^
(31) eeh,([ ,,Y
(32) +« © !#"*$`# !#"()!*
(33) #+ Ç kl~&~`~&vrdhS&V©kgdH~jr\iVÄ~`kgvHV dH^&fNV j©n&vxbeViordg^'h dg^`or_eojrV }\hvx_avxVÄbmk§U'n&bajr_e~&be_ehikjr_edg^UWd9&n&bad]®VV n¸#dH~G\ivxkg^`Vo<Vi^Hjx_aVivxo V j odHn`o©bekBdHvrUWVc n&^¦orhS&\iU*kV#ÅNdgvx^&V &v lãX no ® x+xz³i³ F³ x+x+x t$ ¯ D|Osw t$ ² |0sw t$2 |OFsw ³ ³i³ |Osw i¡ |9UWd® UWd d*¢~£4¤ +° ¥ª®,V jc®,Vozj n&^^&dgUs&vxVvV ¦Às&_ajxo]jxV bEtCn&§V s
(34) t$ ¯ wª?º ¤ ¨ ® ¤©s*t"u ºg ÈÇ<kU'n&bajr_e~&be_ehikjx_adH^AUWdn&bed®~7V nj kl_e^`or_po VÈVihjrn&VivNV U*kl^&_eT vxV vr\hn&vo_egV]V ^¦\ ¢`^&_moroxkl^Cj ª?« ¦2¬®£BVj}Vi^¦h klbmhn`bekg^H>j o ª?« ¦{u°¯±¬2x#s*ª?« ¦{u°¯²w º ¬$w t$2³ |9UWd® WzF k U. V. .
(35) ~7dgn&vg¯ klbebekg^Cj§Vº§¦E 7ÉÊbVozj§¬kgh_ebeVVg\ vx_a¢`V vNtCn&Vvª?« £
(36) ¬Vozj]s&_aVi^\ fCklb ©UWd®- Ò VjrjrV'UW\ jrS&dV_aUW~&be_etCn&VjrdHnjrV dg_mo n&^`V#kg&_ajr_edg^«UWdn&bed*®4badHvxo &V#hS`kgtCn&V _ajr\ vkjx_adH^p `9_p b dg~7\ vkl^`&V %VozjNojrvx_mh+jrViUWV ^Cj}_a^&\ vx_aVin&v W®-¨^`dgn`o}&\in&_mordg^`o Vcbmk vxV bmkjx_adH^ Vcvx\ih n&vxvrVi^`hvV WzF ktCn&?V o s
(37) ª?« ¦u~¯2w \º ¬$w tD³ J¤ sgx® u0Fº |w® u0b º ® u Ç kH&_ajr_edg^UWdn&bedB® vxVitCn&_eV vrj ~Gklv hdH^`o\tCn&V ^Cj©&ViohdHUB~Gklvkl_modH^`owk¢G^V \jxV vxUW_a^&Viv o _ab7¬kgnj vrV jrvkl^`hS&ViBv £`¨H® dgn s® s*ª?« ¦CuJ¯w\º ¬Fw È C{}¢`^V§vr\n&_evrVNbek#orn&vr¬kgh V Vj beV}jxV UW~`o©V§h qhbeVNnh _avhn&_aji¨CgVidg^&f'Vj ©n&vxbaVodH^*~&vxdg~7dHorV ^CjwbaV]h kgbeh n&b n&^W^&dHU's&vxV 7V ¦Zt$w³ º©s&_ajxowh dg^&fHvrn >s
(38) ª?« ¦gu"¯
(39) wA\º ¬w t$³ #UWdn&bed]® Ò dH^`o_m\ivrdH^`on&^WV ^Cjr_eV v c7V ¦Zw s&_ajwVjÄ&\¢`^&_moxodH^`o oxk~Gklvrjr_eVBSGklnjxVV jcoxk~`klvrjr_eVsGkgoxoVjrVibabeVioctCn&V w&_a7 s
(40) t¦V j
(41) EUWd s
(42) tÈ ÈuEklvch dg^`or\itCn&Vi^Hj ¨ s
(43) t®w VjcbeV ^&dHU's&vxVc>V ¦§w º s&_ajxo i s
(44) t¦UWd® w
(45) AViojNhdH^&fgvxn WUWdn&bedW®- `Ç<dgvortCn&V]beVc^&dgUs&vrV Vcs&_ajxo V Vozj ~7Vjr_aji¨ b V ¸~&vrVoror_edg^ s
(46) t UWd® oV%hiklbmhn&beV*\V %hikghViUWV ^Cj «b kl_mV% n&^&VWjkls&beVg Âw^ÀorVBsGkgoxkl^Cj'on`v#hVio'hdg^Gozjkjxkljr_edg^`oi¨pHV dg^`f V j ©n&vxbaVodH^dH^Cj~`vrdH~GdCo\]b _ajr\ vkjx_adH^ orn&_ekl^CjrV#&kg^`&o lãX no « ¦{u~¯#¬ s
(47) ª?« ¦u~¯2w \º ¬UBd7 s t wIx#s
(48) ª?« ¦{u°¯²w º ¬_e% s t | } « ¦{u°¯±¬2 « ¦{u~¯#¬Dw WLX k t$³ ª?« ¦ u~¯#¬ } « ¦ u°¯±¬iUWd7 s
(49) t>w Ix } « ¦{u°¯±¬&_a7 s
(50) tD| k±HVi?h ª?« ¦2¬IQ£«V j Ix `|C x s
(51) t|UBd® Çk ¢`fgn&vxV¦ik¦&\ihvx_aj'b klvhS&_ajrVh+jrn`vrV% n&^\ jxklfHV% _jx\ vkjr_edg^À n&^Un&bajr_e~&ba_eV n`v UWdn&bmkl_evrV hdH!^ nÀ kl~`vrTo'bmk¦vxV bmkjr_edg^À&Vvx\ihn`vrvxV ^`h V WLX k+ gVidg^&fAVjB©n&vxbeViordg^dg^CjW\ HV bedg~&~7\%beV n&vWklbefgdHvr_ajrS&UWV%~7dgn`vBVo _eUW~&bekg^Cjxkjx_adH^`oÁ§Ç9ÉVjwdg^Cjw~`kgvwhdH^`o\tCn&V ^Cjdg~jx\ ~GdHn&vEn&^&V vxV ~&vx\iorV ^Cjkjr_edg^BVio^`dgU's`vrVoV ^ M,eeh,([ ,,Ywkl¢`^B&V vrVozjxvrVi_a^`&vrV beV©jrV UW~`oE&Vh qhbeV&V© b dg~7\ vkjrVin&v Çk§U*klyzdgvx_jx\ VoE¹u©Z { kgh+jxn&V bebeV UWV ^Cjw_mo~7dg^&_es&beVio_mo~7dHoxkl^CjwV©badHfg_mtCn&V vkl~`_eVVozjx_a^&\iV bmk fHViojr_edg^#VopvxVjrVi^9n&Vioi¨^&dHn`opnjx_abe_modH^`o<Vo<kH&_ajr_edg^&^&Vin&vo vxVjrVi^9n&Vw~&vxdg~`kgfg\iV~7dgn`v<bmk vr\klbe_eoxkjx_adH^cVioh_evxh n&_jo WdH~G\ivxkljrVin&v w¶n«bekg^&fHkgfgV§Á§ÅNÆ] Ç k n. y x n−i P[n−i+1]. n bits de n+1 poids faibles. n bits de poids faibles. 2. n n+2. Table. T[n−i]. n+2 n+2. P[s−i+1] 2 ws−i Bit de poids. n+1. T *[n−i]. 2. n. y x s−i. P[n−i]. n bits de poids faibles n+1. T[s−i] 2. P[s−i] Table. faible. Table n. n. ϕ(k) = (k*2 ) mod m (a). ϕ(k) = (k*2 ) mod m (b). "$#&%('} }^À\ jxkgfgV _ajr\ivxkljr_edg^À n&^Un&bajr_e~&ba_eV n`vUWdn&bmkl_evrV~GdHn&v¹u Z {' W¬kk#{}befgdHvr_ajrS`UBVV HV dH^&f¦Vj*©n&vxbaVodH^p BW¬sik {NbafHdgvx_jxS&p UWV &V ) _aU V j§9dgs7V beU*kl^«UWd_¢G\g ) _eU Vj dgs7V beUWkg^dg^Cj]~&vrdH~GdCo\'n&^&V'~`vrViUB_eT vxV'kgUB\iba_edgvkjx_adH^¦Vh Vjcklbefgdgvx_ajrS&UWV#&kg^`oNbmk%vx\\ivrVi^`hVJl
(52) nP
(53) }dgnGo§dH^&^&dH^`o _mh_p b _jx\ vkjr_edg^ V ^«s`kHoVcVin¸%jxvxkH_jx_adH^&^&VibabeVctHn`_<oVc\n&_aj ¬kgh _abeV UWV ^CjNVcbmkBgVivxor_adH~^ M,eeh,([
(54) YN\¢`^`_aVc&kg^`>o l , no * « ¦Ju~¯±¬2 s
(55) ª?« ¦Ju~¯2w \º ¬$w ª?« ¦u~¯±¬2 « ¦{u°¯±¬UBd7 s
(56) t>wt$%2³ x « ¦u~¯±¬_eI s*tD| d¢§ª?« ¦2¬ £& Ò V jjxVNkl~&~`vrdhS&V~7V vxUBV jÄbmkcon&~`~&vrV oror_edg^* n&^%kg&&_jx_adH^&^&V n`vw~`klvEvkl~`~GdHvj bmkcodHban&jr_edg^*VNHV dH^&f Vj ©n&vxbaVodH^p ÉÊb^ Vioj}jrdHnjrV dg_mo}~`kHo}^&\hVioxoxkl_evrV#tCn&V# b dg~7\ vkl^`V odH_j§n&^^&dHU's&vxV# V ¦¡s&_ajxo]ozjxvr_mh+jxV UWV ^CjN_e^\ivr_eV n&v *® GÆ]kl^`oNbek%orn&_ajrVH¨ ^&dHn`o]hdg^Go_m\ vxV vxdg^Go tHn`V ViojNn&^AVi^Cjr_eV v§&,V +s&_ajxoi
(57) NdljrdH^`oN&V'~&ben`o]tHn _ebVozj§jrvx_a9_mklbV'UWd_a¢`V v]hV jjrV^&dgn`gV bebeV#vrVibekljr_edg^ Vcvx\ih n&vxvrVi^`hV k¢`^«V#hiklbmhn&beV v x &w -b|9UWd ® o * « +u°¯±¬ s
(58) ª?« +7u~¯2w \º ¬$w /. >w- . 2³ ³ W¬ k ª?« +Zu°¯±¬ « +7u°¯±¬iUWds
(59) t>w%x « +7u~¯#¬±_e%s
(60) t| . . . .
(61) d*¢- Wö¢`fHn&vxV# sik CÇkc~&vx_a^Gh_e~`klbeVN_%hn&bajr\]hdg^Go_mozjxV '\ jrVivrUW_e^&V v baV}^`dgU's`vrVN&V}s&_ajxo©V « +0u¯#¬7k¢`^%V]_eUBVi^`or_adH^&^&V v hdHvrvxVihjrV UWVi^Hj bmkBjxkgs&baVc_eUW~&bmkl^Cjxkg^Hj bmkBdg^`hjr_edg^ Ix | `uwn`_eoxtCn&V £"¤ « +u°¯±¬UBd7s
(62) t{¤rs
(63) tvu0ºcVjZ£"¤ Ix « +7u¯±¬_e%s*t|I¤ ® u ºg¨7^&dgn`o \n&_modH^`o]tHn`Vvª?« +bu ¯±¬Vozj n&^A^`dgU's`vrVBVC¦ wKº's&_ajx o]VjctCn&V"£ ¤ ª?« +gu¯±¬¤ s*tCw¤®u s ÈÇ<kvrVibekljr_edg^AV vr\hn&vxvxV ^`h vV WL k©^&dHn`o~GVivrUWV jNklbedgvo©&V hiklbmhn&beV v n`^&VcsGdHvr^`V orn&~7\ vx_aVin&vxV]~7dgn`v « +7u°¯±¬o £"¤ « +7u°¯±¬ ¤ s t ¯ w ® u Âw^orn&s`ojr_ajrn`kg^Hj bmkB±kgbaVin&v U*k¸_eUWkgbaVcVc® `kl^`o b \itCn`kjx_adH^ h _¼ÊVororn`oi¨C^`dgn`odHsjrVi^&dg^`o©¢G^`klbeV UWV ^C j o £"¤ « +7u~¯±¬ ¤ s t ² w s t u W k Ò VjrjrVNs7dgvx^&V}_e^`_mtCn&VNtCn&V « +0u¯±¬&ViojÄn&^%^&dHU's&vxV} V ¦>w s&_ajxoi HÇk ¢`fHn&vrVNXgkc_ebaben`ojrvxV}b _eUB~`bekg^Hjkjx_adH^%V} b kgbafHdgvx_jxS&UWV}kl_e^`o_ dgsjxV ^9norn&v©n&^h _avhn&_aj VNbm k#¬kgUW_abebaV§ÁN_evjxV¸9¼½Â¤V N_eba_e^¸Ö }^~`vrViUB_eV vkg`_jx_adH^&^&Vin&v 'vrV jrVi^Cn`VN~&vxdg~`kgfg\iV}\jkls&be_jbmk'ordgUWUWV §V x&- . 2³ w s
(64) ª?« +7u¯²w º ¬|©V j§V . ³ 7 Ò S`kgh n&^¦Vio s`_jo} V Ix « +Zu°¯±¬&_aI s
(65) t| orVc\jxV vxUB_e^&V W b kg_e&V# n`^¦U'n&bajr_e~&beV¸V n&v hdHUBU*kg^`\#~`kgv}beViojxvrdH_eoNs&_joNV'~GdH_e&o}dgvrjxo}&V « +7u¯±¬P 7ÇpVio F
(66) -_Fj-*d Y\[ &Vio}¹u Z {2hdH^`o_m\ivr\o _eor~7dHoxkl^CjNV'tCn`kljrvxV V ^Cjrvx\ Vo ¨_ebpVozj ~7dHoxo_es&beV#VchdgUs&_e^&V v beVch klbmhn`b<V Ix |V j bek*orVih dg ^`V kg`_jx_adH ^ b kl_mVc n`^ n`^&_etCn&Vc\ jxkgfgVcVcbedgfg_mtCn&Vg T[s−i] n+2 0 1. LUT. T[s−i] n+1. P[n−i+1]n 0 1. 0 ϕ(1)n−1 ϕ(2)n−1 ϕ(3)n−1 ϕ(4)n−1. 000. T[s−i] n+1. 001. 011 100. ϕ(.)m1. T[s−i] n+1. ϕ(.)m2. Select. 3. T[s−i] n. T[s−i] n. 010. T[s−i] n. P[n−i+1]n−1 P[s−i] n x n−i y n−1 P[n−i+1]n−2. 0 1. x n−i y1 P[n−i+1]0. 0 1. x n−i y0 wn−i. P[s−i] n. 0 1. T[s−i] n−1. ϕ(.). T[s−i] 1. ϕ(.). T[s−i] 0. ϕ(.). 0 1. P[s−i] n−1. T[s−i] n−1. ϕ(.). T[s−i] 1. ϕ(.). T[s−i] 0. ϕ(.). P[s−i] n−1. 0 1. 0 1. 0 1. P[s−i] 1. P[s−i] 1. 0 1. 0 1. P[s−i] 0. Select. (a). P[s−i] 0. (b). CX ÉÊUW~&bekg^Cjxkjx_adH^ n&^\jxkgfgV© _ajr\ivxkljr_edg^'orn&vn&^B¹u©Z {0&V bmk ¬kgUB_ebebaVÁN_evrjrV¸9¼½Â} DW¬kk }jr_eba_morkljr_edg^'&V bmkNs7dgvx^&V _a^`&_etCn`kg^Hj <Vozj n&^«^&dHU's&vxVc&V ¦Jw s&_ajxoi Ws k©ÉÊUB~Gkgh+jN&njxS&\ dHvrTiUWV#orn&v b kgvxhS&_ajrVh+jxn&vrV Vcb dH~G\ivxkljrVin&vi . "$#&% ' p «I + u°¯±¬ . tCn&V.
(67) !#"$%& '(#)*+,.-0/21436587. 9n&~&~7dHordg^GoU*kg_a^CjrVi^`kl^CjEtCn&V ^`dgn`oordgn&S`kg_jx_adH^`ohdg^GhV Hdg_ev<n&^Bdg~7\ vkjxV n&vdlÈvkl^Cjbmk§~GdCoror_as`_abe_jx\©&Vor\ beVihjr_edg^&^`V vbaV©UWd9&n&bad ® ³ & \ior_avx\§~GklvxUB_Èn&^ V ^GoViU's&beV]&V '^&dgUs&vxVio ® ¯ °® ² ° ³i³ ³i°® ´ `9V n`bÖbaVch kgbeh n&bÖV bek'dg^Gh+jr_edg^ À\i~GVi^`&kg^Hj}V ® ³ ¨_ebponBj VWhdg^Gozjxvrn&_evxV'n&^&Vjxkgs&beVBh dg^CjrVi^`kl^Cj§beVio]klbeV n`vxo]V %x |§~GdHn&vchS`kgh n&^AVo]UBdn`ba_EhdH^`or_e\ivr\o 7ÉÊbVioj &Tio]badHvxo§_eUB~7dgvrjxkg^Cj V#_mo~7dHorV v} n&^`VcsGdHvr^&V ~&vx\ih _eorV on&v « +u°¯±¬<k¢G^«&V]vx\i&n&_avxV bekjkl_ebabeV#V bekBjkls&beVg . m.
(68) 9Ð Ì f_F*`[ ,`[\ _`Y [_D],,`j-Y\[. &°i³ ³ ³ °® ªu º . & . £. « +Iu~¯±¬ ¤. £v¤.
(69) bY[§j-Y ec"Y\[. ¯¯ w s
(70) tvus® 0u º s
(71) t u® u0º s
(72) t ² w s*t ¯ u ® us s t ² u-® us. * s
(73) t. £"¤ ª?« +7u~¯±¬ ¤ Yj. Y j$-. *. ª?« + u ¯±¬ Y\j. ¯ ¯ z w &º ¤¤ ®. []Ks*t [] s
(74) tD. º ®. w >¤. ¯ ¯ z w &º ¤¤®. []Ks*t [] s t$. ª. . ²² x#s*t. * x#s*t. w s
(75) tCu)(*|z [ ]²¦®Y[j
(76) ]e w s
(77) tCu
(78) |z [ ]²¦®Y[j%]c`,]e. º 0®. . . . w. *. ¯ µº. ¤
(79) s tCu ¤Ks t z . w g¤. Yj. ¯ 0º. ¤ s
(80) tCu r ¤Ks*t z . vYI2H_$[ Y\[ dM
(81) eY\[g[_F\e]KY _DeY\[ª ?Y j ! B
(82) j#j-Y ]`j-Y[>
(83) eM[" _`YN®##s
(84) t ¯ z $ . « +&u ¯±¬b[ ,j±][ \,ij ,
(85) eM[JY\[§](
(86) H]j[. º% O[
(87) `j%^&K`]KY\[I
(88) e' s
(89) t ² u []²¦ Y\[j²`,]e s
(90) t ² u []²¦ Y\[j]cZ
(91) ]e. W#mk. ¿. W k. Ò VjrS&\idgvxT UWVg¨\iUBdH^Cjrvx\&kl^Go' b kg^&^&V ¸9V {#¨fHkgvxkg^Cjr_ajkl_e^`or_©tCn&V « + u ¯±¬ViojBn`^^&dHU's&vxV{V ¦4w s¦s`_jo Ò dg^Go_m\ vxdg^Go ^&dHn&gVkln-baV%h _avhn&_aj#VWbek¢`fHn&vxV*Xgk& Èuwn&_moxtHn`V*baV*h kgbeh n&bwVIx |]^` V*^&\ih Vioxo_ajrVW~&ben`octCn&V*V n&¸As&_ajxoi¨<hS`kgh n&^&V*VocÇ NR n orVih dg^`«kg`_jx_adH^&^&Vin&v WvrV jrVi^Cn`Vc~&vrdH~`klfH\ V _mo~7dHorVcU*kl_e^HjxV ^`kg^CjN n`^&VcV ^Cjrvx\ Vcbe_es&vrVH
(92) }dHn`o~7dgn&Hdg^`o q hdH^&^&VihjrViv n&^¦o_efg^`kgb f²Y\YM jWö¢Gfgn&vxV#Xis k VjNhiklbmhn&beV v ¯ o0_ f²Y\@Y j r£ Ix « +u°¯±¬&_aI s t | * x#x#ss*tt «« +I+Iu°u°¯±¯±¬±¬±_a_a77ss
(93) tDt |9|9UWUWdd®«®²( o_0f²Y\@Y j º
(94) NdljrvxVdg~7\ vkjxV n&vViojwTiowbadHvxoh kg~`kls`baVV \jxV vxUW_a^&Viv x 0 w -b|9UWd ® ¯ V Zj x w-g|9UWd9® ² §b kl_mV}VovU iUWViovxVioxodHn&vxh Vio U*kjx\ vx_aVibabeViotCn&V]baV h_evhn&_aj&V§bmkc¢`fHn&vxV Xgk& 9ÇpdgvoxtHn`VNbeV§^&dHU's&vxV§&V§UWdn&be_ÖV 9_eV ^Cj©~`ban`o_eUW~GdHvjkl^Cji¨C^&dHn`o©~GdHn&gdH^`o ojrdh·HV v beVioc_egVivxorVio]klbeV n`vxo V Ix | `kl^`o beVio s&bedh ocVWUW\ UWdH_avxVB&n¡¹u Z {' <ÇpVBjxS&\ dHvrTiUBV W~GVivrUWV jckg_a^Go_E b \hdH^&dgUW_eVB n&^¡s&_aj kg&vrVororV ~`klvvkl~&~7dgvrj bmkWodHban&jr_edg^ ~&vr\oVi^Cjr\ VH¨vr\n&_morkg^Cj &V]UWdH_jx_a\cbmkjxkl_ebebaV#V bmkWUW\ UWdg_evrV]^&\hVioxoxkl_evrVH Ç dH~G\ivxkljrV n`v}dgn`vr^&_moxorkg^Hj§n&^A^&dHU's&vxV'vV ¦wKº's`_jo§h dg^&fHvrn Cw -¶UWdn&bed®¨7n&^&VWhdgvxvxVih+jx_adH^¦Vioj§^&\hVoroxkl_evrVk¢`^ dgs&jrV ^`_av§baV#vx\iorn&bajxkljNVioxhdHUB~&jr\g 7ÇpV jxS&\ dHvrTiUBVon&_ekl^Cji¨7\ UWdH^Hjxvr\#`kl^`oNb kl^&^`V¸V#]¨&dgn`vr^&_ajNbeVioN_a^dHvrU*kljr_edg^`o ^`\ihVoroxkl_evxVio WbmkWhdg^GhV ~&jr_edg^«n¦h_evxh n&_j vxVior~GdH^`oxkls&beV]&V#hVjrjrVcdH~G\ivxkljr_edg^p Ì* 9Ð + f_F*`[ ,`[,\_`Y - % F£&°i³ ³i³°Q ® u0º Y j -
(95) B/. ]j e,j±]K, ^%#i]KYZ
(96) e . _`,j±]K,10243C5\76\eYC_D°,c?d eYgª?« £
(97) ¬7^Y ¦Jw º d\]j#[>j(Y 8 _`YZª?« £,¬2©x >w -b|²c?F^Ö® w 99® B: Y\[Mj%_D°Y `j±]KY\eg[ ,j±][ \,][ ,ijI§e Y\
(98) j#]K
(99) [_D],,`j-Y; * s []s
(100) t$ ¯ w & º ¤0® ¤ s*t$ ¯ w s*t$ ² u0º £"¤ ¤ º []s
(101) t$ ¯ w s*t$ ² ¤0® ¤rs
(102) tCu¤º Ç<dgvortCn& V kl~&~`kgvjx_aVi^Cj 4£&° ºH ° s ¨&Vin¸hdgUW~`kgvxkljrVin&vxo}^&dgn`o§~7V vxUBV jjxV ^Cj]VB\jxV vxUW_a^&Viv§bmkh dgvxvrVh+jx_adH^ vxVitCn&_morVg N^-ordgn`o¼ jxvxkHh+jrVin&vÄbmkcvxVjxvxkg^`hS&VV ^`orn&_ajrV]VZª?« £,¬GV jÄ^&dHn`owdHsjrVi^&dg^`o kl_e^`or_&baV§vr\on`bjkjwVioxhdHUW~jr\ W¢`fgn`vrVNHkk HÆ]kl^`oÄbaVNorVih dg^`Whikgoi¨g^&dHn`o ~7dgn&Hdg^`oÄ^&dgnGoÄ_e^`or~&_avxV v VN b kgvxhS`_jxVih+jxn&vxV jrvkg_ajr_edg^`^&V bebaV§VNb kH&_ajr_edg^&^`V n&v©UWd9&n&bad'® le\n½
(103) Ndgn`o \i&n&_eordg^Go n%jrS&\idgvxT UWV§X tCn&?V o * ª?« £,¬ or_ £v¤ ª?« £
(104) ¬¥¤®° x w -b|UBd ® ª?« £,¬iu-® or_p® ¤ ª?« £
(105) ¬ ¥ sl® Âw^h dg^`ojxkljxkg^Hj tCn& V ª?« £,¬`u®4ViojNozjxvr_mh+jxV UWV ^Cj _e^\ vx_aVin&v §s t ¨&^`dgn`odHsjrVi^&dg^`&o o ª?« £,¬u¡Q ® xKª?« £,¬`u4® |9UWd9 s t xKª?« £,¬UWd97 s t w s t u® |9UWd7 s t ¿.
(106) Ç<kWhdH^`_ajr_edg^«®. l®4~GVinjN\ fCklbeV UWV ^Cj o \ih vr_evxVvo x ® ¤ ª?« £,¬ ¥ s t |°x#s t ¤ ª?« £,¬ ¥ s4® | x® ¤ ª?« £
(107) ¬ ¥ s. . uEklv}hdH^`or\itCn&V ^Cj¨. iUWd7s t iUWd7s t. ¤ ? ª « £,¬ t #x s ¤ ? ª « £
(108) ¬. &_a º ® _e º. ¥ s t |~xKª?« £,¬ Is t | w s t u |°x#ª?« £
(109) ¬ %s t F|. 9UWd ® iUWds
(110) t>w s*t u® |UBds*t or_xKª?« £,¬±_a7s
(111) t? º|©dgnxKª?« £,¬UBd7s
(112) t w s
(113) tCu-® s
(114) t| UBds*t or_a^`dg^ Ç<kW¢Gfgn&vxV'ls-\ih vr_ajNbeVh_evhn&_aj]h dgvxvrVo~7dg^``kl^Cji N^Akg&&_jx_adH^&^&V n`v *vxVjxV ^9n&V#~`vrdH~`klfH\ V#\ jrV vxUW_a^`VcjrdHnj] kls7dgv«bmkordgUWUWV ª?« £
(115) ¬iUWd7s
(116) t w s
(117) tvu® }^AU'n&bajr_e~&beV¸V n&v]hdHUBU*kg^`\#~`kgvNbeV#s&_aj]V~GdH_e`o dgvrj§Vvª?« £
(118) ¬V jNbek%vrV jrV ^9n&VodHvjkl^CjrV#Vb kg`_¼ jr_edg^&^`V n&v}or\ beVihjr_edg^&^&VcVi^`orn&_jxV baVcvx\iorn&bajxkj}h dgvxvrVh+ji x &w -b| * x#ª?« £
(119) ¬ ª?« £,¬. 0 m 2m. >m. 2 n− m P[0]. > 2m P[0]. n. n n. 1 n 0. n. n+1. (xy+w) mod m. (xy+w) mod m bit de poids fort. n+1. (a). (b). Ò dgvxvrVh+jx_adH^BVbª?« £
(120) ¬7k¢G^% dHsjrVi^&_evbx Bw -b|9UWd ® iWLkkª?« £,¬GViojÄ\ fCklb §x %w -b|9UWd ®-¨ ?x %w -b|9UWd ® w ® x >w -b|9UWd® w s® Wis kIª?« £,¬pVozj}\ifHkgb 4x >w -g|9UWd9® dgn {x w -g|9UWd9}® w® . "$#&% ' p. dgn .
(121) . *8 ) '(,
(122) ,) ". Ç dg~7\ vkjxV n&vVU'n&bajr_e~&be_ehikjr_edg^&¼½kH&_ajr_edg^\jxn`_e\h_a¼ÊVioxonGo¦^&\ih Vioxo_ajrVÀtCn&Vb dg~7\ vkl^`&V ªordg_aj_a^&\ vx_aVin&v µ® Ò V jjrV hdH^Hjxvxkg_a^CjxVWo k±gT vxVB~`klvcV ¸9ViUW~&baV*~`vrdHs&ba\iU*kjr_mtCn&V*&kl^GocbaV%hikgocV*b V¸~G² dH^&V ^Cjx_ekljr_edg^¡UWd9&n&bekg_avxVW^&\ih Vioxo_ajxkg^Cjc&Vioc\iba\i±kljr_edg^`o klnh kgvrvx\*onGh hVoror_egVi"o l mn½ pÉÊbwVozjV ^¡VÈVj'_eUW~GdCoror_es&baV%&Vh kgbeh n&baVi v ª?« £
(123) ¬ UBd® « b kl_mV%nÀh_evhn&_aj'V%bmk ¢`fgn`vrV« s< <©_aVi^ tCn _eb ordg_aj#V ^99_morkgfgVikgs&beVW V 7Vh+jrn`V v'Vo'h dgvxvrVh+jr_edg^Go]kg~&vxTiochS`kHtHn`V*U'n&bajr_e~&be_ehikjr_edg^<¨<^&dHn`oc~&vx\\ivrdH^`o \jxn`_eV v#n&^`V%^&dgn`gV bebeV _jx\ vkjx_adH^p }^&V]~&vxV UW_aTivrV]odHban&jr_edg^h dg^`or_mozjxV 'hiklbmhn&beV v©beV§jxV vxUWV « +Bu4¯#¬ÖV]bekvxV bmkjr_edg^V§vx\ih n&vxvrVi^`hvV W¬ k orn&Bv ¦Cw s&_ajxo©Vj _aUW~&bmkl^CjrViv Ix |B b kg_e&V n&^`V jxkls`baV jrvxdg_moV ^Cjxvr\iVioi 9Ç _ajr\ivxkljr _edg^«\¢G^&_aVc~Gkl>v o * « +Iu°¯±¬2 s
(124) ª?« +Iu~¯2wO\º ¬Dw . &w - . ³ ³ WP
(125) k ª?« +u°¯±¬2 « +7u~¯#¬UWd97s*t$ ¯ w bx « +Iu°¯±¬±_a7s
(126) tD ¯ | k±gVh ª?« + ¬2®£#V j bx |©x#s
(127) tD ¯ |UBd ® ~7V vxUBV j©h V ~7V ^`&kg^Cj beV§h kgbeh n&bG&V « + u{¯±¬7orn&Bv ¦ w°s s&_ajxo VjĬkHh_eba_ajrV]bmk#hdHvrvxVihjr_edg^ V ª?« £,¬p^&\hVioxoxkl_evrV Bb dgsjxV ^Cjr_edg^«§V x w -b|UBd ® Ì CÐ fi_F,[\
(128) [' _`Y Y\j -
(129) ej#]KY `2Y\ij Y\j \_`YZ°[\
(130) ]jB_D~Y\ij#]KY e ^*YZ¦?w { º d ]j#[gj-Y\ \_`Y * x#s*t ² w ºgº|z [M]²¦®Y\[j`,]e £v¤ ¤ 5 Z¥ x#s*t ² w
(131) | z [M]²¦®Y\[jB]cZ
(132) ]e Y\[ dM,eY\[&[_F e ]KY\_De Y[ ^*Ygª?« +7u~¯#¬7Y\j ª. 7. « +Iu~¯#¬%[ ,ij%,@,eM[C^%#i]KY[%`,e'. ²² x#s*t u)(*| z . * x#s*t u
(133) | z . []²¦®Y[j
(134) ]e []²¦®Y[j%]c`,]e. . I. Yj. . ² 0u º. st.
(135) ¯ Kº vYbN_$[ IY"2,c?d\e Y>ª?« £
(136) ¬ZeY j(
(137) _DeZ`,e c® ¯ p® Ö® À® Çk\iUWdg^`ojrvkjx_adH^p¨kl^`kgbadHfgn&V h V bebaV njrS&\idgvxT UWVH¨9Vioj©~&vxdg~7dHor\ V &kg^`o©b kl^&^`V¸V Ò Ò V]jrS`\ dgvxT UWV]fHkgvxkg^Hjx_jtCn&V§beV]jrVivrUWV « +gu ¯±¬wVWbek ^&dHn&gVibabeVB_jx\ vkjr_edg^-Viojcn&^-^`dgU's`vrVW" V ¦{wrss&_ajxoi pÇpVo]h dg^`&_jx_adH^`octCn&Vb dg~7\ vkl^`VvAdH_j#oxkjx_eo¬kl_evrVB\jkl^Cj UW dg_e^`o hdg^Cjxvxkg_afH^`kl^CjrVoEtCn&VNhVibabeVioÄ_aUW~7dHor\ Vow~`kgvÄbeV jrS&\idgvxT UWV#H¨g_eb`Vioj U*kg_a^CjrVi^`kl^Cjw~7dHoxo_es&beVN njr_ebe_eorV v©bekcordgvrjr_eVZª?« £,¬ÈhdgUWUWV dH~G\ivxkg^`VZ* n&^&V§^&dgn`gV bebeVNU'n`bjx_a~&be_mh kjx_adH^¼Êkg&_ajr_edg^%oxkl^GowVÈVihjrn&Viv V§h dgvxvrVh+jr_edg^W^&_ÈUWd_a¢`V v©b klvhS&_ajrVh+jrn`vrVN&nh_evxh n&_aji CÉÊb ornDWj}V ^«VÈVjN&V#hdg^GozjkjrViv tHn`V?o * x#s
(138) t ² u
(139) |z ¥ x#s*t ² w ºgº|z or_¦AVozj}~Gkl_ev ª x#s t ² u)(*|z ¥ x#s t ² w
(140) | z or_ ¦AVozj}_eUW~`kl_ev Ò dHUBUWV{ª?« £
(141) ¬ ViojBojrvx_mh+jrViUWV ^Cj_e^\ vx_eV n&v °s®¨<bmkAh dgvxvrVh+jx_adH^^`\ihVoroxkl_evxV ¦b dgs&jrV ^Cjx_adH^°V x ?w -b|UBd ®)o V 7Vh+jrn`V b kg_e&V#n¦h_evxh n&_j}Vcbmk¢`fgn`vrVcls<¨&tHn`V b<tCn&Vcordg_aj® °,s
(142) t$ ¯ w ºH° ³ ³i³° s*tvu¤º Y[v
(143) Y _DeM[&ª Y\j [ ,`jg,j±j(Y ] `j-Y[C,eM[ _`Y x±s
(144) t w s*t$ |z $ w . M e
(145) j(Y _DeCY\[Mj (
(146)
(147) §x w g - |²c?^ , _ Jx >w -b|²c?F^ w . -
(148)
(149) + +« © !#"%$`# !#" <
(150) .0/2143 {}¢`^-V*h dgUW~`klvxV v§baVo§kgbafHdgvx_jxS&UWVio]\ih vr_ajxo]&kg^`o hVjcklvrjr_mhbeVg¨7^&dHn`o k±gdg^GoN\igV bedg~`~G\n&^-~`vrdHfgvklUWUWV Ò fg\i^&\ vkl^CjNn`^&V Vorh vr_e~jx_adH^Á§Å§Æ§Ç0oq9^CjrS`\jr_moxkls&beV n&^\ jxkgfgV _ajr\ivxkljr_edg^Vi^¦dg^`hjr_edg^A n&^-V ^`orV Us&beVVUBdn`ba%_ i®¯°i³ ³ ³i°®¦´ or~7\ih_a¢`\H
(151) NdljrvxV'~`vrViUB_eT vxV#V ¸9~7\ vx_eV ^`h V'hdH^`o_mojrV hdgUW~`kgvrViv}bmkorn&v¬kHhV' n&^h_evxh n&_aj§njx_abe_eoxkl^Cj]bek*s7dgvx^&VB\¢`^`_aV~`klv§b _a^`\ fHkgba_ajr\{W$k h V bebaVN n&^%h _avhn`_jÄV ¸~&badH_jkl^CjwbeV jrS&\idgvxT UWVcg CÇpVoEvr\on`bjkjxowdgsjxV ^9n`oE_e^`_mtCn&V ^Cj tCn&V}beVNhiklbmhn&b`&V}bek#sGdHvr^&V}orn&~7\ vx_aVin&vxV V¸&kghjrV V « +%u~¯±¬p~GVivrUWVj}V vx\in`_avxVckl~&~&vxd±¸_aU*kljr_egV UWVi^Hj n&^ ¬kgh+jxV n&v Vin¸bmk'jxkl_ebebaV#n«h_evhn&_aj}Tio tCn&V beV]^`dgU's`vrVcV UWdn&be_ VozjN orn&~7\ vx_aVin&v WtHnGkjrvx?V Wö¢`fHn&vxV]$ k+ . 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 [tranches]. 310 165 85 50 35 25 17 1. 2. 4. 8 Nombre de moduli. 16. 32. 64. Ä 9n&vr¬kghV« n&^0\jxkgfgV V«b _jx\ vkjx_adH^¤V ) _aUV j9dgs7V beU*kl^Vi^dH^`h+jx_adH^¤n0^&dHU's&vxV &VUWdn&be_}on`vBn`^0h _avhn&_aj Ò Áiggp ÂE¼ ¿ W¬UWd9&n&ba_§fg\i^&\ vx\io*kgba\kjrdH_avxV UWVi^Hjorq9^CjrS&ToVH¨w~&bmkghViUWV ^Cj*VjvxdgnjklfHVV 7Vh+jrn`\iok±HVihÉ9 m eg gg_k wÇ \ihS&VibabeV njx_abe_mo\iVcV ^«dgvdg^`^&\ V]Vioj}badHfHkgvr_ajrS&UW_mtCn&Vg Ò dH^`o_m\ivrdH^`oBUWkg_a^CjxV ^`kg^Hj*n&^0\jklfHV«&V«hiklbmhn&b}&V¦bek^&dHn&gVibabeV«_jx\ vkjr_edg^0\¢G^&_aV¦~Gklv*b \tHnGkjr_edg^ WP*k+ EÂw^µ~&bmk ikl^Cj*n&^ vxV fH_eojrvxV*V ^ordgvrjr_eV%VhVhdgUW~7dHoxkl^CjcVj'Vi^njr_ebe_eoxkl^Cj'n&^UW\h kl^`_eorUWV%V%vxV s7dgn`h bekgfgVg¨Ö^`dgn`o'~GdHn&gdH^`o#h kgbeh n&beV vcn&^^&dHU's&vxV h dg^&fHvrn w- UWdn&bed#® Vi^ +}hqh baVow S&dgvxbadHfgVH lÇpV ^&dHU's&vxV}V}Un&bjx_a~`ba_mh kljr_edg^`o¼½kH&_ajr_edg^`oE&\jrVivrUW_e^&\ VoE~`kgvÄoVhdH^`VVioj . "$#&%('. .
(152) klbedgvowVz +H¨gd¢Vioj bmk vr\tCn&V ^`h V§V}dg^`hjr_edg^&^&ViUWV ^Cj©nh_evxh n&_j 9n&v n`^h_evxh n&_ajÄÁN_evjxV¸9¼½Â}¨H^`dgn`oÄdgsjxV ^&dH^`ow~`kgvÄV ¸9ViUW~&baV º ££ Å ~GdHn&v§&Vio}dH~G\ivxkg^`Vo}V,+>2º»Ws&_ajxoi GuEklv]hdH^`o\tCn&V ^Cji¨`^&dgn`oNVÈVihjrn&dH^`o}Vi^C9_evrdH^¦o_a¸ UW_ebabe_edg^`o] dg~7\ vkjr_edg^Go Bbek*orVih dg^`q V]V ^«njx_abe_morkg^Hj}n`^&_etCn&ViUWV ^Cj#±cjrvkl^`hS`Vioi ©
(153) #
(154) À !#" w !#". ÇpVoNdH~G\ivxkljrV n`vxo§\ih vr_ajxo yzn`oxtCn _mh_dH^`h+jx_adH^&^&V ^Cj n&^&_mtCn&V UWV ^Cj ~GdHn&v]n&^AVi^`oViU's&beVB~`klvrjr_mhn&be_eV v V'UWdn&be_EVj ^&\ih Vioxo_ajrVi^Cj Vio§jkls&beVio]~&vx\ihiklbmhn&be\ VoNkln-UWdgUWV ^Cj]&VbmkfH\ ^&\ivxkljr_edg^nhdVBÁ§Å§Æ§ÇÄ ÖÉÊborV vkl_ajNjxdgnjxVdH_eo§_a^Cjx\ vxVioxorkg^Hj VW_eor~7dHorV v n`^ dg~7\ v¯ kjxV n&vh kg~`kls&beV§&V§h kgbeh n&beV v©baV]~&vxd9&n&_j *UWdn&bed'® ~7dgn&v©^ _aUW~7dgvrjrV tCn&V bebaV]klbeV n&vV]®4ozjxvr_mh+jxV UWV ^Cj©h dgUW~&vx_eorVNVi^CjrvxV s
(155) t$ Vj?s
(156) tÈ Æ§kg^`o#bmkhdH^`hben`or_adH^ÀV%beV n&vBklvrjr_mhbeVg¨gVidg^&f¦V jB©n&vxbeorV dH^¡orn&fHfgT vxV ^Cj#V hdg^Gozjxvrn&_evxV*baVocjxkgs&beVio'h dg^CjrVi^`kl^Cj'beVio klbeV n&voEV Ix |wk±±kg^CjE&V}\ s&n&jrV vÄbek Un&bjx_a~`ba_mh kljr_edg{^ l ,X nP lÉÊbmowh dg^`or_e&T vxV ^CjwtHn`V baV}h d jwV}h V ~&vr\ jrvkl_ajrViUBVi^CjVozjÄkHh h V ~jkls&beV o_ baV "U iUBVUWdn&bedc®_a^CjxV vxC_eV ^CjEbedgvo&V ~&banGo_eV n&vodg~7\ vkjx_adH^`oonGh hVoror_egVioi¨hdHUBUWVH¨~`klvV¸ViUB~`baVH¨lbadHvxo n&^&V V ¸~GdH^&V ^Cjr_mkjx_adH^ UWd9&n&bekg_avxVg iÉÊbeo<^&VÄ~`vrdH~GdCoVi^Hj<h V ~7V ^``kl^Cjkgn`hn&^`VwUW\jrS`d9&V \ klben`kljr_edg^#Vo<hd9V *h _aVi^Cjxo Ix |+ ±Âw^'h dg^`ojxkljxkg^Hj<tCn&V IxK£| £&¨ IxzFº |B®s
(157) t©UWd ® ®s*tvu-®¨ ¯ ¯ Ix#s| s t ¯ UWdQ ® * s
(158) s tt ¯ uu sl®® oror__ s Ks
(159) t$t ¯ w z º> ¤¤w ® >º ¤¤¤® #s ¤rt s
(160) ¯ tCz $u¤ º Vj Fs t u ® o_ s t$ ¯ w s tD ² w º&¤¤® ¤ s t u0º Ix |x s t |9UWd ® Fs
(161) tCuH® o_ +x#s*t¯ ¯ w s
(162) tD|z%( w >º ¤¯ ® ¤rs
(163) tD ¯ w s
(164) t$ ² º ¤0® ¤ +x#s
(165) t w s
(166) tD|z%( Fs
(167) tCu (l® o_ s*t$ w > ^&dgnGo~7dgn`gdg^Goh dg^`ojrvxn&_evrVwbmkNjxkls`baV©_aUW~&bmkl^Cjkl^Cjbmk}dH^`h+jx_adH ^ Nb kl_mVV©h dgUW~`kgvxkljrV n`vxoi¨iV©U'n`bjx_a~&beV¸Vin&vxoi¨±V©ordgnGozjxvxkHh+jrVin&vo Vj &V}U'n&bajr_e~&be_aVin&voE~`klv Viowh dg^`ojxkg^CjrVioÄh klbmhn`bekg^HjEbeViowjrV vxUWVio ® Vj (® gÇpV}h _avhn&_ajÄvx\iorn&bajxkl^CjwVNh VjrjrVNkg~&~&vxd9hS`Vo k±g\ vxV vkl_aj jrdHnjrV dg_mo~&banGo©h dgUW~&beV¸V tHn`V] b \jxkgfgV Vch kgbeh n&bÈ_abebanGozjxvr\]~`klvbmk#¢Gfgn&vxVisV j bek'vr\tCn&V ^`h V§U*kl¸9_eU*klbeV]&V§dH^`h+jx_adH^&^&ViUBVi^CjV b dg~7\ vkjxV n&vc\i~GVi^`vkl_aj#VWb \jkl~7VW&VB~`vr\ jrvkl_ajrV UWVi^Hj
(168) }dgnGo§~`vrdH~GdCodH^`oN_mh_En`^&V*UW\jrS`d9&VW&VWhiklbmhn&bvx\ih n&vxor_egV&Viochd9V B¼ h_eV ^Cjxo Ix |§~GdHn&kl^Cj#o VÈVihjrn&Viv]~GklvklbebaTibaViUBVi^Cj kgn¸~&vxV UW_aTivrVoN_ajr\ivxkljr_edg^`o VocklbefgdHvr_ajrS&UWVo§VBU'n`bjx_a~&be_mh kjx_adH^-UWdn&bmkl_evxV ~&vx\ih\V UWUWV ^CjN\hvx_jo Ò dH^`or_e\ivrdH^`o©n&^&V]dg^Gh+jr_edg^ ~GklvklUW\jxvr\iV§~`kgv ®4VjN\ ¢`^&_eVc~`klv t$ ² %x4 ® | sl® us t ® ³ s ³ ¯ ³ ¡ Ç \ klben`kljr_edg^µV Bx¬® |BVi^µUWkljr\ivr_eV b VozjWjrvx_eC_mklbeV ow_ebNon Bj%Vi^¤V 7V jV\h klbeV vW® n&^`V¦~GdCo_ajr_edg^0gVivxoWbmkfHklnGhS&Vjxdgnj V ^^&\ fHba_efgVkl^Cj§beVBs&_j#VW~7dg_m&o§dgvrj]® t$ ¯ Ò VjrjrVdg^`hjr_edg^~GVivrUWV j§jxdgnjxVdg_mo]baV*hiklbmhn&b_ajr\ vkjx_©V ¨ph dgUWUWV b _a^`&_etCn&VBbaV jrS&\idgvxT UWVcon`_akl^CjN\iUWdg^Cjrvx\]`kl^`o b kg^&^&V ¸9V#Æ Ì CÐ Y\[>
(169) Y _DeM[g[_`MMY[M[],Y[ ^*Y Ix |4 K%º 0[\YvM, _DY `j%e M\_DeM[],Y\c?Y `j § . ,]K^Y?^*Y ?e Y\
(170) j#]K
(171) [_D],,ij(Y * Ix vu0Fº |OuBx¬® | [] %x vu¤º |0uBx¬® | £ %x | Ix vu0Fº |OuBx¬® |Ow® [], : IxK£|®£. Ç<kÀ¢`fgn&vx~V mgkÀ_ebaben`ojrvxVA b kgvxhS`_jxVih+jxn&vxV-nªh_evxh n&_j fg\i^&\ vkl^Cjbmkjkls&beV¡hdg^CjxV ^`kg^Hj baVoh d9\V %h_eV ^Cjxo Ixzº |+¨ Ix#s|%V j Ix |+ ÉÊ^&_ajr_mklbeV UWV ^Cji¨7beVioNjrvxdg_mo}vxV fg_mojrvxVio hdH^Cjr_eV ^&^&Vi^Cj§bmk*klbeV n&v]& ÈÇpdgvo}nA~&vrViUW_aViv§hqh baV' S&dHvrbedgfHVg¨7hV'_mor~GdCo_ajr_aw\ klben&V,%xºF|7 IxK£|uBx¬® |,wW®ÃV jUW\iUBdHvr_morV beV vx\iorn&bajxkljE`kl^`obeViovrVifg_mozjxvrVoÄ©Vjw` Ò V©Vivr^`_aVivVozjEnjx_abe_mo\~7dgn&v_eUB~`bekg^HjxV vbeV©UW\h kl^`_eorUWV©V vr\ jrvxdHkHh+jr_edg^< `ÇpdHvxo}&VioN&V n¸_eT UWV#Vj§jrvxdg_mo_eT UWV'h qhbeVio} S&dHvrbedgfHVg¨&beVioN±kgbaVin&vo Ix#s|Vj %x |NodH^Hj§vrVo~7Vihjr_egV UWVi^HjNhiklbmhn&be\ Vo }^&V¡~7Vjx_jxVn`^&_jx\Vh dg^Cjrv gbeVAfH\ ^&TivrV-n&^Kor_efg^`kgb _e^&_ajr_mklbe_eoxkjx_adH^Okl_e^`or_ tCn&VbeVio«o_efg^Gkln¸ VhS`kgvrfHV UWV ^CjVo vrVifg_mozjxvrVo .
(172) T[s−i]n+1:n n. 2. n n. 11. Registre 2. 10. Registre 1. 01. 0. 00. Clk Ld1 0. Registre 1 n. ϕ(k) = (k*2 ) mod m. ϕ(1). Ld2 ϕ(2). 0. Registre 2 Ld3. Unité de contrôle. 0. Registre 3. Ld3 Ld2 Ld1 Clr. n. 2m−2 m. Registre 3. (a). ϕ(1). ϕ(2). ϕ(3). (b). Ò gk beh n&b`_ajr\ivxkljr_aÖ&V}bekcdg^`hjr_edg^ © WLkkE{NvxhS`_jxVih+jxn&vxV fg\ ^`\ vklbeV}n%h_evxh n&_aji 2WsikwZ]\i^&\ vkjx_adH^%Vio o_efg^Gkln¸WVNhdH^CjrvgbeVg W¢`fgn&vxV msik &¹_a^`kgbaViUWV ^Cji¨&n`^Un&bjx_a~`baV ¸9Vin&v}hdHUWUWkg^`\ ~`kgv©beVio &V n¸s`_jo}V ~GdH_e`o©dgvrjxo V « +%u°¯±¬<or\ beVihjr_edg^&^&V]bmkkgbaVin&v \o_evx\ V'&kg^`o§bekWjkls&beVg ÈÇ<khdgUW~&beV¸_ajr\'&n-h_evhn&_aj§kl_e^`or_<dHsjrVi^CnAVioj§_e^\ vx_aVin&vxV %hVibabeV n&^A\ jxkgfgV'&Vh kgbeh n&b&V'bmk*vrVibekljr_edg^ Vcvx\ih n&vxvrVi^`hV]Vj}bmkWhdg^Gozjxvrn`hjr_edg^«V bekBjkls&beV ^ k~`ban`oN _eUW~`kghjNon`v bekvx\itCn&V ^GhVcV dg^Gh+jr_edg^`^&V UWV ^CjNV b dH~G\ivxkljrVin&vi ÂEjxn`_edg^`o<U*kl_e^HjxV ^`kg^Cjpb \ Hdgbenjr_edg^ V « +u7¯±¬lbadHvxoÖVoÖ~&vxV UW_eT vxVioÈ_ajr\ vkjx_adH^`opVE b klbefgdgvx_ajrS&UWVEVwUn&bjx_a~`ba_mh kljr_edg^ UWd9&n&bekg_avxV \ ¢`^&_p~Gklv bmkvxV bmkjx_adH^Vcvx\ih n&vxvrVi^`h"V W¬ k+ Ò dgUWUWV b dg~7\ vkl^G&V %Vioj}on&~`~GdCo\#ojrvx_ehjrViUBVi^Cj_e^\ivr_eV n&v B®¨^&dgnGo ork±Hdg^`otCn&V £"¤ « +%u\º ¬¤® ukgv}hdH^`o\tCn&V ^Cji¨orV n&beV bekBklbeV n`$ v Ix#£| £_e^CjrVivr9_eV ^Cj©bedgvoV bekB~&vxV UW_eT vxV§\ jxkg~GV VZj £v¤ ª?« +7u\º ¬¤® Ç<dgvoÄ V}bmk'orVih dg^`&V _jx\ vkjr_edg^<¨g^&dHn`oÄdHsjrVi^&dg^Go £v¤ « +u s,¬¤ ® ¥rs
(173) t ¯ w°s
(174) tu-ºH gÉÊb&¬kgnj©&dg^`h§_eor~7dHorV vÄV IxK£|¨ IxFº |EV j Ix#s| k¢G^«&V#h klbmhn`baViZ v ª?« +u s¬½ `Ç<kWordgbenjr_edg^¦hdH^`or_eojrV BvxVjxkgvx&V v beV#\ s&n&j§VcbmkWU'n&bajr_e~&be_ehikjr_edg^ n&^¦h q9h baV# S&dHvrbedgfHVck¢`^ Vchiklbmhn&beV $v IxzFº | bedgvo n&^`V§\ jxkl~7V V ~&vr\ jrvkl_ajrViUBVi^Cji
(175) }dgnGo \iklben&dg^`o V ^Gon&_ajrV Vi^~`kgvxkgbabeT bebV ª?« +BuÀ\º ¬ÖVj Ix#s|+¨~&n&_mIo ª?« +%u s,¬ V j %x | ÇkWUB\ jrS&dVV'h kgbeh n&b< V %x |\ jxkls`ba_eV'&kg^`o}hV'~`klvklfHvxkg~&S&V o kl~&~`ba_mtCn&Vc\ fCklbeV UWV ^Cj WbmkBdg^`hjr_edg^ _e^CjrV vxgVi^`kl^Cj}&kg^`o b kgbafHdgvx_jxS&UWV \¢`^`_`~`kgvEbek vxV bmkjx_adH^BV vx\ih n&vxvrVi^`hV&WP*k+ HƧ\¢G^&_eoxordg^`oVi^`hdHvrVn&^&VdH^`h+jx_adH^x®4|jrVibabeVNtCn&Vx¬® |Bµ® u§s*t$ ¯ Ç<VjrS`\ dgvxT UWVon&_ekl^Cji¨©dg^Cj bek¤\ UWdg^Gozjxvxkljr_edg^ Vioj kg^`klbedgfHn&V h V bebaV-n jxS&\ dHvrTiUBVA`¨©~GVivrUWVj baVh kgbeh n&b§_ajr\ivxkljr_a'VAbmk dH^`h+jx_adH^ § Ì 9Ð "$#&%('`m p. Y[>,,Y _DeM[&[_` MY\[M[M],Y\[ ^Y gx |4 s [\YvM, _DY `j%e M \_DeM[],Y\c?Y `j § . , ]K^Y?^*Y\[>eY\@,j±]K,[&[_D],,`j-Y\[. µºF|Oux®4| µºF|Oux®4| wÀ® s*|OuBx¬® | s*|OuBx¬® | wÀ®. * gx Cu gx C u * gx C u gx C u. gx |. ¤º. ® [M] gx vus*|OuBx¬®. [] bx vu |OuBx 4| £ [ ]
(176) [M]2,. | £. ¯ º ÉÊbE¬kgnj ^&dgn`gVikgn&\jrVivrUW_e^&V v#tCn&VibabeVio kgbaVin&vxoc&V bx `| odH^Hjc^&\hVoroxkl_evrVo]badHvxocVo ~&vxV UW_aTivrVo _jx\ vkjx_adH^`ock¢`^¡V%orq9^¼ hS&vxdg^`_eorV vhdHvrvxVihjrV UWVi^Hj©bekfg\i^&\ vkjx_adH^V bmk#jkls&beV§V j bekU'n&bajr_e~&be_ehikjx_adH^p
(177) }dHn`o\in`_eordg^`oV]bmk'vxV bmkjx_adH^V vx\ihn`vrvxV ^`h V WP*k V jNn jrS&\idgvxT UWVcBtCn&V * x#s*t ² w ºgº |z or_ ¦AVozj}~`kg_av £v¤ « +Iu¤º ¬ ¤ x#s t ² w
(178) | z or_¦AVozj}_eUW~`kl_ev 9_ ¦AVioj}on`~G\ivr_eV n&vdHn\ifHklb Bh_e^`t7¨_abÖVozj¬kHh_ebaVcV UWdg^CjrvxV v tCn&V « +%uº ¬ ¤ s t$ ¯ uÀºH
(179) }dgnGo \in`_eordg^`o©vxVior~GVh+jx_aHV UWV ^Cj V x±s t ² w ºHFº |z ¤ Fs t$ ¯ u0º $: bxK£|®£°Y\j
(180) bx F|®s*t$. i.
(181) Vj x#s t ² w
(182) | z ¤ s t$ ¯ u¤º tCn&V"s*t$ ¯ º or_O¦Vioj]~`kg_av]Vj s*t$ ¯ 2º£o_0¦ÀVozj _aUW~`kg_av Ò Vio]_e^&\ fCklbe_jx\io]odH^Hj orkljr_moz¬kg_jxVio§badHvxoxtCn&V ¦ (%V j orV n`baVo§beVio klbeV n&vowV gxK£|¨ bxºF|V j bx±s*|_e^CjrV vx9_aVi^&^&Vi^HjwbedgvoEV bmk ~&vxV UW_aTivrV_ajr\ivxkljr_edg^*V b kgbafHdgvx_jxS&UWVg ÉÊb&¬kgnjwU*kl_e^CjrV ^Gkl^CjÄ_mozjx_a^&fHn&V v V n&¸ h kHoV ^dg^`hjr_edg^¦V bmkW±kgbaVin&v}V tD ¯ &w - tD ¯ o _ £ ¤ ¤ s*t"u s¨ÈklbedgvZo £{¤ ¯ w - ¯ ¤ s
(183) t"u ºVj ^&dgn`o]\n&_modH^`o§&V'bmkvxV bmkjx_adH^V'vx\ih n&vrvxV ^GhJV W½
(184) k§kl_e^`o_ p O tCn&VcVcbmkW\¢G^&_jx_adH^¦VcbekdH^`h+jx_at$dH ^ tCn& V t$£v ¤ ª?« +u¤º ¬ ¤ s*tCu0ºg uEkgv}hdg^Go\tHn`V ^Cji¨ £"¤ « +7us¬ ¤ s t ¯ us w s t usw º (gs t$ ¯ w s t$ ¯ u ² . ¯ V j}baVchiklbmhn&bp V ª?« +7us¬^&\ih Vioxo_ajrV beVioklbeV n&vo V bx |~7dgn&v °F£`° ³ ³i³° ( V s
(185) tCu0>º ¤ J¤ ¨`^&dgn`o\ jxkgs&ba_moxodH^`otHn`V s
(186) tCu-® ¤ ª?« +Zu0\º ¬¤ w 7º u®4VjNtCn&V p Ç<dgvortCn&> « +Iu s,¬¤ x w Fº |0us® * s
(187) t ² w ºgI º usx±s
(188) t$ ¯ w º |[o_ ¦-Vioj}~`kl_ev ¤ s
(189) t ² w busx#s*t$ ¯ w Fº | o_ ¦-Vioj}_aUW~`kg_av * & » s
(190) t$ ¯¯ w or²_ ¦Vioj~Gkl_ev » s
(191) t$ w ( or²_ ¦Vioj_eUW~`kl_ev & ÇkB±kgbaVin&v}V bxL» |©VozjNdH^`h]^&\ih Vioxorkg_avxV]badHvxoVcbmkWoVhdg^GV _jx\ vkjr_edg^«Vc b kgbafHdgvx_jxS&UWVg Ƨ\¢`^&_moxodH^`Io jrVib<tCn&V £"¤ ¥ x±s t ² w º |z ÉÊbÄVioj#\ 9_e&V ^Cj'tCn&Vg¨<orB_ s*t§uO{º ¤ ¤ ¨bek klbeV n`v'V « +gu s,¬ÄVioj#_a^&\ vx_aVin&vrVWdHn\ifHkgbaV ¦C» s
(192) t$ ¯ uKº*Vj gx¬» | ^ _e^CjrV vx9_aVi^Cj~`ban`o}badHvxon¦h kgbeh n&bp V ª?« +7us¬½ Ò Vio _egVivxorVio h dg^`or_m\ vkjx_adH^`oÄ^`dgn`o~7V vxUWVjjxV ^CjNV dHvrUn&baViv baV]jxS& \ dHvrTiUBVcorn&_e±kg^C&j o CÐ Ì ,[]K^&\e
(193) [ @eY ,j±]K, ^*Y°e M _DeeY MY)0 3
(194) YMJ¦ (®Y\j&£ ¤ ©¤ ¥ x#s
(195) t ² wÃFº |z 7 M,`[Mj±e_` j#]K
(196) ^*Y° . .
(197) . !. #". %$. j(d\@Y ^%#i][ [
(198) `j&; \,2\j±]K, gx | `Y\_Djg[ . Y'&bY j±_`Y\e
(199) e, 6 Y c"Y `jv,_(Je Y\cv] 6\e Y[ \j- `Y[ ^*Y{°cv_DHj#] 2H]KM,j±]K,i+*^*^
(200) ]j#]K
(201) c"F^,_D@,]eY ) Y[>,,Y _DeM[ ^*Y bx±s*| Y j
(202) bx |g[ ,ij
(203) H_MY[ &
(204) ej#]e ^Y gxK£| Y j bx F| ,eM[ ^ . _D2Y, j- `YC^Y72e j#e
(205) ]j(Y c?Y\ij * ¬* ) Y[>,,Y _DeM[ ^*Y bx | Y j
(206) bx (|g[ ,ijIY [_D]j-Y M, _D MY\[B`,e , 6 Y c"Y `j ª?« +Zu
(207) Y " c Y ` j . C Y M
(208) \ D _ 0 * ^ Y b. x | Y j g. x * & | [ . ' Y b &. Y ± j ` _ > Y [ ] v c D _ H ( j
Documents relatifs
Dans chaque ville et pour toutes les lignes de tramway considérées, il faudrait disposer des trafics avant et après ouverture d’une ligne et de plusieurs projections d’évolution
Le fait d’écrire à partir d’une oeuvre d’art visuelle ressemble à une forme de contrainte; nous croyons par ailleurs que cette pratique a eu un effet durable sur l’écriture
variables d'ordre psychologique: soit f.amiliale ou personnelle, soit so- ciale, soit professionnelle, soit une combinaison de ces facteurs. Ce patient a récupéré
For the two coolest objects in Figure 3.6, HD 4539 and PG 1716+426, both at Ta < 27, 000 K, there is a good match between the predicted synthetic spectra and the FUSE data: the
lui inculquer le registre li'lquistioue, les l1abitudes alimen- taires et vestimentaires ainsi que le comportement des bour- geois de Québec. Pierrette Paul doit
La conciliation emploi-famille : des mesures à développer dans les milieux de travail Tableau 4 : Les dispositions favorisant la conciliation travailfamille dans les
Figure 2.4 Predicted (continuous lines) and observed values (filled circles: stands aged from trees with root collar; open circles: stands aged from trees without root collar)
Cette posture permet de dégager deux idées centrales : le poids des milieux professionnels locaux dans l’intermédiation et le transit des idées, modèles et pratiques
