RN-codes : algorithmes d'addition, de multiplication et d'élévation au carré
21
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. RN-codes : algorithmes d’addition, de multiplication et d’élévation au carré Jean-Luc Beuchat et Jean-Michel Muller. N° 5438 Décembre 2004. ISSN 0249-6399. ISRN INRIA/RR--5438--FR. Thème SYM. apport de recherche.
(3)
(4)
(5)
(6) ! #"$ %
(7) & ')(*
(8) + +,
(9) & -./0 '1 /2 #"$3436570 4'8, 93 :<;>=<?A@BC9DFEG;HCDJIK=ML%;NL%:<;>=M? @4OQPRDJI9;HSTOUC9SVSR;HW X6Y[Z]\_^`Hacbedf`>gihkjlZ4\_^mhnhkg>\poMqNrtsvuHw[^]h xzy{q}|$^4j~Tyl]}sy{^ n [MqNykjni^y{^]&Y^4y&&Y[^<9}N d7]4^4\po[yl^ }dJ ^mh. 94]J[}N ~ [y{q <^4ylj$gq ¡ j{Y[^q y{s s}r£¢/q>q}jlY.y{^]4q>[st[_svh6jlYMJjnj{Y[^¤y&h$jnq i¥§¦4^]ylq_is stjn¡¨q rrtqJ©s[ .ª0shT[^]4^]h{hl ylsrg*«ª }_¬>sv^n¬N^4y&hl X6Y[svh® rtrqJ©nh9j{q[y{qJ¬ ^/jlY}j®j{ylw[M4Jj{st0jlY[^c¢q>q jlY_yl^mqiis[q}¡A0>w[\poM^]y®¯%sh®^]uHw[s¬° rt^]Hjzjlqy{q wis[¯ jlq jlY[^6[^m}y{^]hkj] x}±7q y{[^]ylw[p}p² ¥§b³°b´w[rtr^4yKs>¬ ^mh$j{stHJjl^m0jlY^/<qNhlstjlsq rRµJy& ist¶·zµ}>w[\poM^]yKhkgihkjl^4\ h9hlY}y{stnjlYsh9y{q w[Mist [y{q <^4ylj$g¸}.]}rrt^m¸j{Y[^4\¹nº¥»qiis[Hh4®X6Yshy{^]hl^] y{&Y¸yl^]MqNykjnsvhni^]¬ q}j{^]¸jlq_j{Y[^h$j{wig%q}¡K [istjlsq Aµ[\w[rtjls[rs]Jj{stqN¼µM hluHw yls[_}r qNylstjlY[\ h¡¨q yy& [s½¶%¾Gnºn¥§4q>[st[Hh¿¨sV ^Nz¢/qHq jlY
(10) yl^mqiis[Nh&À 9Á)^hkYqJ©ÂjlY}jnstHj{^4 ^]y}y{s½j{Y[\_^jlsv0}%rq svcw[[stj{h q <^4y&Jj{stqNhz}rrqJ©Ãj{q [GqNy®\wr½j{st[rg_¢q>q jlY y{^]qiis[Hh4 Á^T rhlqi^mhl4ylsoM^7}r q y{stjlY[\ hj{ Ä>st[p i¬J}Hj&} ^6q ¡<j{Y[^n[y{q <^4yljls^]h q}¡jlY[^qNyls s rÅ¢/q>q}j{Y%y{^]4qiistj{q ^4^4y&Jjl^cq [jls\Gs¦4^m
(11) Y}y&i©6}y{^Tq <^4y&Jj{q y&h4 Æ*ÇHÈÅÉlÊË JÌ NÎÍ q \_[w[jl^4yn ylstjlY\G^4jlsv}µ[nºn¥§4qiistNh]µ[¢q>q jlY.yl^mqiis[µ>y{q w[Mistpj{qGjlY[^[^m}y{^]hkj. ϼаÑÓÒ£ÔVÕ»Ö4ÔÑÓÒA×&ØÓÒRÙ6×kÚ&×&ÑÓØÛ×&Ü°ØÓÕz×&Ò ×/ÝRÕ$ÒRÕl×&ÝßÞ§ÐÝßÕ$à Ù Ý¨Ô Ù&áÔRаÕKâi×&Ü}Ù ÝR×{ÔVÙ ÑÓÝßÕ®ã°ÕzØvä åçæmáÙ Ýßèn×{ÔRÑÓé]ê°Õã°ê0ë[×&ÝR×&ØÓØÓìkØÓÑÓÒRèÕ íî î&ïð¨ñ ñlò ò òóçô{õmö4÷lø{ù4úlõóvû ü4ñ&ý°þ$ÿ. Unité de recherche INRIA Rhône-Alpes 655, avenue de l’Europe, 38334 Montbonnot Saint Ismier (France) Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52.
(12) ¹ / % 9& . ! )"k
(13) %&'#(* % + +,
(14) & -
(15) 40' /2 )"k33657 ', /3 [^*[y{q [y{sjl*iw y{^]4q>} ^.i^´¢/q>q}j{Y }h h{¡¨q y{\G^*qNyls s rt^
(16) ^]hkj¸uHw[^.rt^*[y{^4\_s^4y&Y[s Åyl^*[qN >w[r7uNws hlw[stjw[ª
(17) ^mh$j ¡¨qNy{44\_^4Hj*«ª
(18) ^jy{]s[y{qiuNw^4\_^4Hj]
(19) »r^mh$j ¡ßNsrt^*i^*\_q Hj{yl^]y_uHw[^Nµ/}h_w[ j{^4rny{^]4q>} ^NµKjly{q MuNw^4y_^]hkj muNwst¬J}r^4Hj yly{q MistyK wGrtwh®[y{Z]h] x9N±7q y{[^]ylw[G^j/² ¥§b³ b*w[rrt^]yzq Hjzy{]4^4\_\_^4Hjzj{wisTis¬ ^4y&hzhlgih$j{Z4\_^]hz[^Hw\G]y{}jlsq *&Y[ s Åyl^mhhks [mh0}ylj{ ^m}Hjc4^jkj{^ [y{q [y{st4jl_^jr^]h0qNHj}[<^4r]hnºn¥§4qii^]h]7 h^4j}yljlsvr^ µ¼q whq wh0sHjl4y{^]h{hlq h i^mhT rtNq y{s½j{Y[\_^]hn is½j{stqN*^j0i^\wr½j{st[rsv4Jj{stqN*MqNw[y7[^]hTTº¥»qii^]hT^4´oM hl^¾*¿ß ^mh$jl¥ }¥§isy{^i^]hTyl^mqi[}N^]hi^G¢/qHq jlYMÀ ºTq wh\_qNNj{ylqNh0uHw srz^mh$j<qNh{hkso[r^_ wij{strsvhk^]yrt^mh0w[[stjlmh}y{s½j{Y[\_j{suHw[^mhi^]h0[y{qi^mhlhl^4w[y&hcMqNw[y is½j{stqN[[^]yq w)\w[rtjls[rst^]y r^]h7nº¥»qii^mhnqNhksvi]ylmh4®ºTq whT[y{q <qNhlq h6]N}r^4\_^]Njni^mhT rtNq y{s½j{Y[\_^]h/jlsy&}Hjn}yljls i^]h7hk<]4s½¤M4s½j{]hn[^]hTTº¥»qii^]hnMqNw[y N4[]yl^]y6[^]hqNM]y{}jl^4wy{h/\ Jj{4y{st^]rh] Ë ] É % ~ny{s½j{Y[\_j{suHw[^i^mhq y&isJj{^4w[y&h]µ>Tº¥»qii^]h]µiy{^]qi[ ^0i^¢/qHq jlY¼µM}y{ylqNis£}w
(20) [rwh[y{Z]h.
(21) .
(22) "!
(23)
(24) #$%'&(
(25) ) )* +# ,-.$/0$/%1 2 3! 45647//0$/%8*9:;4. p' A
(26) ,
(27) z 4' ºnqNwhq whGsNj{4y{^]h{hkqNh )i^mhG] hp}yljlsvwrts^4y&hp[^%y{^4[y{]hl^4Hj&Jjlsq Mh F&Ys Åyl^mh_hks [mhp[^]hG[q \po[y{^]h]?>q y{\_ rtsvhl4^]h}y ~A¬>s¦4s^4[svh^4´ @
(28) A[ B C§µH4^]hyl^][y{]hl^4Hj{}jlsq hz [ y{ sh{hl st^]HjK[$| s\_[rs4s½j{^4\_^4Hj/}hrt^7yl^mqi[ ^ni^c¢q>q jlYDB /CÅq w[ h®r^]h ylmhkwr½j&Jj{hK¡¨q w[y{[svhK}yKr^]h®}r q y{stjlY[\_^]hK[^Tis¬HsvhlstqN `[XEB [µ
(29) @CV ºTq}j{q hK^]qNyl^uHw[^nr^]hz[yl^]\_stZ]yl^mh9si]^]hz}hz4^jljl^Tisy{^] j{stqN yl^]\_q Hjl^]Nj Í }wM&YH9g B / C§ F¼^y{^]4q>} ^0i^¢/q>q}j{Y.}hnh{p¡¨q y{\_^q y{stNst rt^_¿ [^ h4q i¡¨qNiy{^°¬ ^m7i^mhsvi4^mhni s <]yl^]Nj{^]h]µhlq w[¬N^4Hjyl^] y{q w[<4^mh hkqNwhr^ [q \ÎiD^ ; G 5H I
(30) KJL /M 8 / NO4µuNws®hkqNHjrt^mhhk^]w[r^]h0wij{strshl4^mhi^qNh/|$q wy{h<q wyylm}rshl^4yi^mh\wr½j{st[rs^4w[y&h{À qNhksvhkjl^ y{^4\_[rv 4^4y[ h/r ]4ylstjlw[y{^0o[s}syl^0 w[.[q \po[y{^7r^]h6hlq whk¥§&YMQ Pç[^]h/i^0rp¡¨qNyl\_^ R[ªNª ª ªNTª S5SGSlªNª7M}yGª RR
(31) RR
(32) R(SGS5SUR Vª}µ q
(33) W9Vªcimhks ^Tr^c&Y[ s Åyl^G«ª F svi4^0i^0¢q>q jlY4j{}stjn ^]h{hl°gN^4y®i^c¡ß sty{^7 [}y&Q Pvjly{^nr^0[rtwMh<qNh{hlsto[r^7[^c¦]4y{qNh[}h/r my{s½j{w[y{^ w[.\w[rtjls[rs]Jjl^]w[yT}¤
(34) ^ <^m jlw^4yTi^mh6\w[rtjls[rs]Jj{stqNh[rwh6y&}[svi^4\_^]Nj^4j X°qNw
(35) °¬N^]7\_q shni^0\ Jj{4y{st^]rR [^ ylqN[ylsj{ i^]hy{^]4qi[}N^]hci^¢/qHq jlY#^]hkjuHw[^_rt^ yl^]\Gs^4y&Ys Åyl^ q #>w[r®uHw[s®hlw[stjw[ ªG^mh$j¡¨qNy{44\_^4Hj Vª_^jy{]4s½¥ [y{q>uHw[^]\_^4Hj] »rA^]hkjTy{^4rvJj{st¬N^4\_^4Hj6¡ß 4str^ ^4´imiw[syl^puNw^rt^[q \po[yl^qNoijl^]>w
(36) ^].jly{q uHw Hj }yljlsyTi^ s\_MqNykj{^uHw[^4rrt^ MqHhkstjlsq Aµ isvhkqNh4^4rrt^%i^¸<q sv[hpQ¾ Y>µ rt^mhos½j&hi^%MqNs[h¡ß}so[r^]h w[³jl^4r/y{^]qi[ ^_^]hkj .w[^%isvhkj{}M^¸iwÃ[q \po[yl^ s[stjlsv}r st[¡¨4y{st^]w[yl^ Q¾ Y/Z\[9]
(37) /%^ _* %`U 52; G 5H &0;/%-a*^ 5b 4:a5* 7//6 5%^)c1;/%- /d/*d,e.*f ,e;g Í ^jkj{^.[ylqN[y{st4jl¸^mh$jGjly{Z]h stHj{4y{^]h{hl Nj{I^ ]¼^4rrt^ ]¬>s½j{^ jlqNwh0rt^mh[y{q o[rZ4\_^mhistj{hi9^ /*^h56 I/;%e ç x9 ±7q y{[^]ylw[^j²M ¥»b Ab´w[rtr^4iy B AC®q Hjy{]^]\_\G^]Hj j{wisTr^]h®[st¬N^4y&hzhkgihkjlZ]\G^mh &Y[ s Åyl^mh®hlstN[]hzuHw[s< ykj&} ^]HjK4^jljl^TylqN[ylsj{ µ}^4jr^]h®q Hj}M^]rtmh®nºn¥§4q>[^]hn¿ß
(38) q W_ ¬>s^4Hji^ L* %e Kj) /; vÀ »rAqNNjnylqNMqHhki^mhn}r qNylstjlY[\_^mh/[^qNH¬N^4y&hksq
(39) i^[qN\o[y{^]h6my{s½j&hn[}Mh^]hThkgihkjlZ4\_^mh¬ ^]y{h6^4jTi^4[wsh rt^mhnhlg>hkjlZ]\_^]h6whlw[^4rvh] »rh6qNNjn^]Hjly{^0 wijly{^]h/qNoijl^]Hw.r^0ylmhkw[rtj{}jnhkw[s¬J}Hkj ] Ç o$Æ¸Ë / p Ç rqEs Ç f tvu^wew^x\yQz{ |· }ihQ 5
(40) %^* K4;#$%9* 0.~ 54 G lkm HË / n 1 { £ · E L,'&O/6 ( H' H Z\[ Z- H Ze S5SGS
(41) HS Z3[ Ze SGS5S * %_LG 5T 5 5* 6 K 5%^T · ª H ·*« ª S e «i ¾ ¾ · f¾ &26/ SGS5S: S S5SGS_7Q H «ª R ª i (* %Dk:
(42) ) ( O 5 5* 6 K 5%^2 { A 6 f (: %e%9%* 5%^O Z3 [ +
(43) Ze%e Ze/. j %-4 5 Z\[ Z- · ,' 5 * ,^45$ * *E4Uc9'& /6 ¡¢ Z\[ Ze Z- S5SGS; S Z3[ Ze S5S5S£ * %`LG 9 { n 5* 6 5 5%^ ¤ j/*f 6 k f ; 5/%O 7//} * h 5/+*^ % ¥4$ 5* ; /*94UH/6 c§· ¦}¾ ¨ © ª ª
(44) §· ¦}f¾ &6/ 6 (,e: $ : (; )%e%d%^* "a5* § * Oci ;/j
(45) 2
(46) k +%- ,
(47) ,^Q 54 ci 5.*
(48) xK}y^¶i^]\Grt^Nµi^4.o hl^¾iµ¼ ª R
(49) R VªJª Vªc^]hkjnw[*nº¥»qii^Nµij{ isvhuNw^ Gª R
(50) RR Vª Vª µMuNws¼yl^][y{]hl^4Hjl^crt^ \ «]\G^0[qN\o[y{^0[^r ^mh$jnNh4 ¬zoNhk^ µJjlqNwijl^mhzr^]hKmy{s½j{w[y{^]hK[^7q \_<q ylj{ Nj®uHw[^r^]h&Y[ s <y{^]2h VªNµ R0^4jTªnhkqNHji^]h®nº¥»qii^mhT¿¨^4j®q _<^4wij/yl^][y{]hl^4Hjl^]yz}shks jlqNwhr^]h6[qN\o[y{^]h&À ¬z.oNhk^_ ª R['µ 'S5ª >f¾ HV VGª 'fV ®V ^ V VGª R'H¾V V5ª _qNh$j{s½j{w[^r^i4o[w[jTi^0r_y{^4[y{]hl^4Hj&Jjlsq %i^ ¯M}yw[*nºn¥§4q>[^ 7 hrv hkws½j{^ µ[[qNwh[^pq Mhksvi4y{q Mh/uHw[^[^]hnnºn¥§4qii^]h^]*o hl^¾.¿ß ^mh$jl¥ }¥§isy{^0[^]hy{^]4q>} ^mh6i^¢/q>q}jlY<À ºTq}jly{^o[wij i^]hi^n\_q Hj{yl^]yzuNw sr^mh$j shlni^n\ [stw[rt^]y®i^mhz[qN\o[y{^]h9y{^4[y{]hl^4Hj{]hK[ h®4^nhkgihkjlZ4\_^NN^]\_ y{uHw[qNhjlqNwij }o<q y&_uHw[^nrv [y{q [y{st4jli^mhzhks ^]hK}rtjl^]yl]hKi^mhz&Y[ s <y{^]hK w[ nºn¥§4q>[^o[stM}syl^6<^4y{\G^4j®i^r^]hK\_4\_qNylsvhk^]yKi^\ }[sZ4y{^q \_N j{^¿¨srhk'w °Gj i^T\_]\GqNylsvhl^4yKr^7hlstN[^niw[y{^4\_s^4y®&Ys Åyl^T[q i¥§>w[rVµ [w[svhz^]hlw[s½j{^7i^Tyl^][ylmhk^]Hjl^4y®rt^mOh ±pª y®r^no[stj7ª µN^j®r^]h®¦4]ylqHhK}yrt^To[s½j RNÀ Í ^jljl^y{^4[y{]hl^4Hj{}jlsq 4q \_N jl^6^mh$jKsNj{4y{^]h{hl Hjl^ 7w[jlsrtsvhk^]yz<q w[yKw[ hkjlqi&ÄJ}N^^]p\_4\_qNsty{^hk^mq M[}syl^N°xqNw[yK\ }[s[w[r^4y ^mh4q>[^]hzrq y&h{uNw^n[q wMh®^ <^m jlwq h/i^]hq <4y&Jj{stqNhz ylstjlY[\_4jlsvuNw^]h]µN[qNwhzy{^4ylmhk^]Nj{q h®w[[qN\o[y{^ ²i´^ ³´&Y[ s <y{^]h r s[^ i^_i^4wi¶¬ ^]jl^]w[y{h7i ^ ³³o[stj{´h ²9µÂ^4´j ²8Z³jl^]rhuHw[^ ²¶·²9µ#« ²1Zz Í q Hjly&}syl^]\_^4Hj ¸r%yl^][y{]hl^4Hj{}jlsq h/;¸vU 5/7/ B Q CVµ [q wMhTs\_MqHhkqNhTw[^py{^4ylmhk^]Nj&Jj{stqN*w[[svuNw^Giw)&Y[ s <y{)^ Rµ
(51) hl°¬Nq syn¯ µ ¯ Z ¹R[ Í ^jkj{^G4q Hjly&}sHjl^H}y&}Hjlstj7uHw[^ ²9µ ^j ²1Z#[^qNNj{st^][[^]Njn ww[^0&YMQ Pç[^i^ ªi^rtqN[ w^4w[yhlw[M]yls^4wyl^ Gw[¼ ºL»¼Ç 1q Ç ¼ zk{ /?³D¾½\¿r %e h5; Gª R
(52) R'VªJGª RR Viª 5®ª Vª ª Vª}
(53) ª Vª}ª VKª '! 457Q 5%^O; #,^ 07/ K % Gª RR VªJ ª R
(54) R VL ª À Gª R
(55) RÁR[Â ª R
(56) R
(57) ÃR D« ÀR
(58) R
(59) RÁGª Â RR
(60) R[Ã ª fª Vª}ª Vª}ª Vª ª VLª À ª R[ ª ÁR[Â Gª RGª ÃR I« ÀRGª RÁGª Â R[ ª R[Ã ª <. =. Ä(Å. ÄÆ. Ä(Å. ÄLÆ. 7 hpr)hkws½j{^
(61) i^
(62) ^4j_ ykj{s4rt^Nµ [qNwhG[y{]hl^4HjlqNhi^mh_}r q y{stjlY[\_^]hp [istjlsq Q¿ßhl^] j{stqNh *^4j_HÀ^4j_i^%\w[rtjls[rs]Jj{stqN ¿ßhl^]jlsq .HÀ[ºnq wMh6[ylqNMqHhkqNh¤ rt^]\G^]HjTuNw^4rvuNw^]h6q [jls\Gsvh{Jjlsq Mh6MqNw[yst\_[rv}Hj{^4yr ]rt]¬°}jlsq .}w.4 yly{G¿Rhk^m j{stqN*NÀ ¹æ. Ç"Ç È'ÉÊËÌ.
(63) . :%^#¿v*^J *^ '2 5 /%D$: ^ I* } . %
(64) & 4'T5 /'G5 69 A'1 '
(65)
(66) FA^74 r4w[rÅ w¸[qN\o[y{^ i^ M³³ª z&Ys Åyl^mh®^]h/;¸vU 5/7/ /ylmhkw[rtj{ Hji^cr [[s½j{stqN¸i^7[^4wi¶¸nºn¥§4qii^]hi^³´&Ys Åyl^mh 4q hkjlstjlw[^Tw[4NhK}yljlsvwrts^4y®[^nr }r qNylstjlY\G^ is½j{stqN ^4_j{^4\_h®4q hkj{ Hjz^49h/;¸vU 5/7/ z^4j®M^]yl\_^4j®w[[^Thls\Grtst¤M4}jlsq i^pr q <4y&Jj{^4w[y7[y{q <qNhl[}Mkh BÓQC§µ <~n¤i^Gij{^4y{\_st[^]yTr^]h7o[stµ j{h7MqHhkstjlst¡ßh7[^ µ<µ [q wMhT is½j{stqN[[qNh²9µ ^4j!)µ µ6eFA^pqi[ ^ Niq ij{F[qNwhH}y&}Hjlstj¸uHw[^hlsc¯ ª ¿¨y{^]hlM^m jls¬ ^]\_^4H#j " ªmÀµ/ rtqNy{h¯ µr[ ¶R ¿¨y{^]hl<^] j{st¬N^4\_^4H$j " µ?[ ¢RHÀ /x9 y 4q hl]uHw[^]Njmµ<hlsr [[s½j{stqNFi^¯ µ ^ji%^ " µ N4[Z]yl^w[[^Gyl^4jl^4>w[^%¿ß} ¥ [ ¥§£Åhls ¯ µ " µ ª°À µÅ^4jkjl^_i^]yl[sZ4y{^[^Ghk^]y{[ylqN}N4^ uHw[^ w[[^<qNhlstjlsq ¼[ºTq wh6qNoijl^][q h stMhks & µ ¯ µ " µ & µ ¯ µ Z\[(' " µ Z3[ & µ µ?[ ߯ µ µr[ ' " µ µr[ *)%¯ µ " µ -,+.- R S5S5S ³.« 0ª / FA^ \ «4\_^[y{stMsM^h [[rsuHw[^¸4¬>s[^4\_\_^4Hj}w³]}rvw[r®[1^ Zzr F N[istjlsq °¬ ^m_q >¬N^4y&hksq ^4¾ h/;¸vU 5/7/ h ^ <^m j{w[^ r s[^_[^¸/¾ ³ 4^4rrtwrt^mh '/ ¥
(67)
(68) ¿ßq 3w 2T~cÀc^ji^4¾ $³´« 5ª T<q yljl^mh0rtqN svuHw[^]7h 6c9 8
(69) ¿R}r q y{stjlY[\_^
(70) mÀ 3 FA^ &Y[^4\_s)4ylstjlsvuHw[^ i^ r qNM]y{}jl^]w[y4q Hjls^4Hj sthlsw[^n<q yljl^nrq NsuHw[^ :96c ^4j®w[[^7MqNykj{^nrq svuHw[9^ 6c¿¨¤ w[y{^mÀz^j®r^7]}rvw[r<h ^ Å^]jlw[^NµN w[<q sHj i^¬>w[^0j{Y[4qNylsvuHw[^ µ>^]%j{^4\_hn4q hkj{ Hj] <; Ë > =¨ m ¼~[istjlsq *°¬ ^m74q >¬ ^]y{hlstqN^48 h/:/¸vU 5/7/ º pÅ4 >Ç T^]wi¶
(71) Tº¥»qii^]2h ²Î^? j ! i^ ³)&Y[ s <y{^]h @ Ë ° = Ç `iq \_\_ ^ i^ ²Î^?j ! qii]^0^48 h/;¸vU 7Q T°¬ ^m% $³K ª 6&Y[ s Åyl^mh ^ DµGF ²9µ ' !kµ6Hµ D)ZIF ²1Z ' !KZ®µ JµGF ²9µLKM!)µ ^47j JKZNF ²1ZOKM!KZ
(72) r }svi^ i^/¾ ³ ACB Í r4w[r^4y^4) y{ rtrZ4rE 4^4rrtwrt^mh ' ¥
(73)
(74) 5QP ^ LµTFVU µ Z3[ ¾ 4¾ WJ µ Z-YX ) D µ Z3[ZX [ 'O[ µ ^\j ZLFVU Z Z3[ ¾ 4¾ WJ Z Z-YX ) D Z Z3[ZX [ 'O[ Z r }svi^ R B Í r4w[r^4y/^4 y{ rtrZ4rS [^4¾ M³
(75) « ª /<q yljl^]h6rq NsuHw[^mh 6 P y+3 x+3 y+2 x+2 y+1 x+1 y+0 x+0. y−3 x−3 y−2 x−2 y−1 x−1 y−0 x−0. HA. HA. HA. HA. HA. t +4 = v+3. u+0 t +3. 0. t +2. t +1. HA. HA. HA. t −4 = v−3. t +0. u−0 t −1. t −2. t −3. 1. t −0. Additionneur en base deux traditionnelle. 1. Recodage de Booth. d5 d4 d3 d2 d1 d0. 0 s+4. s+3. s+2. s+1. s+0. Addition et conversion en borrow−save. . Soustraction. . s−4. s−3. s−2. s−1. s−0. ¸~[[s½j{stqN.[^0[^4wi¶.nº¥»qii^mhi^uHwJj{yl^0&Y[s <y{^]h]. ]\^`_a ?b ced. Ù ê°ÒAÞkÙ æmÒVÑÓã°ì$ÝßÕkÝßÙ æ°ÒÔRÙ êYfßÙ ê°ÝßÒã°Õ$ÒAà ٠ݨÔRÕkÒ ØÓÙCg ÑÓé]êmÕkÒ hãmÕkê]ÖcÕ$æÔVÝßì$ÕkÒ å d ÇAåji.
(76) .
(77) "!
(78)
(79) #$%'&(
(80) ) )* +# ,-.$/0$/%1 2 3! 45647//0$/%8*9:;4.
(81) % '
(82) Î
(83) , ¹ / % F }r qNylstjlY[\_^pjlwMist wM}y&} y&}Y[^[y{]4]i^]Nj4q hkjlstjlw[^Grv%yl^]\GsZ4y{^Gj{ M^ i^ r [istjlsq #i^ i^]wi¶nºn¥§4q>[^]h0i^ ³ &Y[s <y{^]h ² ^Sj ! iºnqNwh®4q >¬ ^]ykj{sh{hkqNhK^]hkws½j{^Trvy{^4[y{]hl^4Hj{}jlsq d h/;¸vU 5/7/ iL^ ²¹.! ^4%q \_[r4\_^]Nj i^]wi¶ r s[^ w[[^hlq whkjly& jlsq ¼µH[w[svhqNh$j{ylwshlq hr^0Tº¥»qii^ h{hkqi4st p4^0ylmhkw[rtj{}j] Z)j&}Hjw[%\_q j[ ^ $³  ª o[stj{h]µ M³ Ã0¾ o[stj{h hkqNHj[m^mhlh{}syl^mh 7r my{s½j{w[yl^6i^ $S« kZ A^]p4q \_[r4\_^4Hj 7[^4wi¶£°xK}yKqNhkmuHw[^4Hj]µ Lµ07« Z
(84) ^]hkj9]N rt^]\G^]Hj w[G[q \po[y{^ iE^ M³ ³0¾ ®o[s½j&h4`is£[q wh6 [[rsuHw[qNh®r^0yl^mqi[ ^7[^c¢/q>q}j{Y G^0ylmhkwr½j&Jj]µH[qNwh6q oij{^4[qNhw[
(85) Tº¥»qii^0q \_<q ylj{ Njmµi[}h ^]ykj&}sh4Nh4µ $³K 0¾ 6&Y[ s Åyl^mh4 ºL»¼Ç 1q Ç u3z { $ %² V5ª RR
(86) R8 5 ! V5ª R
(87) RR£ *bk:
(88) k f ; 5T¿k! / H/M f K ¤ ; 5U* %e
(89) µ RR
(90) RR
(91) R£ Z9 ª R
(92) R
(93) RRr%e 3¥/~ ,Lµ 5 kS « Z (G 4 L 5%i/ L,e645 5%^§ck
(94) 5*
(95) &\%e/*f (
(96) h5U 5%e%' (D Lµ$ $S« kZ T|RR
(97) R
(98) RR
(99) R-%ª Gª RR
(100) RR ¿" k; G 5H ) ´JG 9
(101)
(102) D * %D%-/ Kh: 1 VGª R
(103) R
(104) RRD
(105) ® : (;
(106) ^* U #¥/~ &T6 a5*^ L² ! ª R
(107) R
(108) R'&?%-/*f
(109) h5U 5%e%' D ¢RGª R
(110) RR
(111) RRR
(112) RR
(113) R
(114) R| 5 ª VGª R
(115) R
(116) RR( K/;a5*^/%^ 9a*^ 9:4 * .j/K
(117) 5 40H/} K % * %e £40* ; d * ® : (; 1 ª R
(118) R
(119) RR b*q Hj{ylqNh®uHw strÅ^mh$j/jlqNwJ|$q w[y&h/MqHhlhlso[rt^ci^0qii^4y 1² ! r }svi^c[^ $³) ª &Y[s Åy{^]h]HºTq wh/wij{strshlq Mh®w[^7[y{q ylsj{ i]4q w[rv}Hjs\G\_mis}jl^]\G^]HjTi^r_i4¤[stjlsq *iw*nºn¥§4q>[^04q hlsvi4y{c}hn4^jn}yljlsvr^ Ë q> = 1¼ z){ -² O* %_k:
(120) )
(121) L³¾: (; &O/6 )ª ²`ª ²2 ¾ Z\[ F¼^¸jl^4y{\_^%i^ M³9Q0¾ os½j& h µ « Z [ ykj{st^]Nj_}shls - «T¾ SGS5S ¾ /H? FAq y&hluHw[^ ^]hkjGhkjly{sjl^4\_^]NjG[4HJj{s½¡$µ [q wMhTimiw[svhkqNhTi^pr¸i¤s½j{stqNFiwF4q \_[r4\_^4Hj ¸i^4wi¶*uHw[)^ µr [ ª xK y7qNhl]uHw[^4Hjmµw[[^¡¨q svhTr^y{^]qi[ ^i^ ¢/qHq jlY ^ <^m jlw µ µr[ R^j®j{q whr^]hzq \oyl^mhzi^Tr ^4 hl^4\po[rt^ - «T¾ S5S5S «0ª /Thk^ny{^4ylmhk^]Nj{^4Hjz y®w[¸Tº¥»qii^Ti^ M³# ª &Y[ s <y{^]h][`>w[[<qNhlq h/\ }sHjl^]}HjuNw^ + - R S5SGS ¾ «³ª /H Í q \_\_´^ µr[ | |R[µ>rt^0y{^]4q>} ^ci^0¢q>q jlY%[^ 2[y{q>[w[s½j Fq w[¬N^]}wÃw[ nºn¥§4qii^
(122) [3 ^ $³95ª &Ys Åyl^mh4§ F¼^.hl^4w[r4NhG[ylqNo[r4\ Jj{suHw[^
(123) hkw[y{¬>st^]Hjp y_4q hl]uHw[^]Hjprq y&hluHw[^ e¾ ] }[rtsvuHw[ kR"ª ÀR( SGÁS5 SRà r^Ty{^]4qi[}N^ni^7¢/q>q}j{Y 4Z4y{^nrt^7[q \po[y{^Ti%^ $³0¾ z&Y[s Åy{^]h7ª V\ª ÀR( S5ÁSG S;Rà ^T\_qiis½¤<4Jj{stqN¸i^mh4^4rrtw[r^]h i^y{^]qi[ ^0i ^ µ ^4
(124) j Z <^4y{\_^j6jlqNwijl^4¡¨q svhni^0yl^][y{]hl^4Hjl^]y¾ _r s[^0[w.Tº¥»qii^i1^ M³ ª 6&Y[ s <y{^]h0rª ÀR(S5ÁSG SRà ] . hls3 µr[ |Rp^4jL ª_¿R} ¥ ¥»£" U¾ À ! Z\[ hl3s µr[ | R.¿R} ¥ ¥»£" +.- R S5SGS ¾ « ª /JÀ R hls[q HV µr[ $ H)_ H Z3[ # R V Z3[ hkhk"ss[ q µr[ ª ¿ß ¥ [ ¥§¼" +.- «T¾ S5SGS «ª /°À. µ . . . Z . ª. H µr[ HV 3Z [. . F¼^yl^mqi[}N^*i^¢/q>q}j{YÂ[]4^]h{hkstjl^F stMhksci^4wi¶Â4j{}N^]h[^F<q yljl^]h¸rtqN svuHw[^]h ^4j
(125) h ^ <^m j{w[^F]N}r^4\_^]Njmµ® w[ <q sHj
(126) i^´¬>w[^ jlY[]q y{suHw[^NµN^] jl^]\_h4q hkj{ Hj] Í ^4j/4j{ ^Ti^c4 r4w[rM^mh$j6qNh$j{s½j{w[7[^% R¾³³ª zMqNykj{^]h®rtqN svuNw^]h®~7ºTpµi w[^7<q yljl^TrtqN svuNw^ 6c^4j0i^ /¾ ³ s>¬ ^]y{hl^4w[y&h] F }r qNylstjlY\G^_ y{]hlw[\_^pr^]hci s <]yl^]Hjl^]hc4j{}<^]h0i^Gr N[istjlsq Fi^_i^]wi¶Fnº¥»qii^mhc^j0r¸¤ wyl^% strrwh$j{yl^ r }y&&Y[s½j{^]jlw[y{^G[^ r q <4y&Jj{^4w[y4q y{yl^mhk<q }Hj] »rz^]hkjqNhkjlstjlw[ w[ is½j{stqN[[^]w[y0jly& istjlsq [^4rn¿R [istjlsq [^4w[y yl^4jl^]Hw^n[y{q 4^Nµ is½j{stqN[[^]w[y®[y{¤¶>^Nµt]4&À9^j[^Trq svuHw[^nhlw[[[r4\_^4Hj&}syl^n<q wy®^ Å^]jlw[^]y®rv4q >¬ ^]y{hlstqNp^]9 h/;¸vU 7Q 4µ ij{^4y{\_st[^]y7r^p4q \_[r4\_^4Hj i^4w[¶*[^¸S« Z³^4jcy{^]4q>[^4ynr^y{]hlw[rtj{Jjm<c}MhTr^[sy{^pi^mhT4Nh4µ<[q jly{^}r qNylstjlY[\_^}|$q wij{^w[[^ MqNykj{^ :96cµiw[[^<q yljl^ 6cµ[[^4wi¶
(127) MqNykj{^]h~Tº7 ^jT[^4wi¶
(128) st>¬ ^]y{hl^4wy{h/}w*&Y[^]\Gs.y{s½j{suHw[^i^r N[istjlsq [^4w[ym w[MqNstHjc[^p¬>w[^prtqN svs^4rVµ<4^jc rtNq y{s½j{Y[\_^h °¬NZ4y{^[rtwMhc4q $ij{^4wi¶£- FA¸[yl^]\_stZ]yl^4j{ M^Gi^pr }r q y{stjlY[\_^p[]4^]h{hkstjl^ ^4rrt^hl^4wrt^ uHwJj{yl^_shkjly{w j{stqNh4 »r®^]hkj[ylqNo}ort^]\G^]Hjc[rwh6|$wMis4st^]wi¶)[^4}rvwrt^]1y M²9µ « ²1Z v W!µ « !KZ
(129) r si^ i^ i^4w[¶¸hkqNwh$j{y{N j{stqNhK^j w[[^0 [[s½j{stqN¼µNw[sh6i^7yl^mqii^]y®r^7y{]hlw[rtj{Jjm [^0}wij{yl^Thlq rwij{stqN¸hk^]y{ s½j6 jlwMist^]yr stHjl] y&Jj{stqN¸i^]h i^4w[¶%j&}N^]h6i^rq svuHw[^ [[s½j{stqN[[^4rvh[ h6w[.[y{q>4^]h{hk^]w[yqN\G\_^]y{4s r<^4jTi^qN\Grt4jl^4ynhlq p|$^4w* sh$j{ylwM jlsq Mh4 ºL»¼Ç 1q & Ç % z(' /%^ $ 4:/%' k
(130) OkG ´
(131) 5* f (; NT%D 3! .*f 0; 56 T¥/%e50$/%^%- K 5%^
(132) ´;! / H/M f K ) "!
(133) /0$/% ,e;,'/ 54G(¿r ,-;:,-$4j4 ¤ %e $a*^ da5*^ ¡,
(134) ,^/0$ %c - (« S5SGS / Nr
(135) * ; © ) /*f 6 i%e h5; b
(136) D 5k %^ 5 Kh6 _ 5 ; j,e;4 5 %U 5%^2cK! $
(137) K 3! * %I
(138) )
(139) f (; 5. ¹æ. Ç"Ç È'ÉÊËÌ.
(140) A. :%^#¿v*^J *^ '2 5 /%D$: ^ I* } . <; Ë >=¨ m ¡u ~[istjlsq *i^i^]wi¶%nºn¥§4q>[^]hi^uHw}jly{^0&Ys Åyl^mh4 º pÅ4 >Ç T^]wi¶
(141) Tº¥»qii^]h2²Î^j?! i^³)&Y[s <y{^]h @ Ë ° = Ç *nº¥»qii^ 1² !!i^E M³iª5 /&Y[s Åy{^]h ^ ² ! ^]8 h:;/¸vU 57Q ¿ß}r qNylstjlY\G^GmÀ ACB Í r4w[rAi F ² !NP R B Í r4w[rAi^ ¹^4*4q \_[r4\_^4Hj _i^4wi¶#M¿ ³ 0¾ o[stj{h&À F Lµ´« ZEP Ë B µ F RSP Z F P ª ³%« ª Ì Ë Ê B Ë É B µ F V 0 H Z3[ P Z F HV Z3[ P Ì Ë B Ç p B µ. F. V µr[ $ )b Z\[ P Z F rµ [ V \Z [ P 0000 1111. 000 0001. 1110. 1101. 001 0010. D négatif. 0011. D positif ou nul. 1100. 0100. 1011. 011 110. 010. 111. S négatif. S positif ou nul. 101. 100. 100. 0101. 1010. 0110 1001. 0111 1000. (a) Représentation de D en complément à deux. (b) RN−code représentant S. < K¶i^]\Grt^ is½j{stqN
(142) [^i^4wi¶
(143) nº¥»qii^mhni^i^4w[¶
(144) &Y[s <y{^]h]. ]\^`_a 7bi¬. %, -./0 '
(145) % , ¹[/
(146) ~n¤*i^\w[rtjls[rst^]yni^4w[¶
(147) nºn¥§4q>[^]h2²Î^j?!µiq wh6<q w[¬Nq h6M}y^¶i^]\Grt^4 r4w[r^4y ².!· M² µ « ² Z W! µ 3« ! Z T|² µ ! µ « ² µ ! Z « ² Z ! µ 1² Z ! Z ¿$°À ^] qN\_[rt]\_^4Hj i^4w[¶#^4j_4q >¬ ^]ykj{styprt^%ylmhkwr½j&Jjp^]Ãw[ nº¥»qii^Nv FA´ylqN[ylsj{´¸H}y&}HjlstjGuHw[^%rt^
(148) [y{qiiw[stjG}}yljls^4Hj - «T¾ Ze S5S5S
(149) ¾ Z->/*^jq wh_wij{strsvhkqNh_r) \ «4\_^´Nh$j{w^*uHw[^´st¥»i^]h{hkwMhG}¤Âi^.rt^´4qii^4yhlw[yM R¾³ «ª _&Y[s <y{^]h]TF¼^mh [y{qiiw[stj{Lh ²,! ^4Lj ²,! M^]w[¬ ^]HjT4¬>svi^4\_\_^4Hj7hl^4 r4w[r^4y r }svi^i^\pw[rtjls[rts^4wy{hT[q ´hlstN[]h]x®rtwhls^4w[y&hn[y{q ylsj{]h i^mhnnºn¥§4q>[^]h6<^4y{\G^4jkj{^4HjjlqNwijl^4¡¨q svhi^]hnhls\Grtst¤M4}jlsq h6q}j{ o[r^]h] . .
(150) . "! # %$& ' (!*) ºTq wh6imy{st¬Nq h/w[[svuHw[^4\_^]Njr^4}rvwrÅ[^´²,! 6'FA^]h6[y{q [y{sjlmh/^jn}r qNylstjlY[\_^mh®[y{q <qNhl]h/h [[rsuHw[^]Nj]N}r^4\_^]Nj ²(! 6 Ë q >= u\z){ r² * %iLG G { £¯ ª&/}/ n¯ µr[ ¯ Z\[ R9 5 ¯ |Re Í ^4jkjl^*ylqN[ylsj{
(151) hl^*i][w[s½js\_\_]isvJj{^4\_^4Hj¸i^.rv)[¤[stjlsq  w[Ânºn¥§4q>[^ z~n[rtsvuHw[4^´ w ]}rvw[rn[^b²H ! 6 µK^]rtr^ sisvuHw[^7uHw[^Thls[¯ " + ª µ jlqNwhzr^]hzjl^4y{\_^]hz[^nr¡¨q y{\G^¯ Z3[ "-H, µ ¯ µr[ "->, µN¯ , " + Z\[ ^4j¯ , " + µr[ hkqNNjz>w[rvh]µ #. + - R S5SGS ³«Fª / ¿¨¤ w[y{^ HÀ '¬®rtr^<^4y{\G^4jn4H}r^4\_^4HjTi^y{]iwsty{^i^\_q stjlsrt^q \oyl^i^ylqiiw[stj{hn}yljls^4rvh _ [[s½j{stqN[[^4yp¿ç¤Nw[yl^N]À'¬z ^ <^4j]µ ¾ µ + ߯ " + ³¯ µr[ " + Z\[ ¾ µ + ߯ " + )%¯ µr[ " + Z\[ . å d ÇAåji.
(152) .
(153) "!
(154)
(155) #$%'&(
(156) ) )* +# ,-.$/0$/%1 2 3! 45647//0$/%8*9:;4. ºnqNwhhlq w[Y s½j{q h0N[istjlsq [[^]yrt^mh0[y{q>[w[s½j&h ykj{st^]rh0^] / /U 57Q %r }svi^i^qN\_[yl^mhlhl^4wy ¾0 n}[<^4rq MhuNw^^]h q <4y&Jj{^4w[+ y&h/ 4Z4y{^4Hjw[*&Y[s Åyl^0i^hkqN\_\G^ + + /^j7+ !« ®o[stj{hi^0y{^jl^]>w[^hkqNykj&}Hjl^ + ykj{styi^ o[stj{h[^c\ «]\G^ MqNsh ^4jTiE^ « ®os½j&hni^y{^jl^]>w[^0^4Hjly&}Hjl^ Z\[ B]QC ¾ + Z + + Z3 [ + Z + Z3[ »rK^]hkj0[q pstHj{4y{^]h{hl Nji^Gyl^] y{q w[<^4yTjlqNwh0rt^mh7jl^4y{\_^]h0i^G \ «4\_^_MqNs[h[^]hc[y{q>[w[s½j&hc}yljls^4rvh0[}h0i^mhc¬ ^]jl^]w[y{h0i^Go[stj{h + , + S5S5S + uHw[sKqNh$j{s½j{w[^4y{q Hjcrt^mhT^]Nj{yl]^]hci^]h04q \_[y{^]h{hk^]w[y{h¿ß}r qNylstjlY\G^G¸^4j7¤M w[y{^pN]À źnqNwhcq oij{^4[qNhc}shks R¾ ³%« ª /\_Z\q}[ j&hi^qN\_MqNykj&}Hj6[^ ª M³r¦}¾ o[stj{h] . .
(157) . .
(158) . . . . . !. . . . (7) + + λ 0 = x1 y6. x+3 y+0. x+4 y+5 x+4 y+4 x+4 y+3 x+4 y+2 x+4 y+1 x+4 y+0 + + x5 y5 x+5 y+4 x+5 y+3 x+5 y+2 x+5 y+1 x+5 y+0. . . . x+i y+j = 1 x+2 y+5 x+3 y+4 x+4 y+3 x+5 y+2. x+1 y+5 x+2 y+4 x+3 y+3 x+4 y+2 x+5 y+1. x+0 y+5 x+0 y+4 x+1 y+4 x+1 y+3. x+0 y+6. x+0 y+5 x+0 y+4 x+0 y+3. x+0 y+2 x+0 y+1. x+1 y+5. x+1 y+4. x+1 y+2. x+1 y+1. x+2 y+6. x+2 y+5. x+2 y+4. x+2 y+3 x+2 y+2 x+2 y+1. x+2 y+0. x+3 y+6 x+3 y+5. x+3 y+4. x+3 y+3. x+3 y+2 x+3 y+1 x+3 y+0. x+4 y+5 x+5 y+4. x+4 y+4 x+5 y+3. x+4 y+3. x+4 y+1 x+5 y+0. x+6 y+3. x+6 y+2. x+6 y+1. x+4 y+2 x+5 y+1 x+6 y+0. x+i y+j = 0. x+0 y+3 x+0 y+2 x+1 y+2 x+1 y+1. x+0 y+1 x+0 y+0 x+1 y+0. x+5 y+6. x+2 y+3 x+2 y+2 x+2 y+1 x+2 y+0 x+3 y+2 x+3 y+1 x+3 y+0 x+4 y+1 x+4 y+0 x+i y+j = 0 x+5 y+0. x+6 y+6. x+6 y+5. x+1 y+1. x+1 y+6. (7). x+4 y+6 x+5 y+5 x+6 y+4. (2) + + λ 0 = x0 y2. x+2 y+5. Λ. (a). x+5 y+5 x+5 y+4.
(159) . ". x+0 y+5 x+0 y+4 x+0 y+3 x+0 y+2 x+0 y+1 x+0 y+0 x+1 y+2 x+1 y+1 x+1 y+0 x+2 y+1 x+2 y+0. x+1 y+5 x+1 y+4 x+1 y+3 x+2 y+5 x+2 y+4 x+2 y+3 x+2 y+2 x+3 y+5 x+3 y+4 x+3 y+3 x+3 y+2 x+3 y+1. x+4 y+5. .
(160) . x+2 y+2 = 1. x+3 y+5 x+4 y+4 x+5 y+3. . . . . . (7) + + λ 2 = x5 y2. x+5 y+2. x+1 y+3. x+4 y+0. x+0 y+0. x+1 y+0. (0) + + λ 0 = x0 y0. (2) + + λ 1 = x2 y0. x+6 y+1 (c). (b). ]\^`_ aM
(161) b ~n[[rsv4Jj{stqN%i^7r[y{q [y{st4jl0iÅ¿ßNÀ/`>sů µ " µ ªNµ j{q whr^]hjl^]yl\_^]h/i^7r¡¨q y{\G^c¯ µ " , µN¯ µ " , µ>¯ µ " µ 4^ j/¯ µ " µ hkqNHj >w[rh]A¿ßoMÀ6b*w[rtjls[rs]Jjlsq *i^7!µ£ $ª R
(162) R[ª R[ª5 }y2 ²9 µ1 #R[ªGRªGRª ¿ß]À6n^4 y{q wM^]\G^]Hj[[ ^]h/jl^]yl\_^mh6¯ µ , " +µ [ ^4jn¯ µ , µr[ " +µ Z\[ 7 h7r%hkw[stjl^NµMq whci4q}jlqNh7rt^G&Y[s Åy{^pi^GMqNshc¾ i^Gr%hkqN\_\G^pi^]hc[y{qiiw[stj{h7}yljls^4rvhn y ; - FA^]h7[y{q [y{st4jlmh i^]hTº¥»qii^]h6<^4y{\_^jljl^4Hjn[rwhlst^]w[y&hhks\_[rtst¤M]Jjlsq Mhi^r is½j{stqN[[^]w[y / /U 5/7/ 6sNj{^4y{¬ ^]}Hj[}h6r^4 r4w[r¼i^ ²,! ¿ç¤Nw[y{^c>´À ] bFA^&Y[s Åy{^0[^MqNs[h7¾ [7iw.[y{q>[w[s½jn^mh$j]N}r [ [ bF [y{q [y{sjlG N y{ Hjlstj7uHw[^phls $ª 5ª µ< rtqNy{h $R
(163) R [^y{^j{^4>w[^p 44y{4^[ hTrv¸4q rq [^i^<q sv[h ¾ ^]hkjT}shks£ylqN}N4^0 w[^hk^]w[r^c<qNhlstjlsq
(164) ^jnr^o[s½jT[^0hlq \_\_^ ^]hkjTi¤s¼}y #. . $
(165) . . %
(166) '
(167) [ )&
(168)
(169) [ . ºTq wh6\_qNNj{ylqNh6i^\ «4\_^uHw[^ ' [ ' *) [ xK}y4q hl]uHw[^]Njmµ L R[µ -+- R SGS5S 4/NrF¼^4 r4w[r®iw &Y[s Åyl^i^¸MqNs[h¡¨q ylji^K²,! ^mh$jrtwhhkw[o[jlsr®^jp^¶i[rq stjl^ rv [y{q [y{st4jlhlw[st¬J Nj{ ^ ] Ë q> = % z){ -² O* %_k:
(170) )
(171) L³¾: (; &O/6 ¾ R¾ « ª \³¾ 3,' ª ² ª ² ! ¾ R¾ « ª 3 ª %e% Í ^jljl^ [y{q [y{sjl_[qNwh<^4y{ \_^ji^\_qNNj{yl^]yuNw ww[^ [ylqN}HJj{stqNi^ y{^j{^4>w[^ [^ [y{qJ¬>st^]Hj[^ r*4q rq [[^ i^ <q sv[h ¾ Ze 7^[rtwMh4µ Ze Z- ¯ Z3[ " Z\[ ^j Z- |R b. #
(172) . . '. $
(173) . ¹æ. Ç"Ç È'ÉÊËÌ. .
(174) .
(175) . &
(176)
(177) .
(178) . :%^#¿v*^J *^ '2 5 /%D$: ^ I* } . <; Ë >=¨ m %c4[]y{}jlsq .i^]h6[y{qiiw[stj{hM}yljls^4rvh<q wynrt^]}rvw[r¼i^²H! 6 º pÅ4 >Ç T^]wi¶
(179) Tº¥»qii^]h2²Î^j?! i^³)&Y[s <y{^]h @ Ë ° = Ç V/¾ ³
(180) « ª5 ®jl^4y{\_^]h + , + Z3[ S5SGS + ACB Fί " P Ze Fί Z\[ " Z\[ P |R « ª Ì Ë R B Ë ¾R /Ì Ë Ë B Ë µr[ F ¯ + " )¸¯ + µ?[ " Ze + P ÊB + + r µ [ e Z p Ì Ç Ë ÉB Ì Ë B Ç p Ë B ª « ª Ì Ë ¾R « ª Ì Ë Ì B Ë Fί + " + )%¯ + " + P B + p Ì Ç Ë Z- µr[ Ze Z\[ A B FÎ ¯ " P ACACB p Ì Ç Ë A R B µr [ ³.« ¾ Ì Ë AË B Ë ¾R )³.« « ¾ Ì Ë AÊ B Ë Fί + " + )%¯ + " + P AÉB + µ" Z µr[ Z- Z3[ µ3 Z µ3 Ze Ze A B Ç p Ì Ë Z Z\[ Fί Z3[ " Z µr[ P AB Ì Ë A Ì B Ç p Z\ [ )³.« ¾ Ì Ë A B Ë ¾R )³.« « ¾ Ì Ë R
(181) B Ë R ACB + µr[ F ¯ µ3 + Z µ3 " Ze + Z\[ )¸¯ µ3 + Z µ" " Ze + Ze P RCR B Ç p Ì Ë Ì Ë R Ë B Ç p
(182) . !.
(183) . ".
(184) .
(185) .
(186) .
(187) .
(188) .
(189) . $
(190) . ".
(191) . U`>s¼¯ Z\[ " Z3[ |Rµi[q wh4q hlsvi4y{q hO²Ã^j?! 4 q \_\_^i^]hq \oyl^mh/[^1 M³
(192) « 5ª /o[s½j&h^jnqNoijl^][q Mh 4¾ V¾ Z3[« 5ª ¾ «#¾ µ38 ¾ Z- lh s3³F^mh$jnM}sy ² ! ! 4¾ V¾ Z3[« 5ª 3ª ¾ « ¾ µ?[vª ¾ Ze lh stq b. ¼^_[y{qiiw[stj²H! U^]hkj stMhks9 w[)[q \po[y{^Gi^O V¾/³´« 7o[s½j&h0^j}wMw[[^_y{^j{^4>w[^G[^_M^]wijk«j{yl^G ][4y{4^Gq w[ylqN}N4^ [}Mh6r_qNrtqN[ [^i^<q sv[hn¾ Z-} bU`>sA¯ Z\[G " ª µ[q wh7q Mhksvi4y{q Mhw[*q <4y&}Mi^qN\_\G^w[*[q \po[y{^i^k³#o[stj{h7^jTr }wij{yl^4q \_\_^w[*[qN\o[y{^ i1^ M³
(193) « ª /o[ stj{Z\h][ ~nshlsRµ 4¾ R¾ « ª 4¾ V¾ Z3[« 5ª ¾ µr[6«#¾ µ36« ¾ µr[v8 ¾ Ze hls\³)^]hkj6M}sy ² ! ! 4¾ R¾ « ª 3 ª ¾ R¾ Z\[« ª 3 ª
(194) ¾ µ? [/«#¾ µr[« ¾ ª ¾ Ze hls[q F¼^c[y{q>[w[s½(j ²(! ³4 j{ Nj6w[
(195) [q \po[y{^ciE ^ V/¾ ³«¾ o[s½j&h4µ[ ww[^7y{^j{^4>w[^c[^0M^]wij «j{yl^c ][4y{4^7q w
(196) [y{q 4^c[}Mhrv qNrtqN[[^i^0MqNshT ¾ Ze b|>st rt^]\_^4Hj]µ[hlsů Z3[ I" Z\[ ª µ>[q wMh/[]iw[svhlq h6i^0rv[y{q ylsj{7GuNw´ ^ ²H!
(197) ¾ Z3[°' FA^7[y{qiiw[stj^]hkj6 sthlsÅw[ [q \po[y{^ciE^ V/¾ ³«³5ª o[s½j& h4 Í qN\_\G^ Ze ¯ Z\[ " Z\[ ª µi}wMw[[^0[y{q M}N}jlsq i^0y{^j{^4>w[^c[^0M^]wijn[y{qJ¬ ^4sty/i^ r_4q rq [[^i^<q sv[hn
(198) ¾ Z-} F.
(199) . å d ÇAåji.
(200)
(201) "!
(202)
(203) #$%'&(
(204) ) )* +# ,-.$/0$/%1 2 3! 45647//0$/%8*9:;4 x+4 y+4. x+4 y+3 x+3 y+4. x+4 y+1 x+3 y+2 x+2 y+3 x+1 y+4. x+4 y+2 x+3 y+3 x+2 y+4. x+4 y+0 x+3 y+1 x+2 y+2 x+1 y+3 x+0 y+4. x+2 y+0 x+0 y+2 x+1 y+1. (5) λ1. HA. (4) λ0. (3) λ1. (2). (3) λ0. FA. (1). λ0. HA. λ0. (0). λ0. HA. (4) c0. (5) c0. s7. x+0 y+0. (2). (4) λ1. (5) λ0. HA. x+0 y+1 x+1 y+0. λ1. λ2 (6) λ0. (6) c0. s8. x+3 y+0 x+2 y+1 x+1 y+2 x+0 y+3. (4). (6). λ1 (7) λ0. (8) λ0. @. s6. s4. s5. s3. s1. s0. x+0 y+1 x+1 y+0. x+0 y+0. s2. (a) n = 5 x+6 y+0 x+5 y+1 x+4 y+2 x+3 y+3 x+2 y+4 x+1 y+5 x+0 y+6. x+5 y+0 x+4 y+1 x+3 y+2 x+2 y+3 x+1 y+4 x+0 y+5. x+4 y+0 x+3 y+1 x+2 y+2 x+1 y+3 x+0 y+4. x+3 y+0 x+2 y+1 x+1 y+2 x+0 y+3. x+2 y+0 x+0 y+2 x+1 y+1. (4). (6). (6). λ2. (6) λ3. (6) λ0. λ1. (5) λ2. FA. Compresseur (4, 2). (5). (6). c6 s 6 x+6 y+6. x+6 y+5 x+5 y+6. x+6 y+3 x+5 y+4 x+4 y+5 x+3 y+6. (10) λ0. (9) λ1. (3) λ0. FA. HA. (2). (1). λ0. λ0. (0). λ0. HA. (4). s4 x+6 y+2 x+5 y+3 x+4 y+4 x+3 y+5 x+2 y+6 (8). λ1. (9) λ0. s2. s3. s0. s1. x+6 y+1 x+5 y+2 x+4 y+3 x+3 y+4 x+2 y+5 x+1 y+6. (8). λ0. (7). λ1. (7). λ2. FA. (10). (8). FA. c0 s 11. (3) λ1. c0. λ2. (11) λ0. HA. s 12. (4) λ0. (8). (10). λ1 (12) λ0. Compresseur (4,2). c5 s 5 x+6 y+4 x+5 y+5 x+4 y+6. λ1 (4) λ1. (5) λ0. HA. c0. c0. (2). λ2. (5). λ1. s 10. c0. c9 s 9. (7). λ0. FA. HA. HA Compresseur (4,2). c8 s 8. c7 s 7. < Í r4w[r¼i^²9µH!µ rq y&hluHw[^_¿ßHÀO³D ® 4^ j¿¨oMÀO³d> (b) n = 7. ]\^`_a b. F¼qNy{h{uHw[L ^ ³D ® ¿ç¤Nw[y{^7HNÀ µHrv[y{q ylsj{c[q wh/H}y&}Hjlstj/uHw }w4w[[^cyl^4jl^]Hw^T ^]hkj[ylqN}N4^T[^7[rwh w[[^c<qNhls½j{stqN} ºnqNwhq o[jl^4q h6}shls. ^j. #
(205) '
(206) [ ). %
(207)
(208) [ )
(209)
(210) & )
(211) [
(212) L$
(213) )&
(214)
(215) [ #
(216) '
(217) [ )
(218)
(219) [ L$
(220) . ¾R[µ -(+.- R S5SGS ½4/NiTZmhnuHw[ ^ ³
(221) [µ[i^mhy{^jl^]>w[^]h/<^4w[¬N^4Hj}yn4q Hjly{^chl^0[y{q ^4y6i^[rwhlst^]w[y&h<qNhlstjlsq h6^4jn[q wh ^ Å^]jlw[qNhr^]h4 r4w[rvh6^48/ /U 5/7/ FA is6°w[rtjl4q hlshkjl^ _is\_^4hlstqN[[^]ynq y{y{^] j{^4\_^4Hj6rt^mhqN\Gyl^mhlhl^4w[y&h
(222) ¾ ºL»¼Ç 1q Ç x"z{ $ %³d
(223) & ² µ $RGª RGª R[5ª 5! µ $ ª R
(224) R[ ª R[ª5 Nr
(225) * ; Hhj
(226) ^¿k!
(227)
(228) #$%d
(229) ®¯ µ " µ 5?
(230) ®¯ µ " µ ,-;/7/Ga5*^ * %e k; j 5%^*^ a5* 3,e;,'G4 *f 5a5*T! cK6_:6/%%e
(231) ,^/$ Q 7¾ /¯ µ " µ ¯ µ " µ ¯ µ " µ ªf F¼qNy{h{uHw[L ^ ³
(232) µ>rt^4 r4w[r£i ^ ^4jni^ yl^muNwst^]ykj/rt^mh/jly{q svh®j{^4y{\_^]h µ [ ^4j µ[}shls¼uNw^7rvGyl^4jl^4>w[^ N4[]yl]^ [}Mh0r*4q rq [[^ i^_<q sv[h¾ ¼ºnqNwhwij{strsvhkqNh}shlsKw[ q \_[y{^]h{hl^4w[Ey M ¾ 7[q Hjr w[[^[^]h0^4Hj{yl]^]h0^]hkj0jlqNwJ|$q w[y&h04H}r^ ¦4]ylq
(233) ¿ç¤Nw[yl^c o<À Í Y 4w[%[^]h/\_q}j{h µ
(234) ³ «³ªcq \_<q ylj{ Nj µr [ 0o[s½j&h4µiq wh64 r4w[rq hrt^mh6q w[rt^m
(235) h r s[^i^p4q \_[y{^]h{hk^]w[y&7h 0¾ FA%hk^]w[r^i6s ° 4w[rtjl_q Mhksvh$j{^ i4jl^]yl\_s[^4y7rt^p[q \po[yl^pi^o[stj{hci^pyl^4jl^4>w[^[ylqN}N4^i^ r_4q rq [[^i^<q sv[hn¾ Z3[)] b `is
(236) [µi[qNwh6yl^m^]¬ q Mhw
(237) w[suHw[^os½jTi^0y{^j{^4>w[^ bÂc}Mhj{q whprt^mh wijly{^]hG4Nh4µ [q whGh{°¬ q MhuNw^¸rvFqNrtqN[[^¸[^%<q sv[h_¾ Z\[ 4q \_<q yljl^w[ q \_[y{^]h{hl^4w[$ y ¦ ¾ 0¾ 0uHw[s N4[Z]ylO^ ¦}¾ c`« co[stj{h + Z3[ A~nshlsRµ rtqNy{h{uHw[^ ^]hkj stymµ¼w[#o[stjpi^y{^ j{^4>w[^_^4Hj{y{ Nj{^i^ rv*q rq [^i^ <q sv[h¾ ^mh$j]N}r G¦]4y{q`>w[MqHhkqNh6}y^4¶i^4\_[r^ |½' FA_qNrtqN[[^i^<q sv[hn¾ ]^mhlhlstjl^0w[*qN\Gyl^mhlhl^4w[y $ ¾ 44y&}Hj ÕKÝRì$ÒVêmØ Ô§×{Ô ÝRÕ$ÒßÔVÕzì»ÚmÑÓãmÕkèèÕ$æÔ£ÚmÝR×&Ñ>à}Ù ê°Ý . . . . .
(238) . . . . . .
(239) .
(240) . . ". . . ". ¹æ. Ç"Ç È'ÉÊËÌ. . ".
(241) m. :%^#¿v*^J *^ '2 5 /%D$: ^ I* } . w[hl^4wr os½ji^Gyl^4jl^4>w[^ \FA¸4q rq [^pi^pMqNs[h¾ 4q \_MqNykj{^w[)q \_[y{^]h{hl^4w[y ® ¾0 7 [[s½j{stqN[}Hj0su
(242) o[s½j&h0i^ MqNsh®¾ ^4ji^]wi¶Gos½j&hzi^ny{^j{^4>w[^^4Hj{y{ Nj{^
(243) F wi^4wi¶_^]hkjz[m^]h{h{}syl^]\G^]Hj9]N r 0¦4]ylqMµ}M^]yl\_^4jkj{ Hj}shls[w[[^n<^j{s½j{^ hks\_[rs½¤M]Jj{stqN
(244) iw*4q \_[y{^]h{hk^]w[y ¾0 c¿¨y{^4\_[rv 4^4\_^4Hjn w[[^4^4rrtw[r^k>i~ y/w[^^]rtrw[r^72T~7À `>sO³ ^]hkj stymµ ^]hkjw[ \_q}jGi ^ ³r¦}® ¾
(245) os½j&h4 Í q \_\_^ rFq rq [^i^<q sv[hp¾ Z\[ q Hj{st^]Njpw[³qN\Gyl^mhlhl^4w[1y $³r¦}¾ 0¾ c^4j N4[Z]yl#^ $³r¦}¾0 1« 6o[stj{hc i^y{^j{^4>w[^ µM[q wMhn[rv 4q h]N rt^]\G^]HjTw[F4q \_[y{^]h{hk^]w[7y M³r¦ ¾ ¾ [ hnrvqNrtqN[[^[^MqNshc¾ - F¼^mh ¬N^] j{^4w[y&h µr[ ^j µ3 4q \_<q ylj{}Hj&Y 4w[ $³r¦}¾« ª 7o[stj{h]µ£st r9¡ß wij^4 4q y{^prN^4yi^mhqN\Gyl^mhlhl^4w[y&h M³r¦}¾ ¾ ¿ß[ q Hj r w[^ i^]hc ^4Hj{yl]^]h7^]hkj7¤[¶i]^ %¦4]ylq>Àn[ hcrt^mhc4q rq [^]hci^p<q sv[h¾ µr[^4j¾ µ3}¼^4\ }y&uHw[q Mhn\ }sHjl^4M}H j0uHw[^ µ3 µ?[ ^4j µ3 µ3 hkqNHji^]h\_q j{hi.^ M³r¦}¾« «U ª o[s½j&h4 »rz^]hkj¡ßNsrt^i^ \_qNNj{yl^]yuHw[^ rv*hkqN\G\_^i^mh + µ3 µ?[ h ^ <^m j{w[^ r s[^ w[FqN\_[yl^mhlhl^4wy M³r¦ ¾c« : 0¾ Í q hlsvi 4y{q hnr4q rq [^i^<q sv[h $³_ ¾ 5ª Í ^4rrt^4¥§4sAy{^ qNs½j M³r¦}¾c« «³0¾ 6os½j&h i^0y{^j{^4>w[^ 7^jqNHjls^4H
(246) j $³r¦}¾n« « ª ®o[s½j&h + µ3 µr[ in [M^]rtqNh6uHw w%qN\_[yl^mhlhl^4wy $³r¦}¾« 0¾ q \_<q yljl^ M³«F¾ « ^]Nj{yl]^]h] »rM^]yl\_^4jciqNr^j{y{ s½j{^4\_ ^4Hj7 i^m%h $³r¦}¾0« «³0¾ ? M³r¦}¾« « ª ( $³r¦}¾0«Ã¾ 1« o[stj{hci^<q sv[%h M³_ ¾ Q ª xK}y7q MhkmuNw^4Hj]µirv hkqN\_\G^[^]h + µ3 µ3 hl^4 r4w[rt^]N rt^]\G^]Hj°¬N^]cw[´4q \_[y{^]h{hk^]w[y M³r¦}¾c« ¾ [ºTq whN4[]ylqNh}shls i^mhNylqNw[M^mhi^¸[^4wi¶#q \_[y{^]h{hl^4w[y&Eh 0¾ µA
(247) q W ³r¦}¾p« µA^4jp}y{:y «jlqNh0r^[y{q>4^]h{hkwMh0rtqNy{h{uHw[^ MµA ^] hkjk¥ J¥»isyl^ ³r¦ ¾T « ' F _q rq [^ i^ MqNshn¾ Z 4q Hjls^4Hjniq M7w.4q \_[y{^]h{hk^]w[y&h $ 0¾ ][4y&}Hj6w[.o[s½jT[^yl^4jl^4>w[^ Z uHw[^ [qNwhnN[istjlsq [[qNh Z ^j [ Z pr }svi^ w[^^]rtrw[rk^ >i~ ¾ Z [ Z Z ~TsthlsVµ9 ww[^¸y{^j{^4>w[^¸ ^mh$j_[y{q 4^ FrqNrtqN[[^¸i^
(248) <q sv[hG
(249) ¾ Z y{ shlq [^4\_^4Hjp }rq Nw[^<^4y{\_^j_r }rrtqi]Jjlsq i^mhqN\Gyl^mhlhl^4w[y&hzrq y&h{uNwL^ ³´^mh$j6st\_ stym F }r qNylstjlY[\_^7py{]hlw[\_^c4^]h6qNhlsi]y{}jlsq hhkwy/r [istjlsq
(250) i^]h/[y{qiiw[stj{h/ ykj{st^]rh] ~TwFj{^4y{\G^ i^Grv¸[y{^4\_sZ4y{^po<q wMr^pN4[]y{ Nj0i^]h4q \_[y{^]h{hk^]w[y&h4µM[q wh0hkjlqi&Ä qNhc[}hcrv¸¬J ylsv}o[r^ . r^p[qN\o[y{^_i^_o[s½j&h0i^ y{^j{^4>w[^ Z\[ N4[]ylmhK[}Mh9r4q rq [[^n[^MqNs[h/¾ Z3[mNc}MhKrt^n] hKN4[]y{ rRµ}4^[q \po[y{^/^mh$j/hks\_[rt^]\_^4Hjz4H}r M³r¦} ¾ « [ X q wij{^¡¨qNsh]µi^jljl^7¡¨q y{\w[r^T^mh$j6sqNyly{^]jl^nrq y&hluHw[^ ³I
(251) [>ºnqNwh6°¬ q Mhz^4%^ Å^j4 r4w[rt7w[¸os½jni^7yl^4jl^4>w[^0hlq ylj{}Hj{^ [ h rvqNrtqN[[^ni^TMqNs[h/¾ Í ^]hkj®MqNw[y&uNwq sq whz[¤[svh{hkqNh . * 9i« ª ~U&YM uHw[^stjl4y&Jj{stqN i^nrvhk^mqNi^ o<q w4rt^Nµ sr£^]hkj/]^mhlh{}sy{^7i^0is\G^]hlstqN[[^4y6r^c4q \_[y{^]h{hk^]w[
(252) y 0¾ iºnq}j{q h6uHw[^crt^4q \_ijl^]w[y ^]hkj/s[stjlsv}rshl ³Kµi[wsh6imy{4\_^4HjlN ¬Kj&}HjTiqN[[w[.sisv^ µ[str }rq y&h^mh$j6¡ßNsrt^i^0\_q Hj{yl^]ynuNw^ ª # ³r¦ ¾ hl\s ³^mh$j sty^4j ³ ¿RNÀ
(253) ³ ¾ ¾ µ?[ hlstq F ^4¶>yl^mhlhlstqNp4s½¥»i^]h{hlwh9^]hkjKjlqNwijl^4¡¨q svh9^4y{ylqN[4^6hk^s ³D
(254) [ FAq y&h[^r w[[svuHw[^ h{h{} ^6[ h9r0o<q wMr^ µJsri¡ß}wijz^]_^ Å^jzsh$j&}4st^]y w[*4q \_[y{^]h{hk^]w[y M 0¾ ¬z.stHjl] y&}Hjn^4jkjl^si¡¨qNyl\ Jj{stqN%[ hr muHwJj{stqN)¿V À µiq whqNoijl^][q Mh®¤M}r^4\_^4Hj. ª ³ ¾  ¾ . . . . .
(255) . .
(256) .
(257) . . . . .
(258) .
(259) . .
(260) .
(261) . . . . " . ". . . ". .
(262) . . %$'&' (" $*)+ , $*".- ." $/".&10. !#" ! . [^
(263) ¡¨q svhpr^]h_uHw}jly{^%[y{qiiw[stj{h ²9µ,!kµ6vµ ²8Z!KZ®vµ ²9µH! ZQ^Kj ²1Z!)µ ]}rvw[r]h]µ r^.ylmhkwr½j&Jj h q oij{st^]Hj_4q i¡¨qNyl\_4\_^]Nj r muHwJj{stqN´¿$mÀNºTq wh [istjlsq [q hz^49/ /U 5/7/ ²9µH!µ)^jO²1Z! Z sthlsuHw[^n^4j²9µ,! Z´^4j²1Z!)µ ¿ç¤Nw[yl^T NÀ^] jl^]}Hj 4q \_ijl^[w%¡ß s½jTuHw[^ ¯ µ " µ ³¯ Z " Z ¯ µ " µ )¸¯ Z " Z ¯ µ Z3[ " µ Z\[ ³¯ Z Z3[ " Z Z\[ ¯ µ Z\[ " µ Z3[ )%¯ Z Z3[ " Z Z\[ ¯ µ " Z ³¯ Z " µ ¯ µ " Z )¯ Z " µ ¯ µ Z3[ " Z Z\[ ³¯ Z Z3[ " µ Z\[ ¯ µ Z\[ " Z Z3[ )%¯ Z Z3[ " µ Z\[ ºTq whqNoijl^][q Mh}shks¼uHw}jly{^7¬N^]jl^4wy{hi1^ V/¾ ³
(264) « 5ª o[s½j&9h D 2 Qµ D 3 µ J 2 ^9j J 43 jl^]rhTuNw^ ¾ D 3 D 42 |² µ ! µ 8² Z ! Z ^4j0¾ J 3 J 2 ² µ ! Z 8² Z ! µ qW [ 3 U 3 N[ 43 Z- NU 3 Ze ¾R[iºTq wh6^ <^m jlwq h/^4Mhkw[stjl^0w[^0hlq wMh$j{y{N jlsq %J¤
Documents relatifs
1) La recherche des victimations dans les établissements scolaires, à commencer par la Martinique et la Guadeloupe (territoires français d’Amérique), puis sur les
Repérer les signes, les traces et les discours de la commande au sein des espaces de l’édition professionnelle, comme nous l’avons fait pour le cas particulier des éditions du
Dans ce contexte, afin de venir en aide aux différentes parties prenantes présentes sur le terrain de la collabora- tion et afin d’appuyer les décideurs dans leur
can observe that it is composed of three layers: (i) service instance layer, which represents the services, denoted by a service instance; (ii) network slice instance layer,
Les deux indicateurs précédents (encadrés 1 et 2) donnent la synthèse des asymétries existant entre les structures d’exportation, par branche et par destination, d’un PECO et
dant, il décrit les utilisateurs de covoiturage comme des personnes « militantes » ou « associatives » et leur pratique du covoiturage s’apparente plus à la mise en commun
Para um diagnóstico preliminar do nível de resíduos presentes nos frutos de mamão comercializados no atacado de São Paulo se montou uma parceria entre os projetos “Otimização
34 Although the radical cyclization route was a useful approach for the synthesis of helicenes due to the facile preparation of the stilbene-type starting material and
