An Algebraic Attack against Augot-Finiasz Cryptosystem
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. An Algebraic Attack against Augot-Finiasz Cryptosystem Pierre Loidreau. N° 5662 Août 2005. N 0249-6399. ISRN INRIA/RR--5662--FR+ENG. Thème SYM. apport de recherche.
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(30) #% ,¤WnU, ¤lnqIRcPRTW2PTEq$j,lhDFE7PTPcle~6E m = (m , . . . , m ) WeXYEq$~eRTB k −1 W6Ea $! (q ) RTWxhY[Z]oE6¢¤*ZacPSR/B_E!Rcaclnq$Pf W6aTD-P m Zq6RcWFRTB_E2m<WnYO?q_WnDFZ]leY m (X) = P m X Weleq$jY[E8q_oWn~eDFRcB m_^QRc E8P Z[RTmB = q_EWn qQ(mE8aTWF) ¢ oW?A W6B_acEjQq Zq$B_leE2RTE8m_P8Z]¢¸opQA P B_^_E2moZapm_leB_q$E8jQaSW6RcEDF?Y[R O ,¤αWnU∈ PS$!E8q$ j_(qP¤RT)W-°$xhleqXY[Z]j oE2lFZ]P6E8oRcWna e . k. u. n. 1. u. u. k. S. u n. n. 1. 0. u. 0,0. 0,k−2. 0. S. n. k−2 i=0. 0. u. 0. 0,i. i. w. y = m + αK + e.. x/YZoE FXacPSR PTB_WnaTRTE8q$PRTB_E£oZm_B_E8aSRcE?RWnq RcB_Eq_WnqQE8aTWmXWIPSZ[RTZWnq$P Wn E U?O aT
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(32) ~#% le YYioW?WnapjQZ[qXlRTE7P2oW6aTacE8PTmXW6q$jQZq_~ RTWRcB_Eq_W6qQ48EacWmXWIPSZ[RTZWnq$P We ¢G?B_EW6UQRclnZ[q$P. ° B_EacERcB_EW6EacY[Zq_Zq_~jQE8q_WeRcE8P PSB$WnaTRTEq$Z[q_~ RTB_EnE7oRTEW6acP8¢ .hEq$oEw ¤E WnUQRpleZq ¢x PSB$WnaTRTEq$E8jkhG oWQjQEZ]P PSRTZY[Y¤lkhG oWQjQE6¢ A B_E8aTE WnacEFUIOWnq_Ej_E8oWQjQZq_~ PfRcEm Z[q RTB_E PTB_WnaTRTE8q_E8j kG oW?j_En°xhY[Z]oEacE8oW6EapP m + αc ¢ G?Zq$oE m = E (m ) leqXj E8 (αp) leq$j PTZq$oERcB_EwjQE8~nacEEWn m Z]PYE8PcP-RcB$leqRcB_EwjQE8~nacEEWn p °/x/YZoE F$apPfR αc = aB_TE7Z~noWB_6E7EPfRhapP oQ(X) WIEoZE=qIRmWe Q(X)Z]P +α °$αp(X) PTB_E2acE8oWU?nO£EapZq6P RcαEac°$mXlnW6q$Yj leRTF$ZWnq$qle¢£YYO A mB_E8q¢ ° U?O oWnq$PTZj_EacZ[q_~:RTB$leR,RcB_E 6¯q RTB$E jQE8PTZ[~6q Wn'RcB_E PSOQPSRTE8D RTB_E PTZE!We*RTB_E2m$^_U_YZo46EOZ]P E8\I^$lnY¥RTW YWn~ U$ZRpP¢ 5hW RcB_EhPTZ[8EWe¥RTB$E/m_^_U_YZ]onEOVB$lnPbRTW2U<Eh\I^_Z[RTE/Y]leac~nE/PTZ[qXoE RTB$EhPSE7o^_acZ[RfOWe¥RcB_nEh×oacO?umQRTWIPSOQ(q)PSRTE8D ZD,m$Y[ZE8P RcB$lR D^$PSRU<E ZqIRTaplnoRcleU$Y[E6°iPTWwRcB$lRlnqIO leRSRcEDFmQR-RTW U_acE8ln RTB_EPTOQPfRcED U?O RTaTB_E7oEW6`iEk acZ[q_m$~aTW6RTB_U_E Y[E8D| PTE8oaTERm$leapleDFERcEapP¤ZP ZqQ E8l6PSZU_YE RTjQB_E8EFoWQPTjQO?Z[PSq$RT~E8D lRTDVRcln^$opPSQRP8¢ acE8PTA ZPSB_RhE7?PSEq_Wl RSRpq:lnoplQRTRcP-l6leop?acE:PPSPS^XRcleopqXB:j_lnlePapjZ[qQm_W6aTaTWQD-oE8lejQRTZ^_WnacqQE8¯PFPSE RhB$jQZE8opBoWQjQacE8Z[q$oW~ nleEqXa,jRcB_EE:acaTEW6acaSac¯WnPSEaTR nE7oRTW6a °/U?ORTacO?Z[q_~RTWjQE8oW?j_Eleq?OZ[qIRcEapoEm_RTE8j oZm_B_EaTRTE?R Zq l jQE7PSZ~nq$E8j oW?j_En¢ _Wna ^_aTRTB_E8a/eaTE EacEq$oE8PiRTWFRcB_E8PTE!m_acW?oE8jQ^$aTE7P¤PTEE }QH° 8 4¢ A B_E8aTE °WnacEn°'RTB_E le^Q¢ RcB_ WnZ[apRTP!Bm_PSac^XWnopm<B W6PTm$E8lejapleRTWDFERpleRTE6apE P8" °qRcB_=E PT2ZE, WnuRT=B_E 7m_°^_RTU$B$Y[leZ]RoZ]6-P EO$! (q ) =Z]PV$!E8\I ^$(2leYiRT)W ° n = 1024 W = 74 c+E y = m + αc + e K =c. S. 0. 0. S. 0. 0. 2. 12. DED ² FHG!I!I!J. u. 84.
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(35) ¤|2+
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(37) A B$ZP¤PSE7oRcZ[W6qZ]PijQE7jQZ]olRcE8jRTW RcB_E!jQE8PcoacZ[m_RTZWnqWnRTB_E!Aacl6oE/W6mXE8acleRTWna¤leq$j PSW6DFEhWeZ[RcP¤m_acWnm<EaTRTZE8P ^XPSE7jA B_Zq E!RTAB_acE l6ojQE/E7PSW6ZmX~nE8q acleWeRTWnWnai^$Za P¤lelRSRpolnYl6opPT¥PT¢ Z]oleY$RTW?WnY¥Wn'leY~nEU$aclnZo RcB_EW6aTO,WnE8abRcB_PE FXq_ZRcPE FXEY]j_P¢¸A B_PE F$q_Z[RTE FXEY]0 lFPTEj R $!γ ,(q. . .), γolnq We U<uE!?EZ[YE8E iDFE7Ej qIRpl6P P¤Wel u$! j_(qZ[DFE)qXPSPSZ^$Wnopq$B lnYXRTBX6lE8R/oRTleWnq?aOPTm$E8Y[lnE8oD,EE8WqI6R E/aα$!We (q)$! °?(q)RcB$loR/lnZ]q P¤U<RTB$E2E^_acEq_Z]E\I?^_Z]EPSYRcO P acZ[RSRTE8q^_q$jQE8a RTB_E! W6aTD a γ +··· +a γ , B$EacERcB_E a ? PYZ[E!Zq RcB_E PTDFlnY[Y FXEY]7j α$!= (q) ¢bA B_E! ^_qXoRTZWnqWn% $! (q ) ZqIRTW $! (q ) jQE F$q_E8j U?O. 84 × 1024 = 84. u. u. S. 2. u. 84. 12. k. u. u. u. 1. u u. 1 1. u. i. u. Z]P $! (q)YZ[q_ E7lea7°PSZq$oEV W6a leYY a ∈ $! (q) °lnq$jw Wna2leYY x, y ∈ $! (q ) ° iE,B$l¡6E ¢ E jQEF$q$E2q_W RcB_E!RTaplnoEKW6mXE8acleRTW6a#" ax + ay § « - +H
(38)
(39) > x 7→ xq. u. q. (a(x + y))q =. q. . . . '&. - . (qu ). % . -/. (q u ),. . -/. (q). (. %. (x) = x + xq + · · · + xq. u−1. 6R¤ZPiE8l6PSOVRTW,PSE8E/RcB$lRPSZq$oEhA'a (x)q = A'a (x) °?RTB$E/RTaplnoE/Wnm<EaplRcWna Rcle6E8P ZRpP leY^_E7P Zq-RTB$EKPT^_U(F$E8Yj !$ Wn !$ ¢%¤, EZq_~/RcB_E¤PT^_D Wn$Y[Zq_E7lea ^_q$oRcZ[W6q$P8°ZRZPlnYPTWlhY[Zq_E8lna' ^$q$oRcZ[W6q¢*$^_aTRTB_E8aTDFWnacEn° W6a/(q)leYY EYEDFEqI(qR ua) RplenE8q Zq RTB_E PT^_U(FXEY]j7!$ (q) ZRhPclRcZP F$E8P¤RcB_E2acEY]lRTZWnq " A'a (ax) = aA'a (x), ∀x ∈ !$ (qu). A B$ZP m_acWnm<EaTRfOZ]P q_WeRRcaT^$EKZq~nE8q_EapleY< Wna a Rcle6Eq Z[q&!$ (qu) ¢ ∀x ∈. ,. Æ É. U VD U.
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(49) (q). u−1. 1. S = (g1 , . . . , gn ) Pk−1 p(X) = i=0 pi X i E. E. . n. n. 1. E #D . &B % &B -/. (q) i = 1, . . . , n c = evS (p) E (c) = evS (P ),
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(52) *ii"_ '/! *-. leqXj *U_ZZ[q_RcP8Zl6°PSRc!Wn~6m_EacRTWnB$m<EW6a!PTE8 jZRcl,B PTnWnEY^QE8m_RcZ[ZW6q_q~ RTlWFPTac^QE8jQ^XooZEEKqIRTR B_E!PfRcm_aTE8^$q_U_~eY[RcZ]oBnlnE8~6OlnZ[PSq$ZPSER 6¯acq WnDRTB$EKPSW6W6D,aTZ~nE Zq$80lnY¥?PTU_OQZ[PfRcRcP2EDjQW°Q x/q^_~6RTWeW R/3096 oYl6PTPTZ]oleY jQE7oWQjQZq_~-lRTRclnopQP8¢¸A B_ZP acE8j_^$oRcZ[W6q o8leq UXE jQE7PToaTZU<E8j Z[q Rf ¤W,PSRTE8m$P8¢ {n¢*Z[apPfRFRcB_EO oWnq$PSRTac^$oR,RTB$EkG oWQjQE acWnD|RcB_E PTOQPfRcED|Zqlm$lnaSRcZo^_YlnaV l¡OnB¢ 5leDFEYOn°iZR PT$!le RTZ]P FXE8PVWnRTB_RTB_EEB?W6O?aTmXZWn~nRTZq$B_lnE7 YPSE7F$P,EY]WeBj `$!aT W6mXWIPSZ[¢RTZWn qZRc}_B°bRcPSB_^XEopB PS^$lem_q mXleW6m$aSR-m_aTU<WIElnZopq_B~RTRcB_leE86OEqo8leZ[qqRpPTleWnnDFE-E ¤PTWn^_ap(U j_FXP2EY]Wnj RTB_E:(q)kGoW?j_E Z[RTB oWIEoZEqIRcP,(qZq $!) (q) aplRcB_Ea,RcB$leqZ[q $! (q ) ¢ A B_Eq%RTB_E8Olnj$j%lnq aTEE7acaTjQW6^$aSoE8nj E7oRcWnRTa ZDFEE8PE7\6 ^XZ[leRTB_YY[W6O:^Q R/Z[jQRTE8B£oaToE7WIlnEPTZ[q$o~ZERcqIB_RcE P!PSZBqE7o^_$!ac Z[Rf(q)Ole¢~IleA Zq$B_PfERPTjQZ[8E7EFoWQWejQ¤Zq_RT~-B_E-lRSm_Rp^$lnU_opQY[Z]P8o¢nE8OZP2RTB?^$P u. u. u. DED ² FHG!I!I!J.
(53)
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(55) ? $ α ∈ (q n ). xh^_~nWnRS¯*Z[q_Z]lnPTFPSOQPSRTED PT^ ¥EapPK aTW6DRTB_E, l6oR!RTB$leR8°'l6P2PSW?W6qlnPKW6q_EVacE8oWnE8acPKleq?OWe RTB_EPTE8oaTER Xm leapleDFERcEapP°*EZ[RTB_E8a ! Wna RcB_E EacaTW6aSnE7oRTW6a e °RcB_Eq RTB_E m_Y]leZq6RcE?R,o8leq U<E Wn^_q$jZ[q m<WnYO?q_WnDFZ]leY[¶RcZ[DFE6¢/xhPhlDFleRSRcEahWn¸ lnoR!giWnacWnqm_acWnm<W6PTE8jl-m<WnYO?q_WnDFZ]leY[¶RcZ[DFE leRSRcl6op acE8oWnE8aTZq_~ FXacPSR α °QRcB_Eq e lnq$j F$q$leYYO-RTB_E2 B$WnYEKm$YlnZ[qIRTE?R8¢ B$E.hacE W ie E8Z]nPiEq$a7Wn°7qQIq$WE8 aTWXZ[q_¢~ G?e^$m_Z]P'mXRTWIW?PSWhEDVRTB$^$EKopB E7l¡Wen_E7ZPTqQjQ acW6WnaTm$D-mXlE8RcabZ[W6bqn¢ E!6¯ZqXqIjQRTEEE7apj2oE8ZmQRRcZ]P¤P'l,E8q_oWnZm_^_B_~6B2EaTRTRTWE?~nR ER'y RT=B$E mw+m<W6αKPTZRcZ[W6+q$eP lnq$j-D-leqXle~nE7PbRcWacE8oWnEabRTB_EKq_Wnq_48EacW mXWIPSZ[RTZWnqXPiWn e ¢¸GQB_EKo8leq oWnDFm_^_RTE/RcB_EKnE8oRTW6a ye We'YEq$~eRTB n ~6Z[6−E8P w WnUQRpleZq_E8jw aTW6D RcB_EoZm_B_E8aSRcE?R y U?OwPT^_m_m_acE8PcPSZq_~ RTB_E w q_W6qQEacW mXWIPSZ[RTZWnq$P!Wn e ¢A B_Z]P B$EacERcB_E2^_q_?q_W qXPleacE mRTeB_E RTB_q$E!Wnm_qQY]leZE8qIaTRTWVEImXR WIPSmZ[RTZ Wnacq$WnP/D We B_¢ ZopB Wnq$EKBXlnPPT^_m_m_acE8PcPSE7j-RcB_E w m<W6PTZ[RTZWnq$PoWnacaTE7PSm<Wnq$j_Z[q_~RTW w e A B_E2EYEDFE8q6R α ∈ $! (q ) ¢ 'ER UXER¸RTB_EPSERWn<E8Y[E8DFEqIRcP¸W6UQRclnZ[q$E8j aTW6DrRTB$E/PS^_m$mXW6aSR U?OVPT^_m_m_acE8PcPSZq_~RTB_E E8Y[E8D,E8qIRcP oWnacaTE7PSm<WnSeqXjQZ[q$~ RTWFRcB_E w q_Wnq_48EacWVm<W6PTZRcZ[W6q$P We e ¢ 6¯q RcB_ZP/ ¤l¡O-S ¤E!~nER me ∈ RS (S)e ¢ w e ye = m e + αK,. u. k. Æ É. U VD U.
(56)
(57) 8. 'ER . UXE lm$leacZ[RfO6¯opB_E7op:D-lRTacZ[Wn PSE8EjQEFXq_ZRcZ[W6q%v?¢C,¤O oW6D,m$^QRTZq_~ e RS (S) W6UQRclnZ[qXP k. e = αKH e yeH = |{z} mH e +αKH. yeH. °Wnq_E. QG Z[q$oEE6EacO6RcB_Zq_~!ZP¸?q_W qFZq,RTB_Z]P¸E8\I^$leRTZWnqFE_oEmQR¸RcB_E/EYEDFE8q6R °n nEho8leq,aTE7oWnE8a U?OVjQW6Z[q$~ RcB_EjQRcZIB$Z]lPTRZ[W6Z]qVPU<q_EW6RfqQ ¤48EEE8acqW$Rc¢ B_E iRcB,oW?WnapjQZ[qXlRTE¤WeXRTB_E 6E8oRcWna yeH leqXj RcB_αE iRcB,oW?WnapjQZ[qXlRTE¤WeXαRTB_E 6E8oRcWna e KH x/ RTEa/RTB$leRhPSB_E2~6ERpP me UIO oWnDFm_^_RTZq_~ 0. $aTW6D. e m e = ye − αK.. I° nEK~nERpP m U?O-ZqIRTEacm<WnY]lRTZq_~2RcB_EKmXW6Y[O?q_W6D,Z]leY m We*jQE~6aTE8E k − 1 WnqRTB_E xhY[YiRTB_E7PSE8EY[E8WnDFm<EEqIapRclP RTZWnWn qXSP e¢leacE-mXW6Y[O?q_W6DFZlnYileqXj Y[W Z[q%oWnDFm_YEQZ[RfOn¢A B_E8aTE WnacE- iEBXl¡nEPTB_W q RcB$lR/~6ERTRTZq_~,RTB_E2q_W6qQEacW,mXWIPSZ[RTZWnq$P Wn e ZP E8q_Wn^_~6BRcWFU_aTE7le-RcB_E PTO?PSRTE8D¢. m e = evSe(m0 ) n−w >k. . . 0. 0.
(58)
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(60) . 6¯qRcB_Z]PPTE8oRTZWnq ¤EPSB$W . e R c $ B l 7 R X ° $ ^ $ q Q j E a T R $ B E F D Q W Q j Zq$lnYiPcopB_EDFEn°RTB$Eq_W6qQ48EacWmXWIPSZ[RTZWnq$PWe e oleq£Z F<U<oElRcW6Z[UQW6q$RclnPZ[q$m_E8acjWnm< W6PTZ[RTE8BjlU?O 6ERTacB_OE,PTleZ[^QDFRcB_m_WnYEapPm_WnacWn¸UXRTleB_U_EVZY[WnZ]acPSRTZ[~nZ]o m<WnYO? 'q_EWnRbDF^$Z]P¸leoY[¶WnRcq$Z[DFPTZj_E2EleaY~nRTB$W6E5aTZ[RTF$B$q_D Z[RT5E¢ F$E8Yj$! ¢ ER¸RTB_E/PT^_m_m<WnaTR Rple6E lnY[YQZ[RcPbEYEDFEqIRcPZq,RTB_E5FXEY]j ) $! (q) ⊂ $! (q ) leqXjY[ER c = E (p) (q∈ RS °< B_E8aTE p(X)S B$lnPoW?EoZEqIRpP/Zq $! (q) ¢KA B_E (S) RcE8B_aTacE Wnm_aT^$nU_E8Y[oZ]RToW6na E8EO KmXleaT=RhWec¸+RcB_EEVWnPTE8oRTaTB_EE,Rh
(61) ZPS P/OQ PSRcRTlnEnDEqB$Wl66P/Eoa W?$!E (q) oZ[E8q6 RpP/Z[RTZB qCW$! q_(q)W6qQ¢ 'EEacRW-AoW?aWnj_apEjQq_Zq$WnlRTE2RcE8RcP8B_ ¢ E ./RcE8acq$l6ooEE W6mXE8acleRTW6a¤Wn% $! (q ) ZqIRT W $! (q) ° WnahlnY[Y E8Y[E8DFEqIR γ Z[q $! (q ) °_Wnq$EKBXlnP Aa (γ) = γ + γ + · · · + γ . apoEm_ZRcqPY[RcZB_q_EVE7leoa Z[m$RTB_ZDFEaTEnRT¢bEIA R B_yZ]P=YE8l6mj_Pi+RTWFαKRTB$E!+ eWnY¢hY[W_ W6Zaq_ln~,qIE8O \I^$EYlERcDFZ[W6E8q q6R γ Zq $! (q ) ° PTB_E2G?o^$lnm_q mXoWIWnPSDFEVm_b^QnRcEVE Z[A'qIRca E(γy) Aa (γy) = Aa (γ(m + αK + e)) . ,¤OYZq_E8lnaTZ[RfOWeRTB_E!Rcacl6oE!Wnm<EaplRcWna Wnq$EKBXlnP A'a (γy) = A'a (γm) + Aa (γαK) + Aa (γe). .hW iE8nE8a8°?U?O oWnq$PSRTac^$oRcZ[W6q WnRTB_E PTOQPfRcED Z[RTBacE8j_^$oE7j 6EOI¯PSZE6°?RTB_E2m_^$U_Y[Z]onE8O B$l6PoW?EF oE?Z[E8RTqIapRhlnoZ[Rcq:E8jRTB_ GE aTW6FXD EY]Cj RTB$$!E! RTap(q)lnoEK¢ Wn,¤m<O Eapm_lacRcWnWnm<a7E¢¸aTA RTZBIE8P/^XPWnA'RTa B_ERTaplnoE2W6A'mXE8a acleRTW6a8°$ln°_q?lnO q$j E8Y[E8D,E8qIRhWe) $!K (q) oleqU<E (γαK) = (γα)K A'a (γy) = A'a (γm) + Aa (γα)K + Aa (γe). u. u. S. k. u. u. q. q u−1. u. DED ² FHG!I!I!J.
(62) {7z.
(63) . 5hW 2°U?OwoW6q$PSRTac^$oRcZ[W6qWniRcB_EkhGoWQjQE6°'A'a (γm) YZE8P!Zq. ¢ 5hleDFE8Y[O6° Z]PKRTB_E-6E8oRcWna 4 o Wnq$PTZ]PfRcZ[q_~ WeiRTB_EFE8¡lnY[^XlRTZWnqWe lm<WnYO?q_WnDFZ]leY m Wn¤jQE8~nRSacEEVY[E7PT(S) PKRcB$leq k − 1 WnmqRTB_EFE8Y[E8D,E8qIRcP Wnj_E~nSac¢EE.hYEE8q$PcP¸oE RcB$γmleq ZP/RTB_E°eRcEBI^XleP Y^$lRcZ[W6U<qEYWeWnq_RT~IB_P¸ERTW mXW6RTB_Y[O?EKq_oW6WQD,jQZ]E leY γm W6Ea ¢ S,¤¢O,A `bB_acWnE m<m<W6WnPTYZO?RcZ[q_W6Wnq-DF}_Z]°leY γm B$lnP RS (S) ZE8]P Y[E8PSDFRclnEU_qIYR E/^_q$jQE8$!ab RcB_kEh−ln1oRTRTB_ZWnEq-q_WnWn<qQγmRcB_E8EhaTW,Aacm<l6W6oPTEZRcW6Z[mXW6q$E8acP leWeRTWn*a7RTE¢B_E25hWe6RcE8RSE/oRTRTWnBXa/lARi(S) Pa TZ[qXoEhAoW6a (0) n ° 6 W a e l ? q , O _ q n W Q q E8qQaTW = 0 T a c a 8 E T P X m 6 W $ q j E _ n l o c R [ Y O T R F W _ q 6 W 8EacWm<W6PTZγRcZ[∈W6q$PhWn(qe ¢K) A B?^$PacE8oW6EacZ[q$~A'a (γe) ~nZnE8Ph^$PKZ[q_ WnacDF(γe) leRTZWnq:W6qRTB$EVq_W6qQEacW-m<W6PTZRcZ[W6q$P Wn eG?¢ ^$m_mXWIPSE2 nE!?q_W /P °_l,q_W6qQEacW,EYEDFEqIR/We $! PS^XopB RcB$lR/Aa °?RTB_E8q γ (q ) (γα) = 0 A'a (γy) = A'a (γm) + Aa (γe), W?W6acjQZZq$P2leYRTE8E8PcP8PK°RTleBXqXlejwq B_°'E8baTE-6EAao(γm) ¢G?Zq$oE ∈W6RS (S) U?O:W6q_E Rc B_B$E-Eacq?E,^_DAa U<(γe) Ea2WniB$q_lnW6P2qQleR2ED,acWWIPfm<R W6wPTZ[RTZq_WnWnq$q_P!48WnE acW Aa o(γe) e l q c a 8 E o E K a ' A a j_E8oWQ jQq Zq_RT~-B$E2PSRT~nEacm'Wn¢ ^_q$jWe*RTB_E7PSE!acED-leacQP¤ iE!B$l¡6E!j_E8PTZ[~6q_E8jRTB_E!wRcB_E! WnYYW Z[q_~FleY~nW6aTZ[RTB$(γe) D" 0 B% + ¸®!q$W q m$lnaclnDFERTE8acP¤Wn'RTB$E PSOQPSRTED°$lnq ZqIRTE8acoEmQRcE8j oZm_B_EaTRTE?R ¢ A+ $! (q ) ° m ∈ RS (S) ° e Z[RTB leR!DFWIPfR w q_W6qQEacWy oW?WnapjQZq$lRcE8P8°'PS^XopBRcB$lR γ ∈ Aa (γy) = m + e ¢ {n¢ `bZ]op^_mapleq$jQW6DFY[Olnq E8Y[E8DFEqIR γ ∈ $! (q ) vQ¢hgiWnDFm_^QRcE y = Aa (γy) }_¢A'acO-RcW % E7oWQjQE y Z[q RS (S) :$¢ B_ZY[E!RcB_E jQE8oWQjQZ[q$~-We y Zq RS (S) lnZ[Y]PjQW l ` Zop ^_m apleqXjQWnDFYOleq EYEDFEqIR γ ∈ $! (q ) U giW6DFm_^QRTE y = A'a (γy) tQ¢ aTERT^_acq γ leq$jRTB$E2nE8oRTW6acP m , e WnU_RcleZq_E7j acWnD RTB_E jQE7oWQjQZq_~$°
(64) PS^$opB RcB$lR Aa m + e = (γy) AlnU_Y[EF{ "xhY[~6WnacZRcB_D W6aaTE7oWnE8aTZq_~Vq_WnqQE8aTW,m<W6PTZRcZ[W6q$P We*RTB_E2E8aTacWnaTnE8oRTW6a e B_E8q_E6EaRcB_ElnY[~6WnacZRcB_D PSRTW6m$P°$Wnq_E2~6ERcP/Rf iW-6E8oRTWnapPPclRTZ]PS OIZq_~Aa = m + e °<leq$j e BXlnPKlR!D,WIPfR w q$WnqQE8aTWoW?WnapjQZ[qXlRTE7P<¢ 5/W iEVB$l¡nERTW PTB_W RcB$lRKRTB_EF(γy) 6E8oRcWna e acERc^_aTq$E8jU?O RcB_E leY~nW6aTZ[RTB$D oWnacacE8PTmXW6q$j_P¤E_lnoRTYO-RTW,RcB_E2nE7oRTW6aA'a (γe) ¢ 'ER leq$j e PT^$opBRcB$lR m A'a (γy) = m + e , B$EacE m UXE8Y[W6q_~6PKRcW RTB_E-khGoWQjQE-leqXj e B$l6PKleR2D,WIPfR w q_Wnq_48EacW oWIW6acj_Z[q$leRTE7P¢A B$Eq PSZq$oE °Q iE!B$l¡6E y = m + αK + e Aa (γm) + Aa (γα)K + Aa (γe) = m + e k−1. 0. 0. 0. k−1. k−1. u. u. k−1. u. . 0. 0. 0. k−1. 0. u. 0. 0. k−1 0. k−1. u. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. Æ É. U VD U.
(65) {6{.
(66) . A BXlRZP8°_U?OjQE7oW6D,m<W6PTZq_~ K = c + E °_ ¤EYF$q$lnY[YO~nER Aa (γm) + Aa (γα)c − m = e − Aa (γe) − A'a (γα)E. A B$EhacZ[~6B6R¤m$leaTRiWnRTB$EhE8\I^$leRTZWnqZ]P l 6E8oRcWnai ZRcB lR¤DFW6PSR W + 2w q_W6qQ48EacWoW?W6acjQZq$leRTE8P8¢¸G?Z[qXoE UXE8Y[W6q_~6PRTW nE7oRc(S) Z[q$oE Aa (γm) leq$j m U<EYWnq$~:RTW RS (S) °¸RcB_EY[E R,m$lnaSRFWehRTB_E cE7\I^$lRcZ[W6qZPKl-RS WnahWn lnq$RSj PT(S) ¢A B_E8aTE WnacE2UIOm_acWnm<EaTRTZE8PhWe¸RcB_EVkhG:oWQjQE8P8°<ZRKZPEZ[RTB_E8ahE7\6^XleY RcW 06°QR-WnZ]aPVBXq$lnWePR-ljQR/ZYE8l6o^_PfR YRFnRc−WkPTE+E 1RTB$q$leWnR,qQZ[E8W6aTq_W,Eo W?WnZ]PTapB_jQE8Zq$PVlRTRcW E8P8o¢ W6q$PSRTac^$oR-lwPTOQPfRcED|acE8PTZPSRTZq_~:RcB_E?q_W q j_E8oWQjQZq_~FlRTRclnopQP8°?RTB_E8qRcB_E2m$lnaclnD,ERTE8acPiD^$PSR/PTleRTZ]Pf O ¢¸A B_EacE W6aTEKZ[qW6^_aolnPTE RcB_Z]P oZW6DFaTacm_E8Y[PTZm<E8PiWnq$RcB$j,lRcR¤W,RTm<B_WnEYY[O?Eq_ R¤WnDFmXleZ]leaTRiY]P WeWn *RcjQB_E8EK~nE8ac\IE^$EKllRcRZ[W6DFqW6Z]PSP R E7\6^XleWYXRT°QW+ 8B_2wEE8acaTW$E7≤%¢ lnP,¤nOF−Z]oP WnkRcq$B_PSE!RTacE^$oleRTYZ^$WnleqRT°?ZWnA'qa (γm) lea¤q$Wej 6 8 E o T R n W m c RcB_Em<WnYOIq$WnDFZlnY p We jQE8 ~nacEE E_lnoRTYO k − 1 ¢A B_Z]P¸Z[DFkm_Y−ZE8P¸2RcB$lR Aa (γα)p ¢¸G?Z[qXoE ° ¤E!B$l¡nE!Aa (γα) = 0 ¢ E2B$l¡6Em_acWnE7j-RcB_E! WnYY[W Zq_~,m_acWnm<W6PTZRcZ[W6q¢ = 0 p 6= 0 4 « « « 2i§ # ? Q « L« : 6 § J « 6# = &
(67) = !+
(68) + ! @ & M> + > =
(69) #H 0. 0. 0. k. k−1. k. 0. k−1. % +H A
(70) B yB>,= m +D αK +e0e % (γα) = 0 . 0. . . . %. - @ " % . . γ∈. - . (q. u. k−1. & #" 7( ( " ? % B> !
(71) " E ( 9
(72) ( & > ). . (γe) = e.
(73) . . . A B$E,$!lnY[~6WnacZRcB_PTD ^$opB W6aYRTB$F$leq$RhjQA'Zq_a ~ q_WnqQE8aT°QW mXB_WIE8PSaTZ[E RTZWnq$Z]PPWeq_ WneR ?RcEq_acWDF Z[q'q$¢ leRTE7P B$EqwW6q_EW6UQRclnZ[qXPlnqwEYEDFE8q6R γ∈ (q ) (γα) = 0 α ? G Z $ q o h E c R _ B / E c R c a 6 l o / E 6 W X m 8 E c a e l T R n W i a Z ¤ P 2 l _ q 6 W Q q E c a. W Y BXlnPEQl6oRcY[O q EYEDFEqIRpP°_DFE7leq_Zq_~FRTBXlR/RcZ[B_q$EE8acleE a¤leleacm_E!m$Y[EZ]_olnleoRTZRTWnYO q-qWe%$! E8(qY[E8D,)E8ZqIq6RcRcP/W Wn$! $!(q) (q°QZRp)P 6EZRcacBq_El Y Rcacl6oE,E8\I^$lnYRcWz_¢,GQ^_m_m<W6PTZ[q_~ RcB$lR γ Z]Pm$Zop6E8j^_m Z[RTB l ^_q_Z[ WnacDMjQZ]PfRcaTZU_^QRcZ[W6qwWnE8a<$! (q ) ° Rcm$B_ZopE?q Zq_γ~F7→^_mγαlVq$jQWnqQE7PToaTE8ZaTU<W,E8EP2YEleDFY]PSWEqIl R ^_q_Z[ WnacDMjQZ]PSRTacZ[U_^_RTZWnq WnE8<a $! (q ) ¢A B?^$P8°RTB$E-m_aTW6U$leU$Z[YZRfOWn $! (q ) PS^XopB RcB$lR/Aa (γα) = 0 Z]PE_lnoRTYOE8\I^$lnY¥RTW γ∈ u. u. u−1. u−1. u. u. u. u. .hW iE8nEaVRTB$EacEo8leq%UXEq_W6qQEacWoW?WnapjQZ[qXlRTE7PZq e B_Z]opB%U<E8oWnDFEz^$q$jQEa,RcB_E:lnoRTZWnqWn RcA WFa (γe) E8aTW ¢ Z[q$j_Em<Eq$jQE8qIRTYOjQZ]PfRcaTZU_^QRcE8joW?WnapjQZq$lRcE8P8°<RTB_EFm_acWnU$lnU_ZY[Z[RfO UXE2Zq¢-G?PS^$Zq$opBoEVlFRTB$ol6EacPSEFE!Z]lnPaTE E8\Iw^$lnq$Y¥WnRTqQW w/q B_E8q_E6EahRTB_Z]P/B$lnm_m<Eq$P8°XW6q_E B$l6PRcWaT^$q:leq_WnRTB_E8aRTZDFE RTB$EVleY~nW6aTZ[RTB$D ¢+6¸Z[RB$lnm_mXE8q$P/W6q$oE DFW6aTEVWnq_E,aT^$q$PKZR le~IleZq¢ x RcEa s RcaTZE8P8°¥RTB_EFm_acWnUXleU_ZY[Z[RfO RTB$leR2 iE,B$l¡nEVq_WeRK Wn^$q$jlnY[Y*RcB_EVq$WnqQ 8EacW oW?WnapjQZ[qXlRTE7P!We Z]P °*jQE8oaTE7lnPTZ[q$~ E?m<Wnq$EqIRTZ]leYY[O ZRcBRTB_E-q?^_DVUXE8a2We¤RTacZ[E7P°*PSZq$oE-lnY[Y RcB_E2 aplel6q$opBjQW6RTD acO:Wele¤acZRTlnB_U_EY[E7elnPY[~6leWnacEacw/q ZRcZq$B_jQD E8mXoE8WIq$PfjQRpP2E8q6lR7R ¢ DFW6PSR2Wnq_E-jQE7oWQjQZq_~Z[qRcB_EkhGoWQjQEFRcB$lR Z]P!ZP n W6mXE8acleRTZWnq$P¤Zq $! (q) ¢ RcB_E A oZB_m_E-B_E8l¡aS6RcEEap?leR~nE ZPoRTWnB?DF^$#Pm_" YEQZ[RfOWebRcB_EF B_WnYE,leRSRplnopEq$lnU_YZ[q_~ RTW acE8oW6EahRTB_EFm_Y]leZqIRTE?R m acWnD P = (q u−1 − 1)/q u ≈ 1/q.. s. 2. 0. DED ² FHG!I!I!J.
(74) {¡v.
(75) . ql¡6Eaple~6E2ZR!ZP!leRKDFW6PSR!E8\I^$leYRTW O(sqn ) U$ZR #" +
(76) % 7 *+H !
(77) ># " !
(78) A@ 5& RS (S)e " k × (n − w) W6mXE8acleRTZWnq$P¤Zq $! (q ) ¢ q $! (q) ¢ #" +
(79) % yeH % KH e " (n − w) × (n − k − w) W6mXE8acleRTZWnq$P¤Z& #" +
(80) % α " q_E!D^$YRcZ[m_YZ]olRcZ[W6q Zq $! (q ) ¢ D Wn!m<
(81) % E aplmRcZ[W6q$& P¤ #W" nE8a/me$!" 6R Z]¢ P df^$PSRVl le~napleq$~nE,Z[qIRTE8aTm<WnY]lRcZ[W6qW6Ea2mXWIPSZ[RTZWnq$P2RTB$leR ZP O(k ) (q). A _ B ] Z. P l T R c R n l p o F . n l i P [ Z F D _ m Y E F D 8 E 6 q c R 8 E j $! (q ) = $! (2 ) leq$B j $! (q) = $!ZRcB(2RTB$E)m$¢ lnac lnDFEE^$RTPTE8E8acj P ~nRcZB_nEE8qJ:U? x OV$!RcB_JwE2x le^QY]RcleB_q_Wn~6ap^$P8le°n~6RcE-B$l R ZZRcP BRTW,l?giq_)` W( PTlnm<m_Em<E8Ejq$jQWeZ[ °_;{ ¤$KEiB_ndfW6¢ Z[q A RTB$B$E E q_J:W6qQ x $!EJ:acW:xrm<oW6W?PTj_ZRcEKZ[W6E8q$q$PVleU$WeY[Zq_e~, RTW-B_E8jQaTE7E PSZ ~nWnq ^$q$RTjB_E ZqleRSYRpE8lnPcopPV¥RT¢ B$lnqtDFZq?^QRTE7P1¢ 6¯q%RTB_E
(82) ¤ [{ % ¢x/^_~6WeR,lnq$j£J£¢*Zq_Z]lnPTn¢:x m_^_U_YZonE8OE8q$oacO?mQRTZWnq£PcopB_EDFEU$l6PSE7P W6q RTB_E mXW6Y[O?q_W6D,Z]leY acE8oWnq$PSRTac^$oRTZWnqm$aTW6U_Y[E8D¢ 6¯q e°<m$le~6E8Pvnv6v vn}n}$°Ivez6zn}$¢ v !h¢<k2¢ ,¤EacY[E8leDFm:leq$j ¢ EY]opB¢/bacacWnahoWnacaTE7oRTZWnq We lnY[~6EU_apleZ]oKU_YW?opoWQjQE7P ¢ (KG`¸lRcEqIR8° 5h^_DU<Ea :$° un}n}$° : 2z$°$#{ 8'nu$¢ 9} 2x¢Qg¤lnqIRTE8ln^QRilnq$j-¤¢?giB$lnU$le^$j¢¥x q_E lnY[~6WnacZRcB_Dr W6*a F$q$j_Z[q_~D,Zq_ZD^_D, iE8Z[~6BIR iW6acj$P¸Z[ql YZq_E8lnahoWQjQE " x/m_m$Y[Z]ole RT ZWnq RT W
(83) JwobYZ[E7oEE ? P/o aTO?mQ 9Rc
(84) W6PTOQPfRcED Mln> q$j RTW-q$lnaTacW ¯PSE8q$PSEU<opB:oWQjQE8PWn YEq$~eRTB:t_{n{n¢. %N( %)(G#% M% &B " % H° :: f;{ !" }nu 2 } 2 '$° 6lnq?^$leacO#{ 88 '_¢ : X¢ ¯G¥¢XgiW6aTW6q¢*giacO?mQRclnq$leYOQPSZ]P Wn*lVm_^_U_YZ]onEO-E8q$oacO?mQRTZWnq PTopB$EDFEUXlnPTE8jW6qRcB_E!mXW6Y[O?q_W6D,Z]leY acE8oWnq$PSRTac^$oRTZWnqm$aTW6U_Y[E8D¢¸B6RTRTm " Em_acZq6R7¢ Z]lnoa8¢ Wnac~ nvez6zn} zn}eu I¢ t X¢ ¯G¥¢XgiW6aTW6q¢*giacO?mQRclnq$leYOQPSZ]P Wn*lVm_^_U_YZ]onEO-E8q$oacO?mQRTZWnq PTopB$EDFEUXlnPTE8jW6qRcB_E!mXW6Y[O?q_W6D,Z]leY acE8o Wn q$ PS( RT)ac ^$+ oRTZWn% qm_ aTW6 U_MY>ED ¢ % 6¯q£ ¤ ¢ ,
(85) le W$ °*
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(88) ^_%DF 9E v 89: 2?°?m$le~6E8P!#{ : v '_¢QG?m_acZ[q$~nEa7°_vez69z :$¢ 9u g¢$ln^_acEleq$j `*¢ W6Zj_aTE7le^¢x q$E m_^_U$Y[Z]onE8OoaTO?mQRcW6PTO?PSRTE8DMU$l6PSE7j£Wnq RcB_Em$aTW6U_Y[E8D Wn ac E8 o Wn qI
(89) % RT ac^$(To5RT& Zq_~ p4m<WnYO?q_WnDFZ]le 9Y]P ¢ 6¯q`¢giB$leac m_ Zq °%$°< J:¢<HhleapRcopaTB E8BIvn^XznPzI°tQln¢ q$j % ¢'x/^$~nWeR7° E8jQZ[RTW6acP8° 32; V ¢ $K^_aT^XPS leDFZlnq$j J ¢ G?^Xj_leq¢ 6¯DFm_acWnE8j j_E8oWQjQZq_~%WekE8E8j G?WnYWnDFWnq lnq$j lnY[~6EU_apleZ]o ~6EW6D,ERTacZo!oW?j_E8P8¢. %)(
(90) #%)(G% M%&B "
(91) #% M> H° :?t ";{ 2nt 2 ;{ 2eu 2?°$#{ 8 88_¢ '92x¢i®!Z]l¡O?Zl6PVlnq$jJ£¢¸Hb^_q_~X¢ giaTO?mQRcWn~6aclnm_B_Z]oB$lnacj_q_E8PcP,U$lnPTE8jWnq%RTB$E:jQE8oW?j_Z[q_~£WnKkE8E8j GQvnznWnz6YWn}_DF¢ WnqoWQjQE7P¢ 6¯q 9
(92) % (T'& n°XnWnY^_DFE ve} 'nzVWe/ °_m$le~6E8Pve}Iv v :6}$° .
(93) %
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(96) #%)( '&. nW. m<EaplRcZ[W6q$P Z[q&$! (q) ¢. " . e. . 2. u. k. . u. 0. 2. u. 84. 12. . . . .
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(107) % M> + ,> e° 6WnY^_DFE2}n}nzVWe/0/ °_m$ln~nE7P v e2 t v96' z_¢?G?m_acZ[q_~6EaT +¸E8aTY]le~X°X{ 8 '_' ¢ {8z9!x¢'X¢$JEq$EE7P¢ +,+ A
(108) 9
(109) %)(T5& .
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(113) Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).
(114). . ISSN 0249-6399.
(115)
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