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A Family of Fast Syndrome Based Cryptographic Hash Functions

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(1)A Family of Fast Syndrome Based Cryptographic Hash Functions Daniel Augot, Matthieu Finiasz, Nicolas Sendrier. To cite this version: Daniel Augot, Matthieu Finiasz, Nicolas Sendrier. A Family of Fast Syndrome Based Cryptographic Hash Functions. [Research Report] RR-5592, INRIA. 2005, pp.18. �inria-00070414�. HAL Id: inria-00070414 https://hal.inria.fr/inria-00070414 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A Family of Fast Syndrome Based Cryptographic Hash Functions Daniel Augot — Matthieu Finiasz — Nicolas Sendrier. N° 5592 Juin 2005. ISSN 0249-6399. ISRN INRIA/RR--5592--FR+ENG. Thème SYM. apport de recherche.

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(45)  mAe)j g k = n−r ue l- eum*\jjZgGue(%j&le q6jUm*o\j6m - j ue n q6jUm*\jum<jZg$6e5m* Iqq6j k* ] ÇcbdE H¨žPH ËQE #A e) ((Hi )1≤i≤w , w, S)u $e <um*  %e jZg('2

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(47) r§yˆ‰]yrjr<kotuŠ`™^„ ¥J„}sutw‚)kob mm‡r tuŠj¥<b3{om"ƒ d0Ât’k&Š PY $

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(49) &

(50) @9 = +1 . = 2 2 2 &$% w Y :<89A

(51)  0  X & 9 9&#' Y 9&$ Y /Y $9A

(52) ZY &$f.$

(53) 8 A9

(54) 'ZY =u2#% Y $ 

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(62) Y 9 = r max +1 . 2 w ∈J1;wK 2 µPmtwk©kn]yr¡­#Št’ŠCª’– «7mo]`„mEm‡]yt’k©a„¡¦lt’axyaÁtwkE{‡bH„<ˆ‰]ybH‚§¨r<{ w0 = α · r ¤<­#]`b3{‡b α ≈ 0.24 t’k©„ˆ3r}Š`knm‡„}ŠLmH¤ „ŠG‚œm‡]`„¡m.Š αr   P  6 Z

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(64) s#-+ g j g

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Ç   )E 3ž[  E ¢ Ç  Ä k‚lb `Š`bH‚¼tuŠ±djb[ˆ mot’r}Š€`¯ €`¤W„+­©bH„}»“tuŠGk~m‰„Š`ˆ3b­#t’s’sEˆ r<{o{‡bHko†Gr<Š`‚“m‡r“„a„mo{‡tŦ §¨r<{)­#]yt’ˆ‰]¼mo]`b3{‡b b ¦ltwk~m‰kE„}Šœ„s’™}r}{‡tumo]yaÁ„pys’b#mor<`Š`‚„ˆ3r}s’sutwkotur<Š ˆ·„¡º:{‡b3™<xys’„}{U­r<{‡‚]`„J¥jt’Šy™„)Šjxys’s7HkoijŠ`‚l{‡r}acb.‰U­#tÅm‡]œ„ ˆ r<ac†ysub3¦ltÅm~i)s’r¡­bH{"m‡]`„Šqmo]G„¡m2r§`mo]yb#p7bHknm2„¡mnm‰„}ˆ‰»KŠJ]yb3{‡bmo]yb#¾S„™<Šyb3{oº:pG„}kobH‚)„¡mom‡„<ˆ‰»7¯J¾¬b kn]G„s’sL]`b3Š`ˆ3b ™}rcr¡¥<b3{©»LŠ`r¡­#Š„mnm‡„<ˆ‰»lk#„Š`‚b3¥¡„s’x`„mobOmo]yb)ŠjxyaqpGbH{#r§­b[„»t’Š`knm‡„ŠGˆ bHk^b[„}ˆ‰]r}Šyb•­#t’sus"™}b3Š`b3{‰„¡mob<¯ µ¶Š`knm‡„}Š`ˆ b[k§¨r}{­#]ytwˆ‰] m‡]yb¼¾S„™<Šyb3{oºPp`„}kobH‚„¡mom‡„<ˆ‰»ˆH„Š ]G„J¥}b°„ˆ r}ac†ys’b ¦ltum~isur¡­©b3{mo]`„}Š mo]`b „J¥}bH{‡„}™}b#„{‡b m‡]yr<kob §¨r<{E­#]ytwˆ‰]¤j­#]ybHŠœx`kotuŠy™kna„}sus’b3{©sutwknm‡k©„s’s7„}sur<Šy™)mo]`bO„}su™<r}{‡tÅm‡]ya+¤mo]`b3{‡b {‡b3a„t’Š`kH¤ t’Š±„J¥}b3{‰„™<b}¤¸r<Šybkor}s’xlmot’r}ŠSt’Š®mo]ybb3Š`‚¸¯) r¡­©b3¥<b3{[¤Wmo]ytwktwkr<Šys’i¼†7r<k‡knt’pys’bœtu§#mo]ybœa„mo{‡tŦSt’kŠyrmqr§ §¨xys’s¸{‡„}Šy»KŠLmo]yb•„s’™}r<{otumo]ya¹ˆH„Šœm‡]yb3Šp7b•„}†y†ys’tub[‚œx`kot’Šy™ r0 = ƒ „}Šy» (H) t’Š`k~m‡bH„<‚œr§ r ¯u r¡­bH¥}bH{H¤<mo]yb †y{‡r}p`„}pyt’sutum~ir§ H Šyrm2p7b3t’Šy™)§¨x`susG{‡„}Šy»twkU¥}bH{oikoa„s’s ˆ·„}pGr<xlm O (2r−n)‰„Š`‚qmo]`bHkob ­b[„»t’Š`knm‡„}Š`ˆ b[k ˆ3„}Šp7b)ŠybH™}s’bHˆ m‡bH‚¸¯ r}ŠGˆ b3{‡Šyt’Šy™mo]ybqµ¶Šl§¨r}{‡a„¡m‡tur<Š“dlb m €bHˆ3rj‚ytuŠy™„¡mnm‰„}ˆ‰»W¤yt’Š`knm‡„ŠGˆ bHk ­#]yt’ˆ‰]“­#t’s’s"]`„J¥<b„sur¡­ ˆ r<aº †ys’b ¦ltum~i°­#t’susE„}s’kor{obH†y{ob[knbHŠLm)„ŠybH™}s’tu™<tup`subc†y{‡r}†7r}{omot’r}Š¬r§^„s’s2†GrLkokot’pysubca„¡m‡{otwˆ b[kAŠ¸tum{‡bHvLxyt’{ob[kOmo]`„m mo]`b3{‡bœt’kq„Š±„}pyŠyr}{‡a„s’sui¼s’„}{o™<b„acr}xyŠLmr}§ knr<suxlm‡tur<Š`k)r}§ sur¡­Á­bHtu™<]LmH¤'kor+mo]G„¡m­#]ybHŠ²ˆ‰]yrjr<kot’Šy™“„ {‰„Š`‚lr<a!t’Šl§¨r}{‡a„¡m‡tur<Šckob m[¤}tumE­#t’susG]`„J¥}b „•sw„{‡™}bH{U†y{or<p`„p`tus’tÅm~iqr§¸p7b3t’Šy™)¥¡„s’t’‚¸¯L\^]ytwk2†y{‡r}p`„}pyt’sutum~i­#t’sus p7b)b3¥}bHŠs’r¡­©b3{©mo]`„}Šmo]`„m^r}§ H Šyr}m#pGbHtuŠy™r}§"§¨xys’s"{‰„Š`»7¯ \^]yb•r}Š`suiœ{‡b3a„}tuŠyt’Šy™†y{‡r}†7b3{om~i­#]yt’ˆ‰]“ˆ3r}xysw‚­©bH„}»}bHŠ„Št’Š`k~m‰„Š`ˆ3b)„™<„}tuŠGk~m#ˆ r<sus’t’kot’r}ŠkobH„}{‡ˆ‰]t’k mo]`bO†`{ob[knbHŠ`ˆ b•r§„¥}b3{‡icsur¡­´­bHtu™<]Lm^ˆ3r}s’sutwknt’r}Š ˆ·k‡„Ji 2w0 ‰ ¯y\^]ytwk^­©„Ji<¤l„qp`{oxlm‡bO§¨r}{‰ˆ b)knb[„{‰ˆ‰]„acr<Šy™ „s’s2s’r¡­ ­©b3t’™}]Lm­©r}{‰‚¼ˆ3r}xysw‚}`Š`‚¬m‡]yt’kqˆ r<sus’t’kot’r}Š¬­#tÅm‡]±„+sur¡­©b3{ˆ3r}ac†ys’b ¦ltÅm~i<¯"Ä kobH„}{‡ˆ‰]S„acr}Š`™+„}sus ­©r}{‰‚yk%r§W­bHtu™<]<mUxy†m‡r 2w0 ­#tus’sy]`„J¥<b}¤¡x`kot’Šy™Omo]yb#pyt’{omo]`‚y„Ji†`„{‰„}‚lrJ¦¸¤¡„•ˆ r<a†`sub3¦jtum~ir}§ O pNw  ­#]ybH{ob Nw ‚lbHŠyrm‡bHk'mo]yb#ŠjxyaqpGbH{Ur§¸JºP{obH™}xysw„{%­r<{‡‚`k"r}§7­©b3t’™}]Lm 2w0 ¯}\^]`t’kU„mnm‡„<ˆ‰»)­#tus’sl]ybHŠ`ˆ b#r<Šysui p7b)r§tuŠLmobH{ob[k~m#§¨r<{ ¥¡„s’xybHk^r}§ w0 kox`ˆ‰]mo]G„¡m Nw < 2 ˆ¨mo]ybk‡vLx`„{‡b•r§%mo]ybq„J¥}b3{‰„™<b€ˆ3r}ac†ys’b ¦ltÅm~i r§•m‡]ybC¾S„™}Š`b3{oº:p`„<knb[‚²„mnm‡„<ˆ‰»u‰ ¯E\^]yb°†y{or<p`„p`tus’tÅm~i±mo]`„m„ŠÍt’Š`knm‡„ŠGˆ b“]`„<kœ„®knr<suxymot’r}Šr§)­©b3t’™}]Lm t’k O  N  „}Š`‚®m‡]jx`kqmo]yb+†y{‡r}†7r}{omot’r}Š±r}§€kox`ˆ‰]±­©bH„»®t’Š`knm‡„}Š`ˆ b[kqˆH„Š²Š`rmcpGb+sw„{‡™}b3{mo]`„}Š 2w0 2   ¤L­#]`t’ˆ‰]¤j„}k­©b ­#t’sus¸kob3bOtuŠm‡]ybOŠyb ¦jm#knb[ˆ mot’r}Š"¤<­#t’sus¸„}su­^„Jilk2p7b€ŠybH™}s’tu™<tup`sub §¨r}{^knb[ˆ xy{‡b (a−1) O 2 †`„}{‡„}ab3mobH{‡kH¯ ¾¬bOˆH„Šœ]ybHŠ`ˆ b•ˆ r<Š`ˆ s’x`‚lb€mo]G„¡m^Šyrq»jŠyr¡­#Š„¡mnm‰„}ˆ‰»cijtubHs’‚`k„qŠyr}ŠŠyb3™<sut’™}t’pys’b †y{‡r}†7r}{omot’r}Šr}§"­©bH„» t’Š`k~m‰„Š`ˆ3bHk^t’Š+r}xy{ ˆ3r}Š`knmo{‡x`ˆ m‡tur<Š¯ #A. . . 0. 0. 2r a+1. 0. w0 r. −r a+1. us t 5t =c8Wv ;)s3; 8Sr28s= † kox`„}sus’iÂ]G„}ko]§¨xyŠ`ˆ m‡tur<Š`k]`„J¥}bC„±kobHˆ3xy{‡tÅm~i r§ „™L„t’Š`knmˆ r}s’s’t’kotur<Š`kH¤Umo]`„mtwk3¤Emo]yb¼pGb[k~m+„¡mom‡„}ˆ‰» twkp`„<knb[‚Sr}ŠSm‡]ybˆ3s’„<kokot’ˆH„s2p`tu{omo]`‚`„Ji¼†`„}{‡„<‚lrJ¦¸¯2µ¶Š±r<xy{qˆH„}kobmo]ytwkq­r<xys’‚®ˆ3r}{‡{ob[kn†7r}ŠG‚°m‡r a pGbHtuŠy™ . . . r 2. | $) -&-+*-,.

(114) \e

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