Arithmetic, Geometry, Cryptography and Coding Theory 2009 by David Kohel, Robert Rolland (ed.)

By David Kohel, Robert Rolland (ed.)

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Nous utiliserons pour cela des r´esultats de van der Geer et van der Vlugt ([19]) et de Maisner et Nart ([10]) sur les courbes supersinguli`eres de genre 2. Par ailleurs, les fonctions bool´eennes vectorielles sont utilis´ees en cryptographie pour construire des algorithmes de chiffrements par bloc. Ces fonctions doivent avoir une r´esistance ´elev´ee a` la cryptanalyse diff´erentielle. Pour ´etudier les attaques diff´erentielles, Nyberg [12] a d´efini la notion de non-lin´earit´e presque parfaite (APN).

On the other hand, if m > h then deg v 2 > g + 1 so by sieving we now have v | pt for at most two values of t. Since the number of monic irreducible polynomials of degree m over k is bounded by q m /m, the number of values of t such that pt is 24 4 WOUTER CASTRYCK AND JOHN VOIGHT divisible by v 2 with v r is at most q g+1 q2 qh q h+1 (2q g+2−4 ) + · · · + (2q g+2−2h ) + 2 + ···+ 2 2 h h+1 g+1 g g+2−h h+1 g+1 q q q q + ···+ + + ··· + = 2 q g+1 + 2 h h+1 g+1 q(2q g+2−2 ) + = ≤ ≤ 2+ 2 g+1 h q g+1 + 2 i=2 q g+2−i + i q g+1 − 1 2 q g+1 + 2 2+ g+1 q−1 2 2 + q g+1 .

En outre, il est essentiel d’avoir un choix ´etendu de fonctions bool´eennes ayant ces propri´etes. La non-lin´earit´e d’une fonction bool´eenne f `a m variables est la distance (de Hamming) de f ` a l’ensemble des fonctions affines a` m variables. Elle est toujours inf´erieure a` 2m−1 −2m/2−1 . Les fonctions bool´eennes atteignant cette borne n’existe que pour m pair et sont dites courbes ([11]). Soit k = Fq un corps fini `a q = 2m ´el´ements. Soit Tr la trace de k sur F2 et soit χ0 l’unique caract`ere non trivial de F2 .

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