Introduction Proof of the first conjecture

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Introduction Proof of the first conjecture Proof of the third conjecture New families A proof of two conjectures on APN functions. Elodie Leducq Institut de Mathematiques de Jussieu Universite Paris 7 AGCT-13 17/03/11 Elodie Leducq A proof of two conjectures on APN functions.

• satisfies ∆d


• institut de mathematiques de jussieu universite


• ∆d ≤


• introduction proof


• apn functions


• fq ?

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• Published by: profil-urra-2012
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• Language: English

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AGCT-13 17/03/11

Institut de Math´ tiques de Jussieu ema Universite´Paris7

A proof of two conjectures on APN functions.

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Elodie Leducq

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n Let q = p , p being a prime number, n an odd integer.

p n d dp i I 3 5 134 (134,160,238,230,206) II 3 5 152 (152,214,158,232,212) III 3 7 40 (40,120,360,1080,1054,976,742) IV 3 7 224 (224,672,2016,1676,656,1968,1532) V 3 7 274 (274,822,280,840,334,1002,820) VI 5 3 14 (14,70,102) VII 5 5 843 (843,1091,2331,2283,2043)

Δ f = max( N f ( a , b ) , a , b ∈ F q , a 6 = 0) = 2 .

Definition If f : F q → F q , for all a and b ∈ F q , we denote by N f ( a , b ) the number of solutions in F q of the equation f ( x + a ) − f ( x ) = b . We say that f is APN if

sn.

We put m = n + 2 1 .

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