Polynomials with maximal differential uniformity and the exceptional APN conjecture

Abstract

We contribute to the exceptional APN conjecture by showing that no polynomial of degree m = 2 r (2 ℓ + 1) where gcd(r, ℓ) 2, r 2, ℓ 1 with a nonzero second leading coefficient can be APN over infinitely many extensions of the base field. More precisely, we prove that for n sufficiently large, all polynomials of F 2 n [x] of such a degree with a nonzero second leading coefficient have a differential uniformity equal to m − 2.This work is partially supported by the French Agence Nationale de la Recherche through the SWAP project under Contract ANR-21-CE39-0012.