Polynomials with maximal differential uniformity and the exceptional APN conjecture

2022/07/28 by Aubry, Yves, Herbaut, Fabien, Issa, Ali
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2207.13945

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.

Related

• Maximal differential uniformity polynomials
• On the Conjecture on APN Functions
• Progress Towards the Conjecture on APN Functions and Absolutely Irreducible Polynomials
• Differential uniformity of polynomials of degree 10
• Polynomial functions of degree 20 which are APN infinitely often