Polynomials with maximal differential uniformity and the exceptional APN conjecture

TL;DR

This paper advances the understanding of the exceptional APN conjecture by proving that certain polynomials cannot be APN over infinitely many extensions, specifically those with degrees of a particular form and nonzero second leading coefficient.

Contribution

It provides a proof that polynomials with degrees of the form 2 r (2 ℓ + 1), under specific conditions, cannot be APN infinitely often, supporting the exceptional APN conjecture.

Findings

Polynomials of degree m = 2 r (2 ℓ + 1) with gcd(r, ℓ) ≠ 2 and nonzero second leading coefficient are not APN infinitely often.

For sufficiently large n, such polynomials have differential uniformity equal to m - 2.

The result narrows the class of potential APN polynomials, contributing to the conjecture's validation.

Abstract

We contribute to the exceptional APN conjecture by showing that no polynomial of degree m = 2 r (2 {\ell} + 1) where gcd(r, {\ell}) 2, r 2, {\ell} 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.