A few more functions that are not APN infinitely often

TL;DR

This paper investigates exceptional APN functions over finite fields, showing that polynomial functions of odd degree are generally not exceptional unless their degree is a Gold or Kasami-Welch number, with partial results for even degrees.

Contribution

It establishes new non-exceptionality results for polynomial functions of odd degree, extending understanding of APN functions over finite fields.

Findings

Polynomial functions of odd degree are not exceptional unless degree is Gold or Kasami-Welch.

Partial results obtained for functions of even degree.

Characterization of exceptional APN functions based on degree.

Abstract

We consider exceptional APN functions on ${\bf F}_{2^m}$, which by definition are functions that are not APN on infinitely many extensions of ${\bf F}_{2^m}$. Our main result is that polynomial functions of odd degree are not exceptional, provided the degree is not a Gold member ($2^k+1$) or a Kasami-Welch number ($4^k-2^k+1$). We also have partial results on functions of even degree, and functions that have degree $2^k+1$.