Bounds on the degree of APN polynomials The Case of $x^{-1}+g(x)$

TL;DR

This paper establishes that functions of the form x^{-1}+g(x) are APN only finitely often, and for degrees of g less than 7, they are APN only in small fields, specifically equivalent to x^3.

Contribution

It proves finiteness of APN functions of the form x^{-1}+g(x) and characterizes their behavior for low-degree g over finite fields.

Findings

Functions of the form x^{-1}+g(x) are APN only finitely often.

For degree g < 7, such functions are APN only when m ≤ 3.

These functions are equivalent to x^3 in the cases where they are APN.

Abstract

We prove that functions $f:\f{2^m} \to \f{2^m}$ of the form $f(x)=x^{-1}+g(x)$ where $g$ is any non-affine polynomial are APN on at most a finite number of fields $\f{2^m}$. Furthermore we prove that when the degree of $g$ is less then 7 such functions are APN only if $m \le 3$ where these functions are equivalent to $x^3$.