Partially APN Boolean functions and classes of functions that are not APN infinitely often

TL;DR

This paper introduces the concept of partial APNness, explores its properties, and identifies classes of functions that are not infinitely often APN, contributing to the understanding of APN function behavior over field extensions.

Contribution

It defines partial APNness, provides characterizations and constructions, and extends results on functions that are never APN over infinitely many field extensions.

Findings

Partial APNness is characterized and constructed.

Certain transformations are shown not to be partially APN.

Classes of functions are identified that are never APN infinitely often.

Abstract

In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field $\F_2$, extending some earlier results of Leander and Rodier.