On Upper Bounds for Algebraic Degrees of APN Functions

Abstract

We study the problem of existence of APN functions of algebraic degree $n$ over ${\mathbb F}_{2^{n}}$ . We characterize such functions by means of derivatives and power moments of the Walsh transform. We deduce several non-existence results which imply, in particular, that for most of the known APN functions $F$ over ${\mathbb F}_{2^{n}}$ the function $x^{2^{n}-1}+F(x)$ is not APN, and changing a value of $F$ in a single point then results in non-APN functions. This leads us to conjectures that an APN function modified in one point cannot remain APN and that there exists no APN function of algebraic degree $n$ .