On APN Functions Whose Graphs are Maximal Sidon Sets

Abstract

The graphs $\mathcal{G}_F=\{(x,F(x)); x\in \mathbb {F}_2^n\}$ of those (n, n)-functions $F:\mathbb {F}_2^n\mapsto \mathbb {F}_2^n$ that are almost perfect nonlinear (in brief, APN; an important notion in symmetric cryptography) are, equivalently to their original definition by K. Nyberg, those Sidon sets (an important notion in combinatorics) S in $({\mathbb {F}}_2^n\times {\mathbb {F}}_2^n,+)$ such that, for every $x\in {\mathbb {F}}_2^n$, there exists a unique $y\in {\mathbb {F}}_2^n$ such that $(x,y)\in S$. Any subset of a Sidon set being a Sidon set, an important question is to determine which Sidon sets are maximal relatively to the order of inclusion. In this paper, we study whether the graphs of APN functions are maximal (that is, optimal) Sidon sets. We show that this question is related to the problem of the existence/non-existence of pairs of APN functions lying at distance 1 from each others, and to the related problem of the existence of APN functions of algebraic degree n. We revisit the conjectures that have been made on these latter problems.