On the Distance Between APN Functions

Abstract

We investigate the differential properties of a vectorial Boolean function $G$ obtained by modifying an APN function $F$ . This generalizes previous constructions where a function is modified at a few points. We characterize the APN-ness of $G$ via the derivatives of $F$ , and deduce an algorithm for searching for APN functions whose values differ from those of $F$ only on a given set $U \subseteq \mathbb {F}_{2^{n}}$ . We introduce a value $\Pi _{F}$ associated with any $F$ , which is invariant under CCZ-equivalence. We express a lower bound on the distance between a given APN function $F$ and the closest APN function in terms of $\Pi _{F}$ . We show how $\Pi _{F}$ can be computed efficiently for $F$ quadratic. We compute $\Pi _{F}$ for all known APN functions over $\mathbb {F}_{2^{n}}$ up to $n \le 8$ . This is the first new CCZ-invariant for APN functions to be introduced within the last ten years. We derive a mathematical formula for this lower bound for the Gold function $F(x) = x^{3}$ , and observe that it tends to infinity with $n$ . Finally, we describe how to efficiently find all sets $U$ such that, taking $G(x) = F(x) + v$ for $x \in U$ and $G(x) = F(x)$ for $x \notin U$ , $G(x)$ is APN.