Paper 2017/907

On differential equivalence of APN functions

Anastasiya Gorodilova

Abstract

For a given vectorial Boolean function $F$ from $\mathbb{F}_2^n$ to itself it was defined an associated Boolean function $\gamma_F(a,b)$ in $2n$ variables by C. Carlet, P. Charpin, V. Zinoviev in 1998 that takes value $1$ iff $a\neq{\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions. In this paper we introduce the notion of differentially equivalent functions as vectorial functions that have equal associated Boolean functions. To describe differential equivalence class of a given APN function is an open problem of great interest. We obtained that each quadratic APN function $G$ in $n$ variables, $n\leq 6$, that is differentially equivalent to a given quadratic APN function $F$, is represented as $G = F + A$, where $A$ is an affine function. For the APN Gold function $F(x)=x^{2^k+1}$, where gcd$(k,n)=1$, we completely described all affine functions $A$ such that $F$ and $F+A$ are differentially equivalent. This result implies that APN Gold functions $F$ with $k=n/2 - 1$ for $n=4t$ form the first infinitive family of functions up to EA-equivalence having non-trivial differential equivalence class consisting of more that $2^{2n}$ trivial functions $F_{c,d}(x) = F(x+c)+d$, $c,d\in\mathbb{F}_2^n$.