Paper 2017/907

On the differential equivalence of APN functions

Anastasiya Gorodilova

Abstract

C.~Carlet, P.~Charpin, V.~Zinoviev in 1998 defined the associated Boolean function $\gamma_F(a,b)$ in $2n$ variables for a given vectorial Boolean function $F$ from $\mathbb{F}_2^n$ to itself. It takes value~$1$ if $a\neq {\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions. This article defines the differentially equivalent functions as vectorial functions having equal associated Boolean functions. It is an open problem of great interest to describe the differential equivalence class for a given Almost Perfect Nonlinear (APN) function. We determined that each quadratic APN function $G$ in $n$ variables, $n\leq 6$, that is differentially equivalent to a given quadratic APN function $F$, can be represented as $G = F + A$, where $A$ is affine. For the APN Gold function $F$, we completely described all affine functions $A$ such that $F$ and $F+A$ are differentially equivalent. This result implies that the class of APN Gold functions up to EA-equivalence contains the first infinite family of functions, whose differential equivalence class is non-trivial.