On differential equivalence of quadratic APN functions

A vectorial Boolean function $F\colon\mathbb F_2^n\to\mathbb F_2^n$ is called almost perfect nonlinear (APN) if the equation $F(x)+F(x+a)=b$ has at most $2$ solutions for all vectors $a,b\in\mathbb F_2^n$, where $a$ is nonzero. For a given $F$, an associated Boolean function $\gamma_F(a,b)$ in $2n$ variables is defined so that it takes value $1$ iff $a$ is nonzero and the equation $F(x)+F(x+a)=b$ has solutions. We introduce the notion of differentially equivalent functions as vectorial functions that have equal associated Boolean functions. The problem to describe the differential equivalence class of a given APN function is very interesting since the answer can potentially lead to some new constructions of APN functions. We start analyzing this problem with the consideration of affine functions $A$ such that a quadratic APN function $F$ and $F+A$ are differentially equivalent functions. We completely describe these affine functions $A$ for an arbitrary APN Gold function $F$. Computational results for known quadratic APN functions in small number of variables $(2,\dots,8)$ are presented.