A classification of differentially nonequivalent quadratic APN function in 5 and 6 variables

A vector Boolean function $F$ from $\mathbb F_2^n$ to $\mathbb F_2^n$ is called almost perfect nonlinear (APN) if equation $F(x)\oplus F(x\oplus a)=b$ has at most 2 solutions for all vectors $a,b\in\mathbb F_2^n$, where $a$ is non-zero. Two functions $F$ and $G$ are called differentially equivalent if $B_a(F)=B_a(G)$ for all $a\in\mathbb F_2^n$, where $B_a(F)=\{F(x)\oplus F(x\oplus a)\colon x\in\mathbb F_2^n\}$. A classification of differentially non-equivalent quadratic APN function in 5 and 6 variables is obtained. We prove that, for a quadratic APN function $F$ in $n$ variables, $n\leqslant6$, all differentially equivalent to $F$ quadratic functions are represented as $F\oplus A$, where $A$ is an affine function.