Characterization of APN functions by means of subfunctions

A vectorial Boolean function $F\colon\{0,1\}^n\to\{0,1\}^n$ is called an APN function if the equation $F(x)\oplus F(x\oplus a)=b$ has at most 2 solutions for any vectors $a,b$, where $a\neq0$. The complete characterization of APN functions by means of subfunctions is found. It is proved that $F$ is APN function if and only if each of its subfunctions in $n-1$ variables is an APN function or has the order of differential uniformity 4 and the admissibility conditions are hold. Some numerical results of this characterization for small number $n$ of variables are presented.