Characterization of almost perfect nonlinear functions in terms of subfunctions

The paper is concerned with combinatorial description of almost perfect nonlinear functions (APN-functions). A complete characterization of $n$-place APN-functions in terms of $(n-1)$-place subfunctions is obtained. An $n$-place function is shown to be an APN-function if and only if each of its $(n-1)$-place subfunctions is either an APN-function or has the differential uniformity $4$ and the admissibility conditions hold. A detailed characterization of 2, 3 or 4-place APN-functions is presented.