On special class of vectorial Boolean functions and the problem of APN permutations existence

The existence of APN permutation of even dimension is an important unsolved problem on vectorial Boolean functions. In this paper, we consider the special set of vectorial Boolean functions, such that the sum of any function in the set and an affine vectorial function is a permutation. We study properties of this set and conditions for existence of its nonempty intersection with the set of APN functions.