On an algorithm generating 2-to-1 APN functions and its applications to “the big APN problem”

Almost perfect nonlinear (APN) functions are of great interest to many researchers since they have the optimal resistance to the differential attack. The existence of bijective APN functions in even number of variables is an important open problem, and there is only one known example of such a function at present. In this paper we consider a special subclass of 2-to-1 vectorial Boolean functions that can allow us to search and construct APN permutations. We proved that each 2-to-1 function is potentially EA-equivalent to a permutation and proposed an algorithm that generates special symbol sequences for constructing 2-to-1 APN functions. Also, we described two methods for searching APN permutations, that are based on sequences generated by this algorithm.