Vectorial $2$-to-$1$ functions as subfunctions of APN permutations

This work concerns the problem of APN permutations existence for even dimensions. We consider the differential properties of $(n-1)$-subfunctions of APN permutations. It is proved that every $(n-1)$-subfunction of an APN permutation can be derived using special symbol sequences. These results allow us to propose an algorithm for constructing APN permutations through $2$-to-$1$ functions and corresponding coordinate Boolean functions. A lower bound for the number of such Boolean functions is obtained.