Permutation polynomials and their differential properties over residue class rings

This paper mainly focuses on permutation polynomials over the residue class ring ZN , where N > 3 is composite. We have proved that for the polynomial f(x) = a1x 1 + · · · + akx with integral coefficients, f(x) mod N permutes ZN if and only if f(x) mod N permutes Sμ for all μ | N , where Sμ = {0 < t < N : gcd(N, t) = μ} and SN = S0 = {0}. Based on it, we give a lower bound of the differential uniformities for such permutation polynomials, that is, δ(f) ≥ N #Sa , where a is the biggest nontrivial divisor of N . Especially, f(x) can not be APN permutations over the residue class ring ZN . It is also proved that f(x) mod N and (f(x) + x) mod N can not permute ZN at the same time when N is even.