On perfect nonlinear functions (Π)

Abstract

Perfect nonlinear functions are of importance in cryptography. By using Galois ring, relative trace and investigating the character values of corresponding relative difference sets, we present a construction of perfect nonlinear functions from $${\mathbb{Z}_4^{2m}}$$ to $${\mathbb{Z}_4^{m'}}$$, where m′ is a divisor of 2m, and a construction of perfect nonlinear functions from $${\mathbb{Z}_{p^2}^{n}}$$ to $${\mathbb{Z}_{p^2}^m}$$ where 2m is possibly larger than the largest divisor of n. Meanwhile we prove that there exists a perfect nonlinear function from $${\mathbb{Z}_{2p}^2}$$ to $${\mathbb{Z}_{2p}}$$ if and only if p = 2, and there doesn’t exist a perfect nonlinear function from $${\mathbb{Z}_{2^kl}^{2n}}$$ to $${\mathbb{Z}_{2^kl}^m}$$ if m > n and l(l is odd) is self-conjugate modulo 2 k (k ≥ 1).