On perfect nonlinear functions (Π)

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 ℤ42m to ℤ_4m, where m is a divisor of 2m, and a construction of perfect nonlinear functions from ℤ_p2n to ℤ_p2m where 2m is possibly larger than the largest divisor of n. Meanwhile we prove that there exists a perfect nonlinear function from ℤ_2p2 to ℤ_2p if and only if p = 2, and there doesn't exist a perfect nonlinear function from ℤ_2kl2n to ℤ_2klm if m > n and l(l is odd) is self-conjugate modulo 2 ^k (k 1).