A note of perfect nonlinear functions

TLDR: Using Galois rings and investigating the character values of corresponding relative difference sets, a perfect nonlinear function is constructed from n to p where 2m is possibly larger than the largest divisor of n.

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