A Note of Perfect Nonlinear Functions

摘要

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 Z_(p~2)~n to Z_(p~2)~m where 1m is possibly larger than the largest divisor of n. Meanwhile we prove that there exists a perfect nonlinear function from Z_(2p)~2 to Z_(2p) if and only if p = 2, and that there doesn't exist a perfect nonlinear function from Z_(2~kl)~2n to Z_(2~kl)~m if m ＞ n and l(l is odd) is self-conjugate modulo 2~k(k (＞) 1) .