A note of perfect nonlinear functions

Authors:
Xiyong Zhang;Hua Guo;Jinjiang Yuan
Affiliations:
Department of Mathematics, Zhengzhou University, Zhengzhou, China;School of Computer Science Engineering, Beihang University, Beijing, China;Department of Mathematics, Zhengzhou University, Zhengzhou, China
Venue:
CANS'06 Proceedings of the 5th international conference on Cryptology and Network Security
Year:
2006

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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 mn and l(l is odd) is self-conjugate modulo 2k(k≥1) .