A Kind of More Secure Orthomorphism Generator

Orthomorphism is a kind of important elementary permutation in symmetric cryptography, which is also a kind of complete mapping. Orthomorphism has been proved to have the perfectly balanced property. Construction and counting of orthomorphism has become one of the focal issues to Chinese and foreign scholars from 1995, however current researches on orthomorphisms pay little attention on their cryptographic properties, such as difference uniformity, nonlinearity and so on. Orthomorphisms with good cryptographic properties can be directly used to construct the cryptographic units in symmetric cryptographic algorithms. In this paper, firstly a problem in a conclusion about nonlinearity of composite functions is pointed out and corrected. Then several cryptographic properties against differential attack and linear attack of normal BDLL orthomorphism generator are analyzed, such as nonlinearity, algebraic degree and difference uniformity. Next, a modified orthomorphism generator based on composite functions is proposed. With the corrected conclusion of composite functions, the modified orthomorphism generator is proved to be able to construct nonlinear orthomorphisms with higher nonlinearity and algebraic degree than previous normal BDLL orthomorphism generators. And the numbers of orthomorphisms which can be derived from the modified orthomorphism generator is also proved to be bigger than that of previous normal BDLL orthomorphism generators.