Cryptography Replacement Nature

This dissertation discusses properties, constructions and enumerations of three kinds of basic permutations. The main results are as follows: Based on Pro. Lu Shu-wang's research work on random permutations, using probability method, we research the variance and the distribution of invariable point in the case of selecting permutations on Zn randomly, analyze the cryptographic security of random permutation, prove the guess that the number of omni-direction permutations on Zn (n is odd)is odd times of n ,and discuss some properties of omni-direction permutations on Zn (n is odd). Based on selecting permutations on Zn randomly, we build a probability model. In sense of selecting permutations randomly, the distribution of distance which is between one point and its neighbor in Zn (called distance for short) and the mathematic expectation, the variance of number of the points with distance a are presented. When distance a and the permutation order n are prime, we give the distribution of number of the points with distance a. Through these results, we analyze the cryptographic security of random permutation, present a new explication on the significance of choosing quick trickle permutation in cipher designs. By using quick trickle Latin square, we give a complementary construction method of quick trickle permutation on Zn (n is even). By using the orthomorphic permutation over vector space F_p~n , we research the rational standard forms of generalized orthomorphic matrices, present some constructions of generalized orthomorphic permutation. Finally, based on generalized orthomorphic permutation, we construct a kind of cryptographic functions and analyze their cryptographic properties.