Property And Construction Of Orthomorphic Permutations

Orthomorphic permutations have good properties of cryptography such as complete balance, uniform distribution of input-output phase difference, used in the design of cryptographic algorithms widely. So it is very important for cryptography to study orthomorphic permutations.The thesis does a lot of works in the property and the construction of orthomorphic permutations systematically, and obtains the main conclusions as follows:1. The thesis studies the properties of orthomorphic permutations by concatenation. It analyzes the properties of three kinds of orthomorphic permutations by concatenation, and some conclusions are obtained by studying the properties such as fixed point, structure and period of cycles. Finally, it is proved that there does not exist the maximal orthomorphic permutations for the kind of permutations.2. The thesis obtains the conditions to judge three kinds of orthomorphic permutation polynomials with special degree over finite fields F2n. Firstly, It is pointed out that there exist some questions about existence of orthomorphic permutation polynomials with degree 2d-1 and degree 2d over finite fields F2n given by Li. By using the congruence theory and the distributive law of degrees for multiplying polynomials, the reasons above are analyzed completely, and the conclusions are obtained repeatedly. Based on the above conclusion, the enumerating form and the existing form of orthomorphic permutation polynomials with degree 4 are analyzed. Finally, the necessary condition of existence of orthomorphic permutation polynomials with degree 2d+1 is obtained, and it is proved that there does not exist orthomorphic permutation polynomials with degree 5.3. The thesis improves a kind of cascade connected algorithm to construct orthomorphic permutations, and gives the new algorithm and the enumeration. Compared with the original algorithm, the enumeration lower bound of orthomorphic permutations by the new algorithm is improved better.