The Research On Permutation Theory In Block Cipher System

Block cipher is an important branch of mordant cryptography, and permutation theory has an important role in studying and designing of block cipher. First a review of developing history and designing principle of block cipher is given and the important function of permutation theory in block cipher is presented in this thesis. Then some cryptographic indices of permutations are discussed. Then we put more efforts on some expressions, enumerations and constructions of orthomorphic permutations, and on the problem of how to make linear orthomorphic permutations non-linear. The results obtained in this thesisoffers some new ideas and methods for studying of permutation theory in block cipher system. The main contributions of this dissertation are as follows:(1) The cryptographic properties of orthomorphic permutations in finite fields are studied; The relationship between orthomorphic permutations and DBS.is studied, at same time , some cryptographic indices of some permutations used in DBS are analyzed and calculated, and furthermore it is presented that these indices are access to that of some orthomorphic permutations with same degree .(2) From the point of mathematical view, some properties of and criteria's for orthomorphic permutations are given. These results can be used as a sufficient preparation for the study of orthomorphic permutations in the later chapters.(3) The problem of the expressions of orthomorphic permutations is studied. It is showed that orthomorphic permutations in F2n are corresponding to orthomorphic permutation polynomials with degree n in finite fields F2n one by one. A conclusion is given based on this expression:The number of all orthomorphic permutations in F2n can be divided by 2n .(4) Linear orthomorphic permutations are studied by using matrix theory.The properties of orthomorphic matrices are given and rational standard forms of orthomorphic matrices are obtained. Using the characteristic of diagonal orthomorphic matricies and combining with Boolean functions, a kind of orthomorphic permutations is constructed, including a kind ofnon-linear orthomorphic permutations. And a lower bound of the enumeration of orthomorphic permutations with order 2" is obtained.(5) The properties and enumerations of maximal linear orthomorphic permutations are discussed, and the problem how to construct non-linear orthomorphic permutations is studied. A method how to construct non-linear orthomorphic permutations is introduced based on maximal linear orthomorphic permutations, and the enumeration of maximal linear orthomorphic permutations is solved out based on polynomial theory in finite fields.(6) The circle construction and corruption of a kind of linear orthomorph permutations are studied. The method for constructing linear orthomorphic permutations is given . And the cryptographic indices of non-linear orthomorphic permutations obtained are calculated.(7) The orthomorphic permutation polynomials in finite fields F8 are studied. We give all orthomorphic permutation polynomials in finite fields F8 , and the enumeration of orthomorphic permutation polynomials are obtained.(8) The orthomorphic permutation polynomials in finite fields F16 are studied. Zech algorithm( also called jacobia algorithm) and the criteria of polynomial roots in the finite fields are introduced. The following results are obtained: it is showed that there do not exist orthomorphic permutation polynomials with degree 2,3,5,6,7 and 14 , and there exists linear orthomorphic permutation polynomials with degree 1 and 4 in the finite fields F16 , so these results show that the degree of non-linear orthomorphic permutation polynomials will distributed in 8,9,10,11,12 and 13.