The Permutation Of Multiple-valued Logical Function

Costas arrays, namely, the best discrete signal designed basing on the delay speciality of time and frequency in Radar and Sonar system,which are first proposed by J.P.Costas in 1966, can exactly mensurate the speed of object nearing or leaving. So there are comprehensive applications in remote control and remote sensing system. In addition because of its excellent speciality of non-circulate correlation and non-linearity, it can be used in the field of multiple-valued logical function and cryptology. So Costas array have become an important researching problem of cryptology theory. Permutation theory has an important role in studying and designing of cryptology system. A cryptology system without any information expansion can be regarded as a permutation. Therefore constructions of permutation with high degree and high nonlinearity have become an important study problem. Orthomorphic permutation is a kind of complete mappings, also is a sort of special Boolean permutation. Orthomorphic permutation which has already been proved has useful cryptology properties, such as high nonlinearity, perfectly balanced properties and so on, can be used to construct S-box. But the study on orhomophic permutation is in the primary stage, a lot of problem need to be solved.This paper has studied the permutation formed by multiple -valued logical function group, Costas arrays, Orthomorphic permutation and has gained the results as following:1) In the first chapter, the main study achievements and results of Costas arrays, Boolean permutation, Orthomorphic permutation have been summarized.2) In the first section of the second chapter, the Boolean permutation on general Boolean algebra has been studied. A sufficient and necessary condition of Boolean Permutation over a Boolean algebra is given, and therefore a kind of Costas arrays is given.3) In the second section of the second chapter, the permutation formed by multiple-valued logical function has been studied. Permutation formed by q-valued logical function group is given; then by using permutation formed by q-valued logical function group, a kind of Bent functions and functions satisfying the strict Avalanche Criterion are constructed; in the end, the algorithm calculating primitive Boolean permutation ( f₀ ( x₀ ,…, x_n-1 ),…, f_n-1( x₀ , , x_n-1))which is determined by x^2n-2 is given.4) In the third section of the second chapter, the Orthomorphic permutation of multiple-valued logical function has been studied. First several properties of Orthomorphic permutation are presented; then a kind of linearity permutation and Orthomorphic...