Research On Cryptographic Functions With Good Properties

The cryptographic function is an important component of a cryptosystem,and its security determines the security of the entire cryptosystem.The ability of a cryptosys-tem to resist known attacks is closely related to the various properties of its crypto-graphic functions.Bent functions are Boolean functions with optimal nonlinearity,which attract extensive attention in the areas of cryptography,coding,and combina-torics.This dissertation studies bent functions and their various generalized functions,including p-ary bent functions,plateaued functions and bent4 functions.The specific research content and innovation points of this dissertation are described as follows:Firstly,we study a special class of p-ary function called Coulter-Matthews bent function.By using a new combinatorial technique,we determine the expression of the dual function of Coulter-Matthews bent function completely.As a consequence,we find many new classes of ternary bent functions not reported in the literature previously.Such bent functions are not quadratic if k>1.Subsequently,we studied the dual functions under four special parameters.Among them,new classes of ternary bent functions with only 8 or 21 trace terms have been dug out.Next,we study a special form of quadratic plateaued function,and consider the enumeration of the quadratic function with prescribed Walsh spectrum.The specific value of the counting function is given by its generating function.We extend previous results,and propose the generic generating functions for the counting functions for all cases by using some number-theoretical methods and combinatorial methods.At the same time,we can get the number of two special classes of functions—bent functions and semi-bent functions.Finally,we study the equivalent characterization of bent4 functions.Firstly,by analysing the relationship between transforms in transform set {H,N}n and Walsh-Hadamard transform,we obtain three sufficient and necessary conditions for Boolean functions to be bent4 when n is odd.The three sufficient and necessary conditions give the relationships between bent4 functions in n variables and special bent functions in n-1 variables,special semi-bent functions in n variables,special bent functions in n + 1 variables,respectively.Then,by studying the decomposition of bent4 functions with respect to codimension one subspaces,we get two new equivalent characterizations of bent4 functions.Therefore,we investigate the transform set {I,N}n which has not been considered before,and obtain some new spectral properties.