Constructions And Analyses Of Cryptographic Properties Of Cryptographic Functions

Cryptographic functions are important parts of many kinds of cryptosystems.To make the designing cryptosystem resisting all existing attacks,the cryptographic func-tion selected for the cryptosystem must satisfy some cryptographic properties,such as balancedness,correlation immunity,resiliency,high algebraic degree,high nonlinearity,high algebraic immunity,low differential uniformity,etc.Therefore,the research and construction of cryptographic functions with good cryptographic properties have im-portant significance both in theory and practical application.This article mainly studies the problem of the construction and analysis of several key cryptographic properties of cryptographic functions,and the research results are as follows.For the three key cryptographic properties—nonlinearity,algebraic immunity and differential uniformity,this paper firstly uses the Schur function,which is an important tool in combinatorial mathematics,to give a new characterization of balanced Boolean functions with optimum algebraic immunity.By employing this characterization,a new proof for the Carlet-Feng function to have optimum algebraic immunity is provided.Meanwhile,three classes of balanced Boolean functions with optimal algebraic immu-nity are constructed.Examples with the high nonlinearity,high algebraic degree and other good cryptographic properties of the three classes are found.Secondly,a class of differential 4-uniform permutations is obtained by the method of dividing the domain of the function into two subsets,with two different permutations defined on each of them.Moreover,the algebraic degree,nonlinearity and other cryptographic properties of the functions are also analyzed.The CCZ-inequivalence of the functions with the 12 known classes of differentially 4-uniform permutations is discussed.Thirdly,five classes of Semi-bent functions and two classes of Plateaued functions are constructed,which are compared with the known ones.Important cryptographic properties of the Budaghyan-Carlet polynomial and Boolean functions of Dembowski-Ostrom type are analyzed.The properties and the number of elements of a set related to the Budaghyan-Carlet polynomials are studied.By studying the component function of the Budaghyan-Carlet polynomial,a class of quadratic bent functions is obtained.The problem whether the Budaghyan-Carelt can become a permutation by adding a linear function is answered.Moreover,it is proved that if the multi-output Boolean function of Dembowski-Ostrom type has only one ze-ro root and the derived function of it has one or four roots,then the function has the classical Walsh spectrum and the Walsh spectrum distribution can be given explicit-ly.Therefore,the walsh spectrum distributions of four classes of APN functions of Dembowski-Ostrom type are obtained.