Constructions Of Two Classes Of Boolean Functions With Good Cryptographic Properties

Recently,with the high development of science and technology,cryptography,as the theoretical basis of modern security systems,is attracting more and more attentions.The analysis and designing of cryptographic algorithms based on Boolean functions is one of the important research interests in the field of cryptography.With the increasing improvement of cryptanalysis,various types of cryptographic attacks are emerging endlessly.In order to resist these attacks,cryptographic scholars have given a series of cryptographic indicators,such as: balancedness,algebraic degree,nonlinearity,algebraic immunity,correlation immunity,etc.Generally,when constructing a Boolean function with good cryptographic properties,the security indicators of the function should be as optimal as possible.However,there is a trade-off among these indexes.For instance,while pursuing a high nonlinearity,the algebraic degree may decline,and the correlation immunity may also decrease.Therefore,it is very important to construct Boolean functions with good cryptographic properties.This thesis mainly studies some cryptographic properties of Boolean functions and the constructing method of Boolean functions with good cryptographic properties.The main results are as follows:1.Based on the idea of ordered integer partitions in number theory,two new kinds of theoretical construction of rotation symmetric Boolean functions with optimal algebraic immunity both on odd variables and on even variables are presented by modifying the support of the majority function.At the same time,the nonlinearity,algebraic degree and algebraic immunity of these functions are also analyzed.The results show that our rotation symmetric Boolean functions have much higher nonlinearity than all the existing theoretical constructions of rotation symmetric Boolean functions with optimal algebraic immunity.Furthermore,the algebraic degrees,and the fast algebraic immunities of the rotation symmetric Boolean functions are also very high in some cases.2.Using the characteristics of the orbits in the 9)-dimensional vector space F9)2over the finite field F2,a construction of balanced rotation symmetric Boolean functions on an arbitrary even number of variables with optimal algebraic immunity is presented by modifying the support of the majority function.Moreover,up to now,there are only several constructions ofbalanced rotation symmetric Boolean functions on 9)variables with optimal algebraic immunity,where 9)is a particular even integer as 26)(6)is positive integer)or 2( is prime).So,this has made a theoretical progress in constructing balanced rotational symmetric Boolean functions on even number of variables with optimal algebraic immunity.3.Using the relationship between Krawtchouk polynomials and the spectral characteristics of the Boolean functions given by Xiao and Massey,a new construction of 1st order correlation immune symmetric Boolean functions is presented by modifying the simplified truth table of the existing symmetric Boolean functions with high correlation immunity.In fact,the problem of constructing 1st order correlation immune symmetric Boolean functions is transformed into the problem of solving the root of a quadratic or cubic equation to finish the construction.