Cryptographic Properties And Construction Of Boolean Functions

Boolean functions and complementary sequence pairs have important applications in the design of cryptosystem and coding theory.As a Boolean functions with the highest nonlinearity,Bent function is widely used in the areas of cryptography,coding and communication.The Walsh-Hadamard transform is a powerful tool for studying the cryptographic properties of Boolean functions,and has been used to characterize a variety of cryptographic properties of Boolean functions.With the gradual deepening of the researches on Boolean functions,the Nega-Hadamard transform was introduced based on the Walsh-Hadamard transform,and the concept of Negabent function was introduced.This dissertation studies the properties and construction of Negabent function and complementary sequence pairs,and the results are as follows:(1)The Nega-Hadamard transform of Boolean functions are studied.Firstly,we generalize the convolution theorem,following with a discussion of the Nega-Hadamard transformation of the composite functions,which were composed of S-box and 8)-variable Boolean functions.Therefore,the generalized Nega-composition theorem is obtained.Secondly,we analyze some results of the Nega-Hadamard transform on combinatorial Boolean functions.Finally,a new proof of the Nega-Hadamard transform of combinatorial function is given by the generalized Nega-composition theorem.(2)We construct a class of Boolean function with n+m+2-variable,and the relationship between the functions and his primary function in their Walsh-Hadamard transform is also evaluated.(3)The Nega-Hadamard transform of a class of Boolean functions is studied by utilizing the generalized Nega-composite theorem,proving the sufficient conditions for the function to be Negabent function.Furthermore,we study the characteristics of the Nega-Hadamard transform and autocorrelation function of two classes of Boolean functions.Based on the results,equivalence conditions for two classes of functions to be Negabent functions is obtained.(4)We study the construction method of complementary sequence pairs.The binary complementary sequence pairs with the length of 2 and 4 are constructed by interleaved technique.The relationship between the complementary sequence pairs and the subsequences is analyzed by calculating the autocorrelation value of the complementary sequence pairs.