On Constructions Of Boolean Functions With Special Properties And Sequences Design

Boolean functions have a wide range of applications in cryptography andcommunications. Algebraic immunity and nonlinearity are important cryptographicproperties of Boolean functions. This thesis investigates the constructions of theBoolean functions with high algebraic immunity of the graphs or high nonlinearity. Newfamilies of binary sequences with low correlation is constructed by using Booleanfunctions. The main results are as follows:1. Algebraic immunity of the graphs of Boolean functions is discussed. Theproblem of algebraic immunity of the graph is converted to the problem of annihilatorsof the single-output assistant function. A new method for constructing single-andmulti-output Boolean functions with high algebraic immunity of the graphs is proposed.This method can give single-output Boolean functions with maximum algebraicimmunity and maximum algebraic immunity of the graphs.2. Based on the theory of quadratic forms and linearized polynomials over finitefields, two families of quadratic Bent functions in polynomial forms are given.Furthermore, the frst family of Bent functions includes the functions proposed byUdaya and some functions proposed by Hu et al..3. This thesis studies the constructions of Semi-Bent functions in even number ofvariables. A new proof of the upper bound on algebraic degrees of semi-bent functionsis given by means of the relation between coeffcients of Boolean functions and theirWalsh transform. Three new classes of semi-bent functions with are proposed by usingNiho exponents. The frst class of semi-bent functions is balanced. It is shown that allsemi-bent functions of the second class attain the maximum degree, and there exists onesubclass with maximum degree in the frst and the third classes of semi-bent functions.It is proved that the semi-bentness of these functions with some restriction is stronglyrelated to Dickson polynomials or Kloosterman sums. Furthermore, several examples ofsemi-bent functions are given by using Kloosterman sums. Owing to the result given byS. Kim et al., the open problem proposed by Charpin et al. can be solved partially. Threeclasses of examples of infnite families of semi-bent functions are obtained.4. The small sets of generalized Kasami sequences are constructed. They have thesame family size and correlation distribution as the small set of Kasami sequences.Moreover, the proposed families include the small set of Kasami sequences as a special family. A family of binary sequences with low correlation and large size is obtained. Itincludes the small set of Kasami sequences as its subfamily.