Study On Some Crucial Topics About Symmetric Boolean Functions And Bent Functions

Boolean functions are frequently used in the design of stream ciphers, block ciphers, error correcting codes and Hash functions. One of the most vital roles in cryptography of Boolean functions is to be used as filter and combination generators of stream ciphers based on linear feedback shift registers. In order to resist different kinds of attacks, Boolean functions used in cipher systems must satisfy some properties, such as balancedness, high nonlinearity, high algebraic degree, and high order of algebraic immunity. In this thesis, we mainly focus on the study of Boolean functions in four topics:the balancedness of elementary symmetric Boolean functions, the algebraic immunity of symmetric Boolean functions on odd variables, the representation and the self-duality of some partial spread Bent functions, the properties of Negabent functions and the construction of Bent-Negabent functions.Firstly, according to the work given by Cusick, Li and Stanica, we will further investi-gate the conjecture about the balancedness of elementary symmetric Boolean functions. By decomposing elementary symmetric Boolean functions, we prove that the conjecture holds in most cases. Our results cover most of the known results, and the method is much simpler than the known methods.Secondly, we will study the algebraic immunity of symmetric Boolean functions on odd variables. Using the tool of weight support, we give the necessary and sufficient conditions for2m+3variables symmetric Boolean functions to achieve suboptimal algebraic immunity. According to the concatenation of Boolean functions and the known results about symmetric Boolean functions with high algebraic immunity on even number of variables, we obtain some necessary conditions for symmetric Boolean functions with suboptimal algebraic immunity on odd variables and construct some classes of symmetric Boolean functions with suboptimal algebraic immunity on odd variables.Next, we will consider the properties of partial spread Bent functions. We introduce two kinds of partial spread Bent functions:Andre partial spread Bent functions and Albert partial spread Bent functions, which are similar with the classic partial spread Bent functions--PSap. We give the representations of these Bent functions and their dual functions, and the necessary and sufficient conditions for these Bent functions to be self-dual.Finally, we will analyze the properties of Negabent functions and the construction of Bent-Negabent functions. We present necessary and sufficient conditions for a Boolean func- tion to be a negabent function for both an even and an odd number of variables, which demon-strates the relationship between negabent functions and bent functions. By using these neces-sary and sufficient conditions for Boolean functions to be negabent, we obtain that the nega spectrum of a negabent function has at most4values. We determine the nega spectrum dis-tribution of negabent functions. Further, we provide a method to construct bent-negabent functions in n variables (n even) of algebraic degree ranging from2ton/2, which implies that the maximum algebraic degree of an n-variable bent-negabent function is equal ton/2. Thus, we answer two open problems proposed by Parker and Pott and by Stanica et al. respectively.