The Properties And Constructions Of Boolean Functions In Cryptography

Boolean functions play an important role in Ciyptology,?especially in stream cipher. The main reason is that at some degree the cryptographic properties decide the security of the system. This dissertation is devoted to the ciyptographic properties of Boolean functions and constructions of Boolean functions, which satisfy certain ciyptographic properties. This paper is divided into six chapters. Chapter one describes briefly the significance of this problem and the main works that the author did. Chapter two summarizes all the different forms of Boolean functions, introduces Walsh spectrum as an important tool in the study of Boolean functions and some important results, which are presented with detailed proof Chapter three completely studies the properties of Boolean functions in ciyptolgy and the relationships among some properties. Some important characteristics of functions which have balancedness, correlation inimunityc propagation criterion or strict avalanche criterion are pointed out respectively; For example, presents the feature of the Walsh spectrum and the algebraic normal forms of functions with balanced correlation immunity. Nonlinearity and its upper bounds of Boolean functions are deep discussed. The relationships between the algebraic degree with correlation immunity, propagation criterion order and etc. are further studied .The trade-off between properties of Boolean functions are emphasized. Chapter four investigations some kinds of special Boolean functions: bent functions, non-repeated homogeneous and so forth. The integrated results about the properties of these functions are obtained. Therefore, their superiority and defect in ciyptology are known clearly. Chapter five studies some ordinaiy methods of instructing Boolean functions. The ciyptographic properties of functions obtained by the introduced methods are analyzed at detail. Chapter six is one focus of this paper, mainly studies the constructions of Boolean functions. Firstly, based on Bent functions and non-repeated homogeneous, one kind of balanced correlation immunity. Boolean functions with high nonlinearity is obtained. Secondly, on the basis of Bent functions and linear functions, one kind of balanced correlation immunity with high propagation criterion order is given. Furthermore, based on choosing parameter in two special functions, the constructing methods of Boolean functions with good cryptographic properties are presented.