Multi-output Boolean Function Of The Nature Of The Study

Block ciphers have been extensively applied in computer communication and the safety field of information systems. In the design and the analysis of block ciphers, multi-output functions play an important role. For example, DES is a standard encryption system, whose central part is 8 S boxes, and each S box is a multi-output function with 6 inputs and 4 outputs in essential. So studying on properties and constructions of multi-output functions is significative not only on theory but also on application.Using the theories of probability, algebra and spectral theory comprehensively, the dissertation investigates the properties and constructions of multi-output functions. Especially we investigate properties and constructions of multi-output Bent functions and multi-output partially Bent functions and get some new results.The main results of this dissertation are as following:1. The properties of the generalized Walsh spectrum of multi-output functions are more studied and some corresponding results are proved. Then the definition and probability expression of the generalized autocorrelation function of multi-output functions are given. And the relationship between the generalized Walsh spectrum and the generalized autocorrelation function of multi-output functions is obtained.2. The independency of multi-output functions is discussed. Then definitions of correlation degree and correlation coefficient of multi-output functions are presented. Relationship between the independency and correlation degree of multi-output functions is proved. Relationship between the correlation degree and balance is also obtained. Especially the paper discusses the correlation degree of a multi-output function and a balanced affine multi-output function. We demonstrate the value range of the correlation coefficient, the relationship between the independency and correlation coefficient of multi-output functions, and the relationship between equality of two multi-output functions and the correlation coefficient of them. At last, we discuss the non-linear degree of multi-output functions by applying the correlation degree.3. It discusses the linear construction of multi-output functions and obtains the generalized autocorrelation function and generalized Walsh spectrum of the linear construction.4. The autocorrelation function, algebra degree, pervasion and number of multi-output Bentfunctions are discussed. From the autocorrelation function, we present two methods to discriminate multi-output Bent functions. It also sums up several equal definitions and gives two methods of constructing multi-output Bent functions.5. Properties and constructions of multi-output partially Bent functions are investigated. Firstly, the dissertation gives the definition of multi-output partially Bent functions and tests the existence. Then the generalized Walsh spectrum and generalized autocorrelation function is presented. Secondly, it discusses the cryptology properties of multi-output partially Bent functions, such as balance, correlation immunity, non-linearity and pervasion. Thirdly, the essential relation between multi-output partially Bent functions and multi-output Bent functions is obtained, containing their function and spectrum. And the method of constructing multi-output partially Bent functions by multi-output Bent functions is given. Lastly, several general methods of constructing multi-output partially-bent functions are presented. Meanwhile the generalized Walsh cyclic spectrum characteristics of these multi-output partially-bent functions are given.