Logic Function. Cryptography Nonlinear Criteria

Using the theories of probability, algebra and number theory comprehensively, we investigate a class of Boolean functions with three-valued Walsh spectrum in the first part of this dissertation: The properties of the extended semi-Bent functions, which are constructed from any two Bent functions, are studied, followed by the structure characteristics of the Boolean functions satisfying propagation criterion with respect to all but two vectors; The definition and cryptographic properties of k-order quasi-Bent functions are proposed whose Walsh spectrum takes on only three values. Some sufficient and necessary conditions are offered to decide whether a Boolean function is a k-order quasi-Bent function; A special method is presented to construct the k-order quasi-Bent functions, whose cryptographic properties are explored by the matrix method, which is different from the method of Walsh spectrum and that of autocorrelation of Boolean functions; The application of this kind of Boolean functions in the fields of stream cipher, communications and block ciphers is discussed, which shows the great importance of the fc-order quasi-Bent functions; Some methodology are proposed to construct the k-order quasi-Bent functions, including the complete construction by using the characteristic matrices of Boolean functions, and the recursive method by two known k-order quasi-Bent functions We further extend our investigation to the ring Zp, where p is a prime, and the similar results are presented as far as the p-valued quasi-generalized-Bent functions are considered.The main contributions of the second part of this dissertation are focused on the cryptographic properties of logical functions over finite field, with the help of the properties of trace functions, and that of p-polynomials, as well as the permutation theory over finite field: The new definition of Chrestenson linear spectrum is given and the relation betweenthe new Chrestenson linear spectrum and the Chrestenson cyclic spectrum is presented, followed by the inverse formula of logical function over finite field; The distribution for linear structures of the logical functions over finite field is discussed and the complete construction of logical functions taking on all vectors as linear structures is suggested, which leads to the conception of the extended affine functions over finite field, whose cryptographic properties is similar to that of the affine functions over field GF(2) and prime field Fp; The relationship between the degeneration of logical functions and the linear structures, the degeneration of logical functions and the support of Chrestenson spectrum, as well as the relation between the nonlinearity and the linear structures are discussed; Using the relation of the logical functions over finite field and the vector logical functions over its prime field, we reveal the relationship between the perfect nonlinear functions over finite field and the vector generalized Bent functions over its prime field; The existence or not of the perfect nonlinear functions with any variables over any finite fields is offered, and some methods are proposed to construct the perfect nonlinear functions by using the balanced p-polynomials over finite field.