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Research on Several Classes of Multi-valued Logical Functions with Excellent Cryptological Properties

Author: YuanYanBin
Tutor: ZhaoYaQun
School: PLA Information Engineering University
Course: Cryptography
Keywords: Generalized Partially Bent Functions p-adic decomposition Perfect Nonlinear Functions k -order strict avalanche criterion Rotationally symmetric function
CLC: TN918.1
Type: Master's thesis
Year: 2009
Downloads: 13
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Abstract


Password excellent performance logic function, design and analysis of the password has important theoretical significance and application value. This article integrated use of probability theory, algebra, number theory and other disciplines basis of the theory of knowledge and spectrum theory, meet the k-th strict avalanche criteria of multi-output Boolean functions over finite fields generalized part of Bent functions and generalized Bent function of the relationship, ring Zpl on multi-output the fully nonlinear function and multi-output generalized Bent function, multi-output and multi-valued nature of the rotationally symmetric function were studied. The paper mainly do the following four aspects: a concept of order k strict avalanche criterion Extension to multiple-output Boolean function, multi-output Boolean function satisfies the strict avalanche criterion and proliferation norms Criterion and gives the necessary and sufficient condition for the function satisfies the strict avalanche criterion of order k; particular, according to the characteristics of symmetric function, were given multiple-output symmetric Boolean functions necessary and sufficient condition to meet the strict avalanche criterion and proliferation norms, and The two combined function satisfies the strict avalanche criterion of order k discriminant formula provides the basis for excellent cryptographic functions for constructing this nature. Second, part of the concept of Bent functions broadened to a finite field, still called Generalized Partially Bent Functions. Generalized Partially Bent functions over finite fields Chrestenson cyclic spectrum characteristics, and accordingly the relationship between characteristics and logic function on a finite field with prime field on a regular basis decomposition function, finite fields Generalized Partially Bent Functions and Generalized Bent Generalized Partially Bent Functions Generalized Bent Function function relationship, the relationship between these two types of logic functions function formula and spectral values, this relationship can be by the relatively small number of the large number of good nature. , The use of p-adic decomposition method, the residue class ring Zpl, the multi-output the fully nonlinear function and multi-output generalized Bent function. First p-adic decomposition sense given the pl value of multi-output function vector representation and Its Operational. Then, get a pl value of multi-output of fully nonlinear function in the sense of p-adic decomposition Criterion. Finally, the the ring Zpl on multi-output Generalized Bent Functions and Generalized Bent multiple-output the fully nonlinear function and multiple-output functional relationship in terms of p-adic decomposition. The conclusions show that the residue class ring Zpl the two are equivalent, some answers the famous scholar Mr. Nyberg About \Fourth, the multi-output and multi-valued rotation symmetric function. The one hand, the first time the concept of rotation symmetric functions extended to the case of multi-output, multi-output rotationally symmetric function of the generalized order the Walsh spectral characteristics and generalized autocorrelation function characteristics; combined cycle spectral characteristics, by constructing a correlation matrix the difference is the necessary and sufficient conditions for such functions to meet the nature of balance, immunity, strict avalanche criterion password; then explored the odd argument multi-output Plateaued rotation cyclic spectrum of the nature of the symmetric function generalized order Walsh, given a looking the odd argument multi-output Plateaued rotational symmetry function by calculating the cyclic spectrum value, reducing the amount of computation. On the other hand, for the prime field multi-valued rotationally symmetric function, first given the function Chrestenson cyclic spectrum characteristics and the autocorrelation function characteristics; incidence matrix polynomial function truth table, simplify algebraic Standard and Chrestenson spectrum linked, and thus gives the function to meet the balance of the nature of necessary and sufficient conditions of correlation immunity, stability and password.

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