布尔函数的密码学性质研究

发布时间：2018-01-10 14:23

  本文关键词：布尔函数的密码学性质研究　出处：《西安电子科技大学》2009年博士论文　论文类型：学位论文

  更多相关文章： 布尔函数 全局雪崩准则 代数厚度 正规布尔函数 等重对称布尔函数 Krawtchouk多项式

【摘要】： 布尔函数在密码学和通信领域有广泛的应用.论文研究了布尔函数的一些性质.取得以下主要结果: (1).将Son关于n元平衡布尔函数的全局雪崩准则(GAC)的结果推广到了任意汉明重量的布尔函数,从布尔函数的汉明角度给出了平方和指标的下界表达式,同时得到了布尔函数的非线性度上界的汉明重量表达式;从Bent函数角度构造了两类平方和指标和绝对值指标较小的布尔函数. (2).基于一个布尔函数的全局雪崩准则(GAC),提出了两个不同布尔函数的互相关函数所对应的全局雪崩准则:平方和指标和绝对值指标,给出了这两个指标的上下界.这个指标推广了Zhang和Zheng提出的GAC指标.同时也得到了两个布尔函数Walsh谱与互相关函数的一些性质. (3).通过研究具有线性结构的布尔函数的性质,利用Walsh谱和汉明重量得到了布尔函数不具有k维线性结构的充分条件,进而给出了具有线性结构的弹性布尔函数新的非线性度上界. (4).基于代数厚度的定义,研究了一些布尔函数代数厚的关系式,得到仿射函数、相关免疫函数、部分Bent和Bent函数的代数厚度上界是2n?1,在此基础上改进了k(2≤k≤n?2 1)次基本对称布尔函数代数厚度的上界. (5).基于线性子空间理论给出了一个布尔函数在给定仿射空间上是k -正规的充要条件,同时给出布尔函数满足k -正规时k和其的汉明重量的关系,进而给出了判断一个布尔函数是否是k -正规的算法,经分析此算法较对所有的k维空间进行搜索计算量小,易于实现. (6).利用Krawtchouk多项式和组合数学讨论了等重对称布尔函数的密码学性质,给出了等重对称布尔函数Walsh谱的表达式,利用此表达式给出了等重对称布尔函数的非线性度,相关免疫性,扩散性,平衡性等,结果表明这类函数不具有较好的密码学性质.
[Abstract]:Boolean functions are widely used in cryptography and communication. Some properties of Boolean functions are studied in this paper. The main results are as follows: The result of Son's global avalanche criterion for n-variable equilibrium Boolean functions is extended to Boolean functions of arbitrary hamming weight. The lower bound expression of square sum index is given from the hamming angle of Boolean function, and the hamming weight expression of upper bound of nonlinear degree of Boolean function is obtained. From the point of view of Bent function, two classes of Boolean functions with small square sum index and absolute value index are constructed. Based on the global avalanche criterion of a Boolean function, the global avalanche criterion corresponding to the cross-correlation function of two different Boolean functions is proposed: the sum of squares index and the absolute value index. The upper and lower bounds of these two indices are given, which generalize the GAC indices proposed by Zhang and Zheng. Some properties of Walsh spectrum and cross-correlation functions of two Boolean functions are also obtained. By studying the properties of Boolean functions with linear structure, the sufficient conditions for Boolean functions without k-dimensional linear structures are obtained by using Walsh spectrum and hamming weight. Furthermore, a new nonlinear upper bound of elastic Boolean function with linear structure is given. Based on the definition of algebraic thickness, the relation of algebraic thickness of some Boolean functions is studied. The affine function, the correlation immune function and the upper bound of algebraic thickness of some Bent and Bent functions are obtained. 1. On the basis of this, we improve the KG 2 鈮，

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