本文關(guān)鍵詞：布爾函數(shù)的密碼學(xué)性質(zhì)研究　出處：《西安電子科技大學(xué)》2009年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 布爾函數(shù) 全局雪崩準(zhǔn)則 代數(shù)厚度 正規(guī)布爾函數(shù) 等重對(duì)稱布爾函數(shù) Krawtchouk多項(xiàng)式

【摘要】： 布爾函數(shù)在密碼學(xué)和通信領(lǐng)域有廣泛的應(yīng)用.論文研究了布爾函數(shù)的一些性質(zhì).取得以下主要結(jié)果: (1).將Son關(guān)于n元平衡布爾函數(shù)的全局雪崩準(zhǔn)則(GAC)的結(jié)果推廣到了任意漢明重量的布爾函數(shù),從布爾函數(shù)的漢明角度給出了平方和指標(biāo)的下界表達(dá)式,同時(shí)得到了布爾函數(shù)的非線性度上界的漢明重量表達(dá)式;從Bent函數(shù)角度構(gòu)造了兩類平方和指標(biāo)和絕對(duì)值指標(biāo)較小的布爾函數(shù). (2).基于一個(gè)布爾函數(shù)的全局雪崩準(zhǔn)則(GAC),提出了兩個(gè)不同布爾函數(shù)的互相關(guān)函數(shù)所對(duì)應(yīng)的全局雪崩準(zhǔn)則:平方和指標(biāo)和絕對(duì)值指標(biāo),給出了這兩個(gè)指標(biāo)的上下界.這個(gè)指標(biāo)推廣了Zhang和Zheng提出的GAC指標(biāo).同時(shí)也得到了兩個(gè)布爾函數(shù)Walsh譜與互相關(guān)函數(shù)的一些性質(zhì). (3).通過研究具有線性結(jié)構(gòu)的布爾函數(shù)的性質(zhì),利用Walsh譜和漢明重量得到了布爾函數(shù)不具有k維線性結(jié)構(gòu)的充分條件,進(jìn)而給出了具有線性結(jié)構(gòu)的彈性布爾函數(shù)新的非線性度上界. (4).基于代數(shù)厚度的定義,研究了一些布爾函數(shù)代數(shù)厚的關(guān)系式,得到仿射函數(shù)、相關(guān)免疫函數(shù)、部分Bent和Bent函數(shù)的代數(shù)厚度上界是2n?1,在此基礎(chǔ)上改進(jìn)了k(2≤k≤n?2 1)次基本對(duì)稱布爾函數(shù)代數(shù)厚度的上界. (5).基于線性子空間理論給出了一個(gè)布爾函數(shù)在給定仿射空間上是k -正規(guī)的充要條件,同時(shí)給出布爾函數(shù)滿足k -正規(guī)時(shí)k和其的漢明重量的關(guān)系,進(jìn)而給出了判斷一個(gè)布爾函數(shù)是否是k -正規(guī)的算法,經(jīng)分析此算法較對(duì)所有的k維空間進(jìn)行搜索計(jì)算量小,易于實(shí)現(xiàn). (6).利用Krawtchouk多項(xiàng)式和組合數(shù)學(xué)討論了等重對(duì)稱布爾函數(shù)的密碼學(xué)性質(zhì),給出了等重對(duì)稱布爾函數(shù)Walsh譜的表達(dá)式,利用此表達(dá)式給出了等重對(duì)稱布爾函數(shù)的非線性度,相關(guān)免疫性,擴(kuò)散性,平衡性等,結(jié)果表明這類函數(shù)不具有較好的密碼學(xué)性質(zhì).
[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，

本文編號(hào)：1405609