【作者】 苏为；

【导师】 唐小虎；

【作者基本信息】 西南交通大学 ， 信息安全， 2013， 博士

【摘要】 布尔函数在密码系统中有着非常重要的应用,密码系统中的很多问题都可以转化成布尔函数相关密码学性质的研究。本论文主要研究了初等对称布尔函数的平衡性问题、奇变元对称布尔函数的代数免疫度、两种Partial Spread Bent函数的表达式及自对偶情况、Neagbent函数的一些性质以及Bent-Negabent函数的构造。首先,根据Cusick、Li和Stanica等人的工作,本文进一步研究了他们提出的初等对称布尔函数平衡性问题的猜想。利用初等对称布尔函数的分解,得到了该猜想在大多数情况下都是成立的,并且本文的结果覆盖了几乎所有的已知结果,分析方法也很简单。其次考虑了奇变元对称布尔函数代数免疫度次优的情况。利用权重支集这个工具并结合对称布尔函数零化子的特点,给出了2m+3个变元的对称布尔函数代数免疫度次优的充要条件；并通过布尔函数的分解与级联,利用偶变元对称布尔函数代数免疫度的一些结果,得到了奇变元对称布尔函数代数免疫度次优或最优的必要条件以及一些奇变元代数免疫度次优或最优的对称布尔函数类。接着介绍了与Regular Spread相近的两种Spread:Andre Spread和Albert Spread,并分析了由这两种Spread定义的Partial Spread Bent函数的函数表达式、对偶函数的表达式以及自对偶的充要条件。最后,本文讨论了Negabent函数的一些性质及Bent-Negabent函数的构造问题。通过分析Nega-Hadamard变换和Walsh-Hadamard变换之间的关系,给出了任意变元的布尔函数是Negabent的充要条件,确定了Negabent函数的Nega谱值分布情况,并介绍了一种构造n元Bent-Negabent函数的方法。这种构造方法可以得到任意指定代数次数的Bent-Negabent函数,这表明n元Bent-Negabent函数的代数次数的最大值为n/2,因此解决了关于Bent-Negabent函数代数次数的最大值及构造具有高代数次数的Bent-Negabent的公开问题。更多还原

【Abstract】 Boolean functions are frequently used in the design of stream ciphers, block ciphers, error correcting codes and Hash functions. One of the most vital roles in cryptography of Boolean functions is to be used as filter and combination generators of stream ciphers based on linear feedback shift registers. In order to resist different kinds of attacks, Boolean functions used in cipher systems must satisfy some properties, such as balancedness, high nonlinearity, high algebraic degree, and high order of algebraic immunity. In this thesis, we mainly focus on the study of Boolean functions in four topics:the balancedness of elementary symmetric Boolean functions, the algebraic immunity of symmetric Boolean functions on odd variables, the representation and the self-duality of some partial spread Bent functions, the properties of Negabent functions and the construction of Bent-Negabent functions.Firstly, according to the work given by Cusick, Li and Stanica, we will further investi-gate the conjecture about the balancedness of elementary symmetric Boolean functions. By decomposing elementary symmetric Boolean functions, we prove that the conjecture holds in most cases. Our results cover most of the known results, and the method is much simpler than the known methods.Secondly, we will study the algebraic immunity of symmetric Boolean functions on odd variables. Using the tool of weight support, we give the necessary and sufficient conditions for2m+3variables symmetric Boolean functions to achieve suboptimal algebraic immunity. According to the concatenation of Boolean functions and the known results about symmetric Boolean functions with high algebraic immunity on even number of variables, we obtain some necessary conditions for symmetric Boolean functions with suboptimal algebraic immunity on odd variables and construct some classes of symmetric Boolean functions with suboptimal algebraic immunity on odd variables.Next, we will consider the properties of partial spread Bent functions. We introduce two kinds of partial spread Bent functions:Andre partial spread Bent functions and Albert partial spread Bent functions, which are similar with the classic partial spread Bent functions--PSap. We give the representations of these Bent functions and their dual functions, and the necessary and sufficient conditions for these Bent functions to be self-dual.Finally, we will analyze the properties of Negabent functions and the construction of Bent-Negabent functions. We present necessary and sufficient conditions for a Boolean func- tion to be a negabent function for both an even and an odd number of variables, which demon-strates the relationship between negabent functions and bent functions. By using these neces-sary and sufficient conditions for Boolean functions to be negabent, we obtain that the nega spectrum of a negabent function has at most4values. We determine the nega spectrum dis-tribution of negabent functions. Further, we provide a method to construct bent-negabent functions in n variables (n even) of algebraic degree ranging from2ton/2, which implies that the maximum algebraic degree of an n-variable bent-negabent function is equal ton/2. Thus, we answer two open problems proposed by Parker and Pott and by Stanica et al. respectively.更多还原