布尔函数

From this,two algorithms to find low degree annihilators of a Boolean function are present:complete searching and stochastic searching.The complete one can not only get all the low degree annihilators of a Boolean function,but also get the algebraic immunity of the function.

利用完全搜索法可以得到布尔函数的所有低次零化子，同时还可以确定该布尔函数的代数免疫度；利用随机搜索法能快速得到布尔函数的任意一个低次零化子。

Additionally, the properties and normal extension construction of normal Boolean function is studied, and checking algorithm of normal is discussed.

另外，本文分析了正规布尔函数的性质和正规扩展的构造方法，讨论了布尔函数正规性的检验算法，这对于Bent函数的分类问题的解决具有重要意义。

Bent function is a class of the highest nonlinearity Boolean function, and Boolean function is the key in the cipher system.

Bent函数是一类具有最高非线性度的布尔函数，布尔函数是对称加密密码体制设计和分析的关键。

Second, in the viewpoint of crypto analysis, the practical requirement for the cryptographic function is to construct the function that can resist all the known crypto analysis and attacks. In advanced, the main cryptographic properties are compared, and the relation among these cryptographic properties is analyzed, including the trade-off ones. Get the conclusion that the construction of cryptographic Boolean function is to find out the proper Boolean function in the whole vector space under the restriction of several cryptographic properties required for the security system.

然后本文从密码分析的角度，讨论了密码函数的实际需求，即构造能够抵抗已有密码分析与攻击的布尔函数；分析了布尔函数的主要的密码学性质，并比较了这些性质之间的相互关系，包括制约的关系；得出了构造布尔函数的实质的结论，即在向量空间中构造全体布尔函数，以所需的若干密码学性质作为限制条件，用有限域上的多项式、Walsh谱、Hadamard矩阵等数学工具找出适当的布尔函数用于加密体制的设计。

In the fourth chapter, firstly it introduces the Boolean functions algebraic expressions of the 2-value clock-controlled stop-and-go generator and Gunther generator. It reveals the balanced property of the two kinds of Boolean functions, and studies the Walsh cycle spectrum and the autocorrelation function. It also obtains the coincidence rate of their output sequences with affine sum of some bits of input sequences, and analyzes their ability of resisting the best affine approximation cryptanalysis and differential cryptanalysis. Secondly, we properly present a new definition of the Best Affine Approximation, namely BAA on the Boolean vector functions, followed by the spectral characteristic of such defined BAA attacks through using the decomposition formula of the union distribution for random variables. A lower bound of such BAA attacks is proposed. Finally, we also study the spectral characteristic of the second kind of nonlinearity of Boolean vector functions, followed by a higher bound of such nonlinearity. Furthermore, the limited relationship between the second kind of nonlinearity of Boolean vector functions and the linear structure of the linear combination of every component is analyzed.

在第四章中，首先给出了2值密钥流"停走生成器"和"衮特生成器"中实际存在的布尔函数的代数表示，揭示了这两类布尔函数的平衡性，随后研究了它们的Walsh循环谱和自相关函数等，得到了它们的输出序列与输入序列中的某些bit的仿射项的符合率，分析了它们抵抗最佳仿射逼近攻击和差分攻击的能力；其次，我们合理地给出了布尔向量函数最佳仿射逼近的新定义，利用布尔随机变量联合分布的分解式考察了相应的谱特征，并给出了布尔向量函数与其最佳仿射逼近的符合率的一个下界；最后，我们还考察了布尔向量函数第二类非线性度的谱特征，给出了布尔向量函数第二类非线性度的一个上界，并揭示了布尔向量函数第二类非线性度与其各个分量的线性和的线性结构之间存在的制约关系。

A necessary and sufficient condition for a fx + t(y + g to be a bent function is disscussed,on which basis a necessary and sufficient condition for a function F = fix + tq + g to be a bent function is given when q is not a balanced-function by use of cyclic walsh spectrum and decomposition formula of walsh spectrum.

讨论了形如fx+τ(y+g的布尔函数是Bent函数的充分必要条件，在此基础上利用Bent函数的循环Walsh谱及谱分解式，进一步给出了当q不是平衡函数时，形如F=fx+τ(yq+g布尔函数是Bent函数的充分必要条件。

Based on the definition of Boolean function on set and positive extension, The sufficient and necessary condition of PBF is proven. At last, an algorithm of generating MSP expression of stack filter positive Boolean function is presented taking advantage of Hasse diagram.

通过定义布尔函数开、闭集和最小项正、负扩展，证明了布尔函数具有层叠性的充要条件是布尔函数开集的正扩展或闭集的负扩展具有不变性，在此基础上，提出了一种通过Hasse图确定层叠滤波器最简正布尔函数的算法。

Based on the definition of Boolean function on set and positive extension, The sufficient and necessary condition of PBF is proven. At last,an algorithm of generating MSP expression of stack filter positive Boolean function ispresented taking advantage of Hasse diagram.

通过定义布尔函数开、闭集和最小项正、负扩展，证明了布尔函数具有层叠性的充要条件是布尔函数开集的正扩展或闭集的负扩展具有不变性，在此基础上，提出了一种通过Hasse图确定层叠滤波器最简正布尔函数的算法。

Using the concept of Boolean functions and combinatorics theory comprehensively, we investigate the construction on annihilators of Boolean functions and the algebraic immunity of symmetric Boolean functions in cryptography:Firstly, we introduce two methods of constructing the annihilators of Boolean functions, Construction I makes annihilators based on the minor term expression of Boolean function, meanwhile we get a way to judge whether a Boolean function has low degree annihilators by feature matrix. In Construction II, we use the subfunctions to construct annihilators, we also apply Construction II to LILI-128 and Toyocrypt, and the attacking complexity is reduced greatly. We study the algebraic immunitiy of (5,1,3,12) rotation symmetric staturated best functions and a type of constructed functions, then we prove that a new class of functions are invariants of algebraic attacks, and this property is generalized in the end.Secondly, we present a construction on symmetric annihilators of symmetric Boolean functions.

本文主要利用布尔函数的相关概念并结合组合论的相关知识，对密码学中布尔函数的零化子构造问题以及对称布尔函数代数免疫性进行了研究，主要包括以下两方面的内容：首先，给出两种布尔函数零化子的构造方法，构造Ⅰ利用布尔函数的小项表示构造零化子，得到求布尔函数f代数次数≤d的零化子的算法，同时得到通过布尔函数的特征矩阵判断零化子的存在性：构造Ⅱ利用布尔函数退化后的子函数构造零化子，将此构造方法应用于LILI-128，Toyocrypt等流密码体制中，使得攻击的复杂度大大降低；通过研究(5，1，3，12)旋转对称饱和最优函数的代数免疫和一类构造函数的代数免疫，证明了一类函数为代数攻击不变量，并对此性质作了进一步推广。

In this paper, first we define a kind of transformation on the sequences of Boolean functions, and then analyze the changes of nonlinearity, algebraic degree and balancedness of Boolean functions under the transformation.

本文首先定义了对布尔函数的序列的一种变换，并分析了进行此种变换后布尔函数的非线性度、代数次数与平衡性的变化情况，最终在此基础上提出一种新的递归构造方法，可以构造出非线性度很高的n元m阶n-m-1次弹性函数。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。