布尔函数代数免疫性质的研究
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摘要
近年来，代数攻击的提出和发展被认为是密码分忻技术中的最重要的突破。其主要特点是采用了基于代数思想的方法与技巧，将一个密码算法的安全性完全归约为求解一个超定的多变元高次方程组(即该系统中的方程个数多于变量的个数)的问题上。几们对超定的多变元非线性方程组生成和求解进行深入研究，取得很大的进展，使得代数攻击已逐渐发展为一种强有力的密码分忻方法。代数攻击的思想方法具有一般性，对公钥密码、分组密码和序列密码算法的安全性都构成了潜在的威胁，特别对基于线性反馈移位寄存器(LFsR)的序列密码具有强大的攻击威力。代数攻击方法在密码算法设计和安全性分忻的研究中发挥了重要的作用，对未来密码算法的设计与分忻产生了深远的影响。
分组密码和序列密码体制中重要的非线性部件都可以用布尔函数来刻画，其密码学性质的好坏直接影响着密码体制的安全强度。代数攻击丰富了密码分忻技术，它不同于所有以住的基于统计的密码分忻技术，而是依据密码内在的代数结构，采用了代数的分忻技术。针对代数攻击，密码学者提出了新的密码学准则即布尔函数的代数免疫。目前，对布尔函数代数免疫的研究主要包括三方面内容：代数免疫和零化子次数的快速计算、最优代数免疫布尔函数构造和代数免疫与其它密码学性质之间的关系。代数攻击的复杂度是和代数免疫成指数关系。这就意味着代数免疫的些微差别，对代数攻击复杂度的影响都是非常大。因此对布尔函数的代数免疫性质的研究具有重要的理论和实际意义。
本文就布尔函数的代数免疫性质进行了研究，取得的主要结果如下：
1．利用布尔函数特征矩阵的代数结构，讨论正规性与代数次数之间的关系，并首次得到布尔函数正规阶与代数免疫之间的制约关系，布尔函数的正规阶较高时就保证布尔函数具有低次零化子，因而通过确定布尔函数的正规阶来计算其代数免疫，并给出了布尔函数具有1次和2次零化子的判别条件。
2．通过讨论布尔函数汉明重量与代数次数之间的关系，发现若一个n元布尔函数厂的零点集O，限制在一簇仿射子空间上都是代数次数≥k的函数，且这一簇仿射子空间的并集包含O则厂必不存在代数次数 3．给出布尔函数多项式表示定义，重点讨论布尔函数的单变元多项式表示，并利用单变元多项式函数的迹表示给出布尔函数的代数次数与其多项式表示的次数之间的关系：一个布尔函数g的代数次数≤d的等价判别条件是其单变元多项式表示g=∑,gix，gi∈F2n，对于w2(i)≥d+1的系数g，都有gi=O。
4．利用Reed-s010mon码的相关结果，讨论布尔函数的代数免疫，得到布尔函数具有最优代数免疫的判别条件，特别当变元个数为奇数时，得到布尔函数具有最优代数免疫的等价判别条件。
5．研究了旋转对称函数(RSBFs)的密码学性质。发现RSBFs的walsh谱和自相关函数值与函数的取值类似，当输入变量进行循环移位变换时，RSBFs的Wslsh谱和自相关函数值保持不变，其中的Walsh谱特性还是RSBFs的一个等价判别条件。进而讨论了RSBFs的相关免疫、代数次数和代数免疫。
6．讨论了具有最优代数免疫的偶数元择多布尔函数的密码学性质。计算了一些已知具有好的密码学性质的RSBFs的代数免疫，发现这些函数也是代数免疫性质较好的一类布尔函数。
In recent years, algebraic attack is considered as the most important breach of cryptoanalysis. The main characteristic of algebraic attack is that it reduces the safety problem of cryptographic algorithm to the problem of solving an over-defined equations system with high degrees (There are more equations than variants in the system). People make some works on how to generate and solve an over-defined equations system and achieve some results on it. Algebraic attack has turned into an important and mature analysis method. Due to its generality, algebraic attack poses potential threats upon public-key cipher, block cipher and stream cipher. This is especially true for stream ciphers based on LFSR. Algebraic attack has made significant effect on the analysis of a cipher system.
The non-linear parts of both block cipher and stream cipher can be realized by Boolean function. The cryptography properties of Boolean fuctions affect the security of a cipher derectly. As a new important cryptanalysis method, algebraic attack brings forth a new requirement for the design of Boolean functions, algebraic immunity. Alebraic attack makes the cipher analysis techmology diversiform. It is quite different from former analytical methods which were mostly based on the idea of probability. At present, the research to algebraic immunity of Boolean functions main include three aspects: the methods of determining the existence of the low-degree annihilator and calculating the algebraic immunity of a Boolean function, the construction of Optimal Algebraic Immunity Boolean Function and some of its cryptographic properties.
The main object of this dissertation is to study the algebraic immunity of Boolean functions, the main results are as follows:
1. By using the algebraic structure of a Boolean function's character matrix, obtain the interdependent relation between the normality and algebraic immunity of Boolean functions, a relatively high normality of Boolean function guarantees the existence of a low-degree annihilator, thus, the algebraic immunity of Boolean function can be determined by figuring out its normality, get the equivalent criteria when the algebraic immunity is 1 and 2.
2. By making use of the relation between the algebraic degree and Hamming Weights of Boolean function and the fact that Boolean function, find that if we let the offset of a n-variables Boolean function 0/ limits on a sequence of affine subspaces, we always have new Boolean functions with degree >k, we determine the function/has no annihilator with degree 3. Present the general definition of multinomial representation of Boolean function, emphases on the discuss of univariate form. By making use of the trace functions representation of the univariate form, obtain the algebraic degree relation between Boolean function and its univarite form: a Boolean functions g have degree 4. By making use of results about Reed-Solomon code to discuss the single variant form of Boolean function, we can obtain the criteria of the Boolean function with optimal algebraic immunity. Particularly, when there are odd variants, we can obtain the equivalent criteria of the Boolean function with optimal algebraic immunity.
5. Study the cryptography properties of RSBFs. Discuss the Walsh spectra and autocorrelation function of RSBFs, when input variants was cycle shifted, the Walsh spectra and autocorrelation function are unchanged. We attain an equivalent criterion of RSBFs.
6. Discuss the cryptographic properties of even variables majority Boolean functions with and the optimal algebraic immunity, and the algebraic immunity of some RSBFs with the best cryptographic properties is computed.
分组密码和序列密码体制中重要的非线性部件都可以用布尔函数来刻画，其密码学性质的好坏直接影响着密码体制的安全强度。代数攻击丰富了密码分忻技术，它不同于所有以住的基于统计的密码分忻技术，而是依据密码内在的代数结构，采用了代数的分忻技术。针对代数攻击，密码学者提出了新的密码学准则即布尔函数的代数免疫。目前，对布尔函数代数免疫的研究主要包括三方面内容：代数免疫和零化子次数的快速计算、最优代数免疫布尔函数构造和代数免疫与其它密码学性质之间的关系。代数攻击的复杂度是和代数免疫成指数关系。这就意味着代数免疫的些微差别，对代数攻击复杂度的影响都是非常大。因此对布尔函数的代数免疫性质的研究具有重要的理论和实际意义。
本文就布尔函数的代数免疫性质进行了研究，取得的主要结果如下：
1．利用布尔函数特征矩阵的代数结构，讨论正规性与代数次数之间的关系，并首次得到布尔函数正规阶与代数免疫之间的制约关系，布尔函数的正规阶较高时就保证布尔函数具有低次零化子，因而通过确定布尔函数的正规阶来计算其代数免疫，并给出了布尔函数具有1次和2次零化子的判别条件。
2．通过讨论布尔函数汉明重量与代数次数之间的关系，发现若一个n元布尔函数厂的零点集O，限制在一簇仿射子空间上都是代数次数≥k的函数，且这一簇仿射子空间的并集包含O则厂必不存在代数次数
4．利用Reed-s010mon码的相关结果，讨论布尔函数的代数免疫，得到布尔函数具有最优代数免疫的判别条件，特别当变元个数为奇数时，得到布尔函数具有最优代数免疫的等价判别条件。
5．研究了旋转对称函数(RSBFs)的密码学性质。发现RSBFs的walsh谱和自相关函数值与函数的取值类似，当输入变量进行循环移位变换时，RSBFs的Wslsh谱和自相关函数值保持不变，其中的Walsh谱特性还是RSBFs的一个等价判别条件。进而讨论了RSBFs的相关免疫、代数次数和代数免疫。
6．讨论了具有最优代数免疫的偶数元择多布尔函数的密码学性质。计算了一些已知具有好的密码学性质的RSBFs的代数免疫，发现这些函数也是代数免疫性质较好的一类布尔函数。
In recent years, algebraic attack is considered as the most important breach of cryptoanalysis. The main characteristic of algebraic attack is that it reduces the safety problem of cryptographic algorithm to the problem of solving an over-defined equations system with high degrees (There are more equations than variants in the system). People make some works on how to generate and solve an over-defined equations system and achieve some results on it. Algebraic attack has turned into an important and mature analysis method. Due to its generality, algebraic attack poses potential threats upon public-key cipher, block cipher and stream cipher. This is especially true for stream ciphers based on LFSR. Algebraic attack has made significant effect on the analysis of a cipher system.
The non-linear parts of both block cipher and stream cipher can be realized by Boolean function. The cryptography properties of Boolean fuctions affect the security of a cipher derectly. As a new important cryptanalysis method, algebraic attack brings forth a new requirement for the design of Boolean functions, algebraic immunity. Alebraic attack makes the cipher analysis techmology diversiform. It is quite different from former analytical methods which were mostly based on the idea of probability. At present, the research to algebraic immunity of Boolean functions main include three aspects: the methods of determining the existence of the low-degree annihilator and calculating the algebraic immunity of a Boolean function, the construction of Optimal Algebraic Immunity Boolean Function and some of its cryptographic properties.
The main object of this dissertation is to study the algebraic immunity of Boolean functions, the main results are as follows:
1. By using the algebraic structure of a Boolean function's character matrix, obtain the interdependent relation between the normality and algebraic immunity of Boolean functions, a relatively high normality of Boolean function guarantees the existence of a low-degree annihilator, thus, the algebraic immunity of Boolean function can be determined by figuring out its normality, get the equivalent criteria when the algebraic immunity is 1 and 2.
2. By making use of the relation between the algebraic degree and Hamming Weights of Boolean function and the fact that Boolean function, find that if we let the offset of a n-variables Boolean function 0/ limits on a sequence of affine subspaces, we always have new Boolean functions with degree >k, we determine the function/has no annihilator with degree
5. Study the cryptography properties of RSBFs. Discuss the Walsh spectra and autocorrelation function of RSBFs, when input variants was cycle shifted, the Walsh spectra and autocorrelation function are unchanged. We attain an equivalent criterion of RSBFs.
6. Discuss the cryptographic properties of even variables majority Boolean functions with and the optimal algebraic immunity, and the algebraic immunity of some RSBFs with the best cryptographic properties is computed.
引文
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