特殊性质的布尔函数构造与序列设计
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摘要
布尔函数在密码学和通信领域有着广泛的应用。代数免疫度和非线性度是布尔函数重要的密码学指标。本文研究了具有大的图的代数免疫的布尔函数和高非线性布尔函数的构造。应用布尔函数设计了二元低相关序列。主要成果有:
1.研究了布尔函数图的代数免疫问题，将多输出布尔函数图的代数免疫问题转化为其单输出辅助函数的零化子问题。给出了一个构造具有大的图的代数免疫的单输出和多输出布尔函数的新方法，此方法可以得到具有最大代数免疫和图的代数免疫的偶变元单输出布尔函数。
2.根据有限域上的二次型和线性化多项式理论，给出了两族多项式形式的二次Bent函数。第一族Bent函数包含了Udaya提出的Bent函数和Hu等人给出的某些Bent函数。
3.研究了偶变元Semi-Bent函数的构造。利用布尔函数的系数和Walsh变换之间的关系，给出了Semi-Bent函数的代数次数上界的一种新的证明。利用Niho指数构造了三类Semi-Bent函数。第一类Semi-Bent函数是平衡函数。证明了所有第二类Semi-Bent函数具有最大代数次数，并且在第一类和第三类函数中均存在一个Semi-Bent函数的子类达到最大代数次数。证明了当限制某些参数的取值范围时，函数的Semi-Bent性与Dickson多项式或Kloosterman和密切相关。利用Kloosterman和，给出了Semi-Bent函数的一些例子。最后，利用S.Kim等人的结果，部分解决了Charpin等人提出的一个公开问题，据此给出了三类无限族Semi-Bent函数的具体例子。
4.构造了广义小集合Kasami序列集，它们与小集合Kasami序列有相同的序列数目和相关分布。并且小集合Kasami序列集是其一种特殊情况。也设计了一个具有较多的序列数目的低相关序列集。它包含小集合的Kasami序列。
Boolean functions have a wide range of applications in cryptography andcommunications. Algebraic immunity and nonlinearity are important cryptographicproperties of Boolean functions. This thesis investigates the constructions of theBoolean functions with high algebraic immunity of the graphs or high nonlinearity. Newfamilies of binary sequences with low correlation is constructed by using Booleanfunctions. The main results are as follows:
1. Algebraic immunity of the graphs of Boolean functions is discussed. Theproblem of algebraic immunity of the graph is converted to the problem of annihilatorsof the single-output assistant function. A new method for constructing single-andmulti-output Boolean functions with high algebraic immunity of the graphs is proposed.This method can give single-output Boolean functions with maximum algebraicimmunity and maximum algebraic immunity of the graphs.
2. Based on the theory of quadratic forms and linearized polynomials over finitefields, two families of quadratic Bent functions in polynomial forms are given.Furthermore, the frst family of Bent functions includes the functions proposed byUdaya and some functions proposed by Hu et al..
3. This thesis studies the constructions of Semi-Bent functions in even number ofvariables. A new proof of the upper bound on algebraic degrees of semi-bent functionsis given by means of the relation between coeffcients of Boolean functions and theirWalsh transform. Three new classes of semi-bent functions with are proposed by usingNiho exponents. The frst class of semi-bent functions is balanced. It is shown that allsemi-bent functions of the second class attain the maximum degree, and there exists onesubclass with maximum degree in the frst and the third classes of semi-bent functions.It is proved that the semi-bentness of these functions with some restriction is stronglyrelated to Dickson polynomials or Kloosterman sums. Furthermore, several examples ofsemi-bent functions are given by using Kloosterman sums. Owing to the result given byS. Kim et al., the open problem proposed by Charpin et al. can be solved partially. Threeclasses of examples of infnite families of semi-bent functions are obtained.
4. The small sets of generalized Kasami sequences are constructed. They have thesame family size and correlation distribution as the small set of Kasami sequences.Moreover, the proposed families include the small set of Kasami sequences as a special family. A family of binary sequences with low correlation and large size is obtained. Itincludes the small set of Kasami sequences as its subfamily.
1.研究了布尔函数图的代数免疫问题，将多输出布尔函数图的代数免疫问题转化为其单输出辅助函数的零化子问题。给出了一个构造具有大的图的代数免疫的单输出和多输出布尔函数的新方法，此方法可以得到具有最大代数免疫和图的代数免疫的偶变元单输出布尔函数。
2.根据有限域上的二次型和线性化多项式理论，给出了两族多项式形式的二次Bent函数。第一族Bent函数包含了Udaya提出的Bent函数和Hu等人给出的某些Bent函数。
3.研究了偶变元Semi-Bent函数的构造。利用布尔函数的系数和Walsh变换之间的关系，给出了Semi-Bent函数的代数次数上界的一种新的证明。利用Niho指数构造了三类Semi-Bent函数。第一类Semi-Bent函数是平衡函数。证明了所有第二类Semi-Bent函数具有最大代数次数，并且在第一类和第三类函数中均存在一个Semi-Bent函数的子类达到最大代数次数。证明了当限制某些参数的取值范围时，函数的Semi-Bent性与Dickson多项式或Kloosterman和密切相关。利用Kloosterman和，给出了Semi-Bent函数的一些例子。最后，利用S.Kim等人的结果，部分解决了Charpin等人提出的一个公开问题，据此给出了三类无限族Semi-Bent函数的具体例子。
4.构造了广义小集合Kasami序列集，它们与小集合Kasami序列有相同的序列数目和相关分布。并且小集合Kasami序列集是其一种特殊情况。也设计了一个具有较多的序列数目的低相关序列集。它包含小集合的Kasami序列。
Boolean functions have a wide range of applications in cryptography andcommunications. Algebraic immunity and nonlinearity are important cryptographicproperties of Boolean functions. This thesis investigates the constructions of theBoolean functions with high algebraic immunity of the graphs or high nonlinearity. Newfamilies of binary sequences with low correlation is constructed by using Booleanfunctions. The main results are as follows:
1. Algebraic immunity of the graphs of Boolean functions is discussed. Theproblem of algebraic immunity of the graph is converted to the problem of annihilatorsof the single-output assistant function. A new method for constructing single-andmulti-output Boolean functions with high algebraic immunity of the graphs is proposed.This method can give single-output Boolean functions with maximum algebraicimmunity and maximum algebraic immunity of the graphs.
2. Based on the theory of quadratic forms and linearized polynomials over finitefields, two families of quadratic Bent functions in polynomial forms are given.Furthermore, the frst family of Bent functions includes the functions proposed byUdaya and some functions proposed by Hu et al..
3. This thesis studies the constructions of Semi-Bent functions in even number ofvariables. A new proof of the upper bound on algebraic degrees of semi-bent functionsis given by means of the relation between coeffcients of Boolean functions and theirWalsh transform. Three new classes of semi-bent functions with are proposed by usingNiho exponents. The frst class of semi-bent functions is balanced. It is shown that allsemi-bent functions of the second class attain the maximum degree, and there exists onesubclass with maximum degree in the frst and the third classes of semi-bent functions.It is proved that the semi-bentness of these functions with some restriction is stronglyrelated to Dickson polynomials or Kloosterman sums. Furthermore, several examples ofsemi-bent functions are given by using Kloosterman sums. Owing to the result given byS. Kim et al., the open problem proposed by Charpin et al. can be solved partially. Threeclasses of examples of infnite families of semi-bent functions are obtained.
4. The small sets of generalized Kasami sequences are constructed. They have thesame family size and correlation distribution as the small set of Kasami sequences.Moreover, the proposed families include the small set of Kasami sequences as a special family. A family of binary sequences with low correlation and large size is obtained. Itincludes the small set of Kasami sequences as its subfamily.
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