高非线性函数的构造及其在序列编码中的应用
发布时间：2018-08-03 09:47
【摘要】：最佳非线性函数即Bent函数和完全非线性函数分别是抵抗线性密码攻击和差分密码攻击能力最强的密码函数,故其在密码学中扮演着非常重要的角色。而且,最佳非线性函数在编码理论、序列设计和组合理论等领域中亦有重要的应用。 本论文的第一个主要研究内容是Bent函数的构造。基于环上的二次型理论和线性化方程途径,本文首先构造出几类新的二次广义布尔Bent函数。结合布尔Bent函数与广义布尔Bent函数之间的关系并将构造广义布尔Bent函数的方法应用于奇特征域中,本文相继得到新的二次布尔Bent函数和二次p-元Bent函数,其中p是-奇素数。而对于高次Bent函数,本文着重研究了具有最佳代数次数的Dillon型Bent函数和Niho型Bent函数。通过对有限域中某些部分指数和的讨论,本文成功刻画出几类新的Dillon型布尔Bent函数和Dillon型p-元Bent函数,并推广了部分已知结果。将研究Dillon型Bent函数的方法运用在Niho型函数上,本文推广了偶特征域中Leander-Kholosha类Niho型Bent函数的结论,并给出了其Bent性的一个简洁的证明。同时,本文证明了所考察的Niho型函数在奇特征域中具有四值Walsh谱且确定了其谱值分布。 本论文的第二个主要研究内容是利用完全非线性函数和几乎完全非线性函数构造最佳循环码。通过利用有限域上低次多项式的因式分解以及不可约多项式次数与其对应方程解之间的关系,本论文成功解决了由Ding和Helleseth提出的一个关于最佳三元单纠错循环码的公开问题。借助于有限域上的二次特征,运用同样的方法,对于正整数m,本论文得到了四类新的参数为[3m-1,3m-2m-1,4]的最佳三元单纠错循环码。更进一步地,通过利用完全非线性函数的性质,本论文亦构造出两类新的参数为[3m-1,3m-2m-2,5]的最佳三元双纠错循环码。而且,本论文亦考虑了上述所得最佳循环码的覆盖半径及其对偶码的重量分布。然而,本论文仅得到部分相关结果,目前仍有较多问题尚未解决。 本论文的第三个主要研究内容是利用广义布尔Bent函数和高非线性Gold函数研究最佳或几乎最佳四元序列集。借助于环上的二次型理论和广义布尔Bent函数的性质,本论文考察了环上一类指数和的性质进而确定了两类最佳序列集的精确相关分布。而且,基于环上二次型理论,本文利用统一的方法得到了一类已知的最佳四元序列集和一类新的低相关四元序列集。另一方面,通过对伽罗华环上Gold函数性质的考察,本论文确定了四元Gold序列集的精确相关分布。而且,依据四元序列与二元序列之间的关系,本文确定了四元Gold序列集的MSB序列的最大非平凡相关值以及四元Gold序列集的Gray序列的精确相关分布。
[Abstract]:The best nonlinear function, that is, Bent function and complete nonlinear function, are the most powerful cryptographic functions to resist linear cipher attack and differential cryptosystem attack, respectively, so they play a very important role in cryptography. Moreover, the optimal nonlinear function has important applications in the fields of coding theory, sequence design and combination theory. The first part of this thesis is the construction of Bent function. Based on the theory of quadratic form over rings and the path of linearization equations, several new classes of quadratic generalized Boolean Bent functions are constructed in this paper. Combining the relationship between Boolean Bent function and generalized Boolean Bent function, and applying the method of constructing generalized Boolean Bent function to odd characteristic domain, we obtain new quadratic Boolean Bent function and quadratic p- element Bent function, where p is an odd prime number. For higher order Bent functions, this paper focuses on the study of Dillon type Bent functions and Niho type Bent functions with the best algebraic degree. Through the discussion of some partial exponential sums in finite fields, several new classes of Dillon type Boolean Bent functions and Dillon type p-element Bent functions are successfully characterized in this paper, and some known results are generalized. In this paper, the method of studying Dillon type Bent functions is applied to Niho type functions. In this paper, we generalize the conclusion of Leander-Kholosha type Niho type Bent functions in the even characteristic domain, and give a concise proof of its Bent property. At the same time, it is proved that the Niho type function has four-valued Walsh spectrum and its spectral value distribution is determined in the odd characteristic domain. The second main content of this thesis is to construct the best cyclic codes by using completely nonlinear functions and almost completely nonlinear functions. By using the factorization of low order polynomials over finite fields and the relationship between irreducible polynomial degrees and the solutions of their corresponding equations, this paper successfully solves an open problem of optimal single error correcting cyclic codes for ternary systems proposed by Ding and Helleseth. By using the same method and by means of the quadratic characteristics over finite fields, four new classes of triple single error-correcting cyclic codes with [3m-1n 3m-2m-1m-1] are obtained for positive integers m in this paper. Furthermore, by using the properties of completely nonlinear functions, this paper also constructs two kinds of optimal binary error correcting cyclic codes with two new parameters [3m-1n 3m-2m-2m-2]. Furthermore, the coverage radius of the optimal cyclic codes and the weight distribution of the dual codes are also considered in this paper. However, only some results have been obtained in this paper, and there are still many problems to be solved. The third main content of this thesis is to study the best or almost optimal quaternion sequence set by using the generalized Boolean Bent function and the high nonlinear Gold function. With the help of the theory of quadratic form over rings and the properties of generalized Boolean Bent functions, this paper investigates the properties of a class of exponential sums over rings and determines the exact correlation distributions of two kinds of optimal sequence sets. Furthermore, based on the theory of quadratic form over rings, a class of known optimal quaternion sequences and a new low correlation quaternion sequence set are obtained by using the unified method. On the other hand, the exact correlation distribution of quaternion Gold sequence sets is determined by investigating the properties of Gold functions over Galois rings. Furthermore, according to the relation between quaternion and binary sequence, the maximum nontrivial correlation value of MSB sequence of quaternion Gold sequence set and the exact correlation distribution of Gray sequence of quaternion Gold sequence set are determined.
【学位授予单位】：西南交通大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TN918.3
，
本文编号：2161361
[Abstract]:The best nonlinear function, that is, Bent function and complete nonlinear function, are the most powerful cryptographic functions to resist linear cipher attack and differential cryptosystem attack, respectively, so they play a very important role in cryptography. Moreover, the optimal nonlinear function has important applications in the fields of coding theory, sequence design and combination theory. The first part of this thesis is the construction of Bent function. Based on the theory of quadratic form over rings and the path of linearization equations, several new classes of quadratic generalized Boolean Bent functions are constructed in this paper. Combining the relationship between Boolean Bent function and generalized Boolean Bent function, and applying the method of constructing generalized Boolean Bent function to odd characteristic domain, we obtain new quadratic Boolean Bent function and quadratic p- element Bent function, where p is an odd prime number. For higher order Bent functions, this paper focuses on the study of Dillon type Bent functions and Niho type Bent functions with the best algebraic degree. Through the discussion of some partial exponential sums in finite fields, several new classes of Dillon type Boolean Bent functions and Dillon type p-element Bent functions are successfully characterized in this paper, and some known results are generalized. In this paper, the method of studying Dillon type Bent functions is applied to Niho type functions. In this paper, we generalize the conclusion of Leander-Kholosha type Niho type Bent functions in the even characteristic domain, and give a concise proof of its Bent property. At the same time, it is proved that the Niho type function has four-valued Walsh spectrum and its spectral value distribution is determined in the odd characteristic domain. The second main content of this thesis is to construct the best cyclic codes by using completely nonlinear functions and almost completely nonlinear functions. By using the factorization of low order polynomials over finite fields and the relationship between irreducible polynomial degrees and the solutions of their corresponding equations, this paper successfully solves an open problem of optimal single error correcting cyclic codes for ternary systems proposed by Ding and Helleseth. By using the same method and by means of the quadratic characteristics over finite fields, four new classes of triple single error-correcting cyclic codes with [3m-1n 3m-2m-1m-1] are obtained for positive integers m in this paper. Furthermore, by using the properties of completely nonlinear functions, this paper also constructs two kinds of optimal binary error correcting cyclic codes with two new parameters [3m-1n 3m-2m-2m-2]. Furthermore, the coverage radius of the optimal cyclic codes and the weight distribution of the dual codes are also considered in this paper. However, only some results have been obtained in this paper, and there are still many problems to be solved. The third main content of this thesis is to study the best or almost optimal quaternion sequence set by using the generalized Boolean Bent function and the high nonlinear Gold function. With the help of the theory of quadratic form over rings and the properties of generalized Boolean Bent functions, this paper investigates the properties of a class of exponential sums over rings and determines the exact correlation distributions of two kinds of optimal sequence sets. Furthermore, based on the theory of quadratic form over rings, a class of known optimal quaternion sequences and a new low correlation quaternion sequence set are obtained by using the unified method. On the other hand, the exact correlation distribution of quaternion Gold sequence sets is determined by investigating the properties of Gold functions over Galois rings. Furthermore, according to the relation between quaternion and binary sequence, the maximum nontrivial correlation value of MSB sequence of quaternion Gold sequence set and the exact correlation distribution of Gray sequence of quaternion Gold sequence set are determined.
【学位授予单位】：西南交通大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TN918.3
，
本文编号：2161361
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