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。
Optimal nonlinear functions, i.e., Bent functions and perfect nonlinear functions, appose optimal resistances to linear cryptanalysis and difference cryptanalysis respectively, thus they play an important role in cryptography. Moreover, optimal nonlinear functions also have significant applications in coding theory, sequence design, combinatorial theory and so on.
The first main topic of this thesis is the construction of Bent functions. From the ap-proach of linearized equations, several new classes of quadratic generalized Boolean Bent functions are obtained based on the theory of Z4-valued quadratic forms. This together with the link between Boolean Bent functions and generalized Boolean Bent functions leads to new Boolean Bent functions, and new p-ary Bent functions are also presented by using the same method in the finite fields with odd characteristic p. For the Bent functions with high degree, this thesis mainly focuses on the Bent functions of Dillon type and Niho type which have the optimal algebraic degree. Through the discussions on some partial exponential sums over finite fields, several new classes of Dillon type Boolean Bent and p-ary Bent functions are characterized, which generalize some known results. By adopting the same techniques, this thesis generalizes the well-known Leander-Kholosha's class of Niho type Boolean Bent functions and a much simpler proof regarding the bentness is also given. Meanwhile, the investigated function of Niho type in the finite fields of odd characteristic is proven to be four-valued Walsh spectra and the spectra distribution is also determined.
The second main topic of this thesis is the construction of optimal cyclic codes from perfect nonlinear and almost perfect nonlinear functions. By using the canonical factoriza-tions of polynomials with low degrees over finite fields, and the relation between the degree of an irreducible polynomial and its roots in finite fields, an open problem proposed by D-ing and Helleseth about optimal single-error correcting ternary cyclic codes is successfully solved, and four new classes of such optimal codes with parameters [3m-1,3m-1-2m,4] are also obtained by virtue of the quadratic character over finite fields, where m is a positive integer. Furthermore, two new classes of optimal two-error correcting ternary cyclic codes with parameters [3m-1,3m-2-2m,5] are also presented by making use of the properties of perfect nonlinear functions. In addition, the covering radiuses of the constructed optimal codes and the weight distributions of their dual codes are considered as well. However, only few results are obtained and there still some problems remain open.
The third main topic of this thesis is the investigation of optimal or almost optimal quaternary sequence families by using generalized Boolean Bent functions and the highly nonlinear Gold functions. According to the theory of Z4-valued quadratic forms and the properties of generalized Boolean Bent functions, a class of exponential sums over Galois rings is investigated and then the exact correlation distributions of two classes of optimal sequences are determined. Moreover, again by the theory of Z4-valued quadratic forms, a class of known optimal quaternary sequences and a new class of quaternary sequences with low correlation are obtained by a unique method. On the other hand, based on the properties of the Gold functions over Galois rings, the correlation distribution of the quaternary Gold sequences is determined. From the constructed quaternary Gold sequences, the maximal nontrivial correlation value of its MSB sequences and the exact correlation distribution of its Gray sequences are also determined in virtue of the relation between quaternary sequences and binary sequences.
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