密码函数的密码学性质分析及构造

发布时间：2019-05-08 16:48

【摘要】：密码函数是多种密码系统的重要组成部分.要使设计的密码系统能够抵抗各种已有的攻击,要求该系统所选用的密码函数必须满足一些相应的密码学性质,如平衡性、相关免疫性、弹性、高代数次数、高非线性度、高代数免疫度、低差分均匀度等.因此研究和构造具有优良密码学性质的密码函数在理论和实际应用上都具有重要意义.本文主要研究密码函数几个关键密码学性质的分析和构造问题,得到了如下研究成果:针对非线性度、代数免疫度及差分均匀度这三类关键密码学安全性指标,本文首先运用组合数学中的重要工具一 Schur·函数,给出了最优代数免疫平衡布尔函数的一种新刻画.利用此刻画给出了 Carlet-Feng函数是最优代数免疫函数的新证明.同时,构造了三类平衡的最优代数免疫布尔函数.发现所构造的三类函数中存在高非线性度、高代数次数等其它优良密码学性质的例子.其次,采用将函数的定义域分为两个子集,且在这两个子集上定义不同置换的方法,得到了一类4-差分置换.研究了其代数次数、非线性度等密码学性质.还讨论了该类函数与12类4-差分置换的CCZ不等价性.最后,构造了五类二次Semi-bent函数及两类Plateaued函数,并与已知构造进行了比较.本文还对Budaghyan-Carlet多项式及Dembowski-Ostrom型函数的重要密码学性质进行了分析.讨论了一个与Budaghyan-Carlet多项式有关的集合所含元素的性质和个数.通过研究Budaghyan-Carlet多项式的分量函数,得到了一类Bent函数,回答了 Budaghyan-Carelt多项式是否能通过加上线性化多项式成为置换多项式这一问题.另外,证明了若Dembowski-Ostrom型多输出布尔函数有唯一零根且其导函数有一个或者四个根,则该布尔函数具有经典Walsh谱,且其Walsh谱分布可以明确给出.由此进一步得到了四类Dembowski-Ostrom型APN函数的Walsh谱分布.
[Abstract]:Cryptographic function is an important part of many cryptographic systems. In order to make the designed cryptosystem resist all kinds of existing attacks, the cryptosystem must satisfy some corresponding cryptographic properties, such as balance, correlation immunity, elasticity, high algebraic times, and high nonlinearity, and the cryptology function chosen by the system must satisfy some corresponding cryptographic properties, such as balance, correlation immunity, elasticity, high algebraic times, and high nonlinearity. High algebraic immunity, low differential uniformity, etc. Therefore, the research and construction of cryptographic functions with excellent cryptographic properties are of great significance both in theory and in practice. This paper mainly studies the analysis and construction of several key cryptology properties of cryptographic function, and obtains the following research results: aiming at the three key cryptology security indexes: nonlinearity, algebraic immunity and differential uniformity, In this paper, a new characterization of the optimal algebraic immune equilibrium Boolean function is given by using the Schur function, an important tool in combinatorial mathematics. In this paper, a new proof that Carlet-Feng function is an optimal algebraic immune function is given. At the same time, three kinds of optimal algebraic immune Boolean functions are constructed. It is found that there are some examples of other excellent cryptographic properties, such as high nonlinearity, high algebraic times, and so on, among the three classes of functions constructed in this paper. Secondly, a class of 4-difference permutations is obtained by dividing the defined domain of a function into two subsets and defining different permutations on the two subsets. The cryptology properties such as algebraic number, nonlinearity and so on are studied. The CCZ inequality of this class of functions with 12 kinds of 4-difference permutations is also discussed. Finally, five classes of quadratic Semi-bent functions and two classes of Plateaued functions are constructed and compared with the known constructions. In this paper, the important cryptographic properties of Budaghyan-Carlet polynomials and Dembowski- Ostrom-type functions are also analyzed. The properties and number of elements in a set related to Budaghyan-Carlet polynomials are discussed. By studying the component functions of Budaghyan-Carlet polynomials, we obtain a class of Bent functions and answer the question whether Budaghyan-Carelt polynomials can be permutation polynomials by adding linearized polynomials. In addition, it is proved that if Dembowski- Ostrom type multi-output Boolean function has unique zero root and its derivative function has one or four roots, then the Boolean function has classical Walsh spectrum, and its Walsh spectrum distribution can be clearly given. The Walsh spectral distributions of four types of Dembowski- Ostrom-type APN functions are obtained.
【学位授予单位】：湖北大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TN918.1;O174

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