Constructions,counts and classifications of cyclotomic involutions over finite fields

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摘要：

由于有限域上多项式f(x)可以唯一地写成x^rh(x^s)+f(0),2009年王强等基于此表示提出多项式指标概念.这一概念自提出之后,在研究多项式值域、特征和、置换多项式等问题上起到了重要的作用.对合在分组密码构造中有十分重要的意义.近两年,有多位学者对对合进行研究,旨在为分组密码构造中S盒的设计提供更多选择.最近,郑大彬等对Fq上形如x^rh(x^s)的对合进行研究,给出了该类多项式是对合的一个充要条件并提出了一种构造此类对合的方法.该方法需要对某方程组,即方程组(3),进行求解.利用对称群中的共轭关系和分块矩阵的思想,首先对郑大彬等的方法进行深层次的分析,给出了方程组解的确切表达式,改进了该构造方法;其次,给出了有限域上任意固定指标、常数项为0的对合的个数;再次,根据指标的大小,对具有显性表达式的已有对合进行分类;最后,确定了几类对合,丰富了已有结果.具体地,针对低指标对合,给出了指标为2和3的较郑大彬等结果更具体的对合条件;针对非低指标对合,利用李康荃等得到的复合逆结果,给出了一类Fq2上形如x^rh(x^q-1)的对合.

Since any polynomials f(x) over finite fields can be written uniquely as x^rh(x^s)+f(0), based on this form, Wang et al. presented a new concept called the index of polynomials in 2009. Since it was proposed, this parameter has turned out to be very useful in studying value set size of polynomials, character sum, permutation polynomials, among others. Involutions play very important roles in the design of block ciphers. For the past two years, in order to provide more S-boxes for block ciphers, several scholars did some research about involutions. Recently Zheng et al. studied involutions of the form x^rh(x^s) over F q providing a necessary and sufficient condition of this polynomials to be involutary and presenting a method to construct involutions with this form. However, the method needs to solve one equation system, i.e.,(3). In this paper, we firstly improve the method of Zheng et al., obtaining the explicit solutions of the equation system using the conjugacy relation over symmetric group and the idea of block matrices secondly we give the number of involutions with any fixed index and constant term 0. Thirdly according to the index, known involutions with explicit expression are classified. Finally, we determine several classes of involutions, enriching the known results. Specifically, aiming at the involutions of low indexes, more specific involutary conditions of index 2 and 3 are given than the results of Zheng et al. For involutions with high indexes, using the compositional results obtained by us before, we give a class of involutions of the form x^rh(x^q-1)over F q 2 .

作者：

屈龙江李康荃

Qu Longjiang;Li Kangquan(College of Liberal Arts and Sicences,National University of Defense Technology,Changsha 410073,China)

机构地区：

国防科技大学文理学院

出处：

《betway官方app 学报：自然科学版》 CAS 北大核心 2019年第4期1-15,F0002,共16页

基金：

国家自然科学基金(61722213 61572026)

关键词：

有限域指标对合分类

finite fields index involutions classification