摘要
置换多项式在代数学、组合学、数论、编码理论、密码学等领域中均有广泛而又重要的应用.近年来,置换多项式的研究取得一系列进展,研究者先后提出Akbary-Ghioca-Wang法则、分段构造法、交换构造法等方法来构造和证明置换.有限域上的置换多项式因其简单的代数形式和优良的密码性质,在密码算法设计中备受关注.在特征为2的偶次扩域上寻求同时具有低差分均匀度、高非线性度、高代数次数等密码性质的代数形式简单的置换成为学者们研究的热点.本文介绍了置换多项式的相关应用和研究背景,给出了一些基本概念和预备知识.研究了有限域F_p^n上形如x^((p^n-1)/d+1)+ax的二项式的置换属性,给出了该二项式是置换的充要条件.在此基础上,构造出四类二项式置换,并利用Dickson多项式和线性多项式理论予以证明.列举了上述二项式置换在特征为2和3的有限域上的相关实例,并由此得到一些完全置换单项式.根据迹函数的性质,通过引入新的参数构造出两类具有特定指数的多项式置换,上述多项式置换推广了一个已知的结果.
Permutation polynomials have extensive and important applications in various areas such as algebra, combinatorics, coding theory, cryptography and so on. Recently, the research of permutation polynomials makes a series of advances. A number of methods have been proposed to construct permutations, including the Akbary-Ghioca-Wang criterion, the piecewise construction method, the switching method and so on. Permutation polynomials over finite fields are particularly interesting in the design of cryptographic algorithms for their simple algebraic forms and good cryptographic properties. It is a hot topic to find permutations with simple algebraic form over a finite field with characteristic two and even number of extension degree, which has low differential uniformity, high nonlinearity, high algebraic degree and other cryptographic properties. In this paper, we firstly introduce the related applications and research background of permutation polynomials. Some basic definitions and preliminaries are presented. Then we study the permutation properties of binomials with the form of x^(p^n-1)/d+1)+ ax over finite fields F_p^n and present a necessary and sufficient condition for these binomials to be permutations. Based on this result, we construct four classes of binomial permutations by applying the Dickson polynomials and the theory of linearized polynomials. We list some related examples of our new binomial permutations over characteristic two or three finite fields.Moreover, we obtain some complete permutation monomials from these binomial permutations. According to the properties of trace functions, we construct two classes of polynomial permutations by introducing some new parameters. Our new polynomial permutations generalize one known result.
出处
《密码学报》
CSCD
2017年第3期291-298,共8页
Journal of Cryptologic Research
基金
国家重点基础研究发展项目(973计划)(2013CB834203)
国家自然科学基金项目(11571005
61472417)
中意智慧城市合作研究室项目(2016YFE0104600)
河南省高校科技创新人才支持计划(16HASTIT039)
关键词
有限域
置换多项式
迹函数
finite field
permutation polynomial
trace function
