• 论文 • 上一篇    下一篇

几类特殊形式的置换多项式

朱喜顺1,陈媛2,曾祥勇2   

  1. 1.南昌大学共青学院,共青城 332300;2.湖北大学数学与统计学学院应用数学湖北省重点实验室,武汉 430062;中国科学院信息工程研究所信息安全国家重点实验室,北京 100093
  • 出版日期:2016-08-25 发布日期:2016-09-26

朱喜顺,陈媛,曾祥勇. 几类特殊形式的置换多项式[J]. 系统科学与数学, 2016, 36(8): 1349-1357.

ZHU Xishun,CHEN Yuan,ZENG Xiangyong. SEVERAL SPECIAL TYPES OF PERMUTATION POLYNOMIALS[J]. Journal of Systems Science and Mathematical Sciences, 2016, 36(8): 1349-1357.

SEVERAL SPECIAL TYPES OF PERMUTATION POLYNOMIALS

ZHU Xishun1 ,CHEN Yuan2 ,ZENG Xiangyong2   

  1. 1.Nanchang University Gongqing College, Gongqingcheng 332300;2.Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062; State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093
  • Online:2016-08-25 Published:2016-09-26

有限域上的置换多项式在密码学, 编码理论和序列设计等领域有着广泛应用, 但目前已知的置换多项式的构造还 很有限. 文章分别给出有限域\,$\mathbb{F}_{2^n}$\,上两类形如\,$(x^{2^{i}}+\eta x+\delta)^{s}+x$\,和两类形如\,$x^{r}+\delta x^s+\delta^{t}x$\,的置换多项式.

Permutation polynomials over finite fields have been an important subject of study for a long time and have wide applications in coding theory, cryptography and sequence designs. However, only some specific classes of permutation polynomials have been described in the literature so far. In this paper, based on the knowledge of finite fields, such as the properties of the trace function, we propose two classes of permutation polynomials having the form $(x^{2^{i}}+\eta x+\delta)^{s}+x$ and two classes of permutation trinomials having the form $x^{r}+\delta x^s+\delta^{t}x$ over the finite field $\mathbb{F}_{2^n}$. The permutation polynomials of the first form are studied along with the work of Yuan and Ding, and the method studying the permutation polynomials of the second form relies a sufficient condition for a trinomial having no root, which is established in this paper. There are rare known classes of permutation trinomials over $\mathbb{F}_{2^n}$ in the literature, and most of them are of the simple form, i.e., all nonzero coefficients equal to the identity. The permutation trinomials presented in this paper have the coefficients which can take any nonzero elements in $\mathbb{F}_{2^n}$.

MR(2010)主题分类: 

()
[1] 古丽斯坦·库尔班尼牙孜, 孟丽君, 田茂再. 配对设计中条件优势比的置信区间构造[J]. 系统科学与数学, 2021, 41(3): 824-836.
[2] 胡建,曹喜望. 自共轭互反多项式的推广[J]. 系统科学与数学, 2020, 40(8): 1507-1516.
[3] 田诗竹,陈媛. 一类幂函数在$\mathbb{F}_{p^n}$上的差分谱[J]. 系统科学与数学, 2017, 37(5): 1351-1367.
[4] 蒋剑军;孙琦. $Z/p^k Z$上的多元置换多项式[J]. 系统科学与数学, 2005, 25(3): 299-305.
[5] 高有;游宏. 特征不为2的有限域上酉群的极小生成元集[J]. 系统科学与数学, 1999, 19(1): 46-050.
阅读次数
全文


摘要