• 论文 •

### 几类特殊形式的置换多项式

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

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

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 $\mathbb{F}_{2^n}$.

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