数学理论与应用 ›› 2023, Vol. 43 ›› Issue (2): 32-47.doi: 10.3969/j.issn.1006­8074.2023.02.003

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几类新的有限域上的 n-­cycle 置换

张志林,袁平之*
  

  1. 华南师范大学数学科学学院, 广州, 510640
  • 出版日期:2023-06-28 发布日期:2023-06-27

Some New Classes of n­cycle Permutations over Finite Fields

Zhang Zhilin,Yuan Pingzhi*   

  1. School of Mathematical Science, South China Normal University, Guangzhou 510640, China
  • Online:2023-06-28 Published:2023-06-27
  • Contact: Yuan Pingzhi (1966–), Associate Professor, PhD; E-mail: yuanpz@scnu.edu.cn E-mail:yuanpz@scnu.edu.cn
  • Supported by:

    This work is supported by National Natural Science Foundation of China (Nos. 12001204, 12171163).

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摘要：

本文给出几类新的有限域上的 $n$-cycle 置换. 首先, 我们给出一个判断 Dickson 多项式是否为 $n$-cycle 置换的方法. 其次, 我们给出一个线性多项式是否为对合的充要条件. 最后, 我们给出几类新的不同型的 $n$-cycle 置换.

关键词: $n$-cycle 置换, 置换多项式, 线性译码 , Dickson 多项式, 线性多项式, 有限域,  , 分段函数

Abstract:

This paper presents some new classes of $n$-cycle permutations over finite fields. At first, we present a concise criterion for Dickson polynomials over finite fields being $n$-cycle permutations.

Then, we give a necessary and sufficient condition for linearized polynomials over finite fields being involutions.

Finally, some interesting new classes of $n$-cycle permutations are demonstrated by considering polynomials of different forms.

Key words: $n$-cycle permutation, Permutation polynomial, Linear translator, Dickson polynomial, Linearized polynomial, Finite field, Piecewise function

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