Local permutation polynomials and their companions

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2025-08-12 DOI:10.1016/j.ffa.2025.102717

Sartaj Ul Hasan, Hridesh Kumar

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引用次数: 0

Abstract

Gutierrez and Urroz (2023) have proposed a family of local permutation polynomials over finite fields of arbitrary characteristic based on a class of symmetric subgroups without fixed points called e-Klenian groups. The polynomials within this family are referred to as e-Klenian polynomials. Furthermore, they have shown the existence of companions for the e-Klenian polynomials when the characteristic of the finite field is odd. Here, we construct three new families of local permutation polynomials over finite fields of even characteristic, and derive a necessary and sufficient condition for each of these families to achieve the maximum possible degree. We also consider the problem of the existence of companions for the e-Klenian polynomials over finite fields of even characteristic. More precisely, we prove that over finite fields of even characteristic, the 0-Klenian polynomials do not have any companions. However, for

e \geq 1

, we explicitly provide a companion for the e-Klenian polynomials. Moreover, we provide a companion for each of the new families of local permutation polynomials that we introduce.

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局部置换多项式及其伴随多项式

Gutierrez和Urroz（2023）基于一类没有不动点的对称子群e-Klenian群，提出了任意特征有限域上的一组局部置换多项式。这个族中的多项式被称为e-Klenian多项式。进一步证明了有限域特征为奇时e-Klenian多项式伴子的存在性。本文在偶特征有限域上构造了三个新的局部置换多项式族，并给出了每个族达到最大可能度的充分必要条件。我们还考虑了偶特征有限域上e-Klenian多项式伴子的存在性问题。更准确地说，我们证明了在偶特征的有限域上，0-Klenian多项式没有任何同伴。然而，当e≥1时，我们明确地提供了e- klenian多项式的伴侣。此外，我们为我们引入的每一个新的局部置换多项式族提供了一个伴侣。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.