More classes of permutation pentanomials over finite fields with characteristic two

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-07-13 DOI:10.1016/j.ffa.2024.102468

Tongliang Zhang , Lijing Zheng , Hanbing Zhao

引用次数: 0

Abstract

Let $q = 2^{m}$ . In this paper, we investigate permutation pentanomials over $F_{q^{2}}$ of the form $f (x) = x^{t} + x^{r_{1} (q - 1) + t} + x^{r_{2} (q - 1) + t} + x^{r_{3} (q - 1) + t} + x^{r_{4} (q - 1) + t}$ with $\gcd (x^{r_{4}} + x^{r_{3}} + x^{r_{2}} + x^{r_{1}} + 1, x^{t} + x^{t - r_{1}} + x^{t - r_{2}} + x^{t - r_{3}} + x^{t - r_{4}}) = 1$ . We transform the problem concerning permutation property of $f (x)$ into demonstrating that the corresponding fractional polynomial permutes the unit circle U of $F_{q^{2}}$ with order $q + 1$ via a well-known lemma, and then into showing that there are no certain solution in $F_{q}$ for some high-degree equations over $F_{q}$ associated with the fractional polynomial. According to numerical data, we have found all such permutations with $4 \leq t < 100, 1 \leq r_{i} \leq t$ , $i \in [1, 4]$ . Several permutation polynomials are also investigated from the fractional polynomials permuting the unit circle U found in this paper.

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特性为 2 的有限域上的更多类置换五元数

设 q=2m。在本文中，我们研究 Fq2 上形式为 f(x)=xt+xr1(q-1)+t+xr2(q-1)+t+xr3(q-1)+t+xr4(q-1)+t 的置换五次方，gcd(xr4+xr3+xr2+xr1+1,xt+xt-r1+xt-r2+xt-r3+xt-r4)=1。我们将 f(x) 的置换性质问题转化为通过一个著名的 Lemma 证明相应的分数多项式以 q+1 的阶置换 Fq2 的单位圆 U，然后转化为证明与分数多项式相关的 Fq 上的一些高阶方程在 Fq 中没有一定的解。根据数值数据，我们找到了 4≤t<100,1≤ri≤t, i∈[1,4] 的所有此类置换。本文还从单位圆 U 的分数多项式中研究了几种置换多项式。

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.