Permutation polynomials over finite fields are an interesting subject due to their important applications in the areas of mathematics and engineering. In this paper we investigate the trinomial $f(x)=x^{(p-1)q+1}+x^{pq}-x^{q+(p-1)}$ over the finite field $\mathbb{F}_{q^2}$, where $p$ is an odd prime and $q=p^k$ with $k$ being a positive integer. It is shown that when $p=3$ or $5$, $f(x)$ is a permutation trinomial of $\mathbb{F}_{q^2}$ if and only if $k$ is even. This property is also true for more general class of polynomials $g(x)=x^{(q+1)l+(p-1)q+1}+x^{(q+1)l+pq}-x^{(q+1)l+q+(p-1)}$, where $l$ is a nonnegative integer and $\gcd(2l+p,q-1)=1$. Moreover, we also show that for $p=5$ the permutation trinomials $f(x)$ proposed here are new in the sense that they are not multiplicative equivalent to previously known ones of similar form.
Permutation polynomials over finite fields are an interesting and constantly active research subject of study for many years. They have important applications in areas of mathematics and engineering. In recent years, permutation binomials and permutation trinomials attract people's interests due to their simple algebraic forms. In this paper, by reversely using Tu's method for the characterization of permutation polynomials with exponents of Niho type, we propose a new method to construct permutation trinomials with coefficients 1. Moreover, we give the explicit compositional inverses of a cla…
In this paper, by analyzing the quadratic factors of an $11$-th degree polynomial over the finite field $\ftwon$, a conjecture on permutation trinomials over $\ftwon[x]$ proposed very recently by Deng and Zheng is settled, where $n=2m$ and $m$ is a positive integer with $\gcd(m,5)=1$.
Let $q$ be a power of a prime and $\mathbb{F}_q$ be a finite field with $q$ elements. In this paper, we propose four families of infinite classes of permutation trinomials having the form $cx-x^s + x^{qs}$ over $\mathbb{F}_{q^2}$, and investigate the relationship between this type of permutation polynomials with that of the form $(x^q-x+δ)^s+cx$. Based on this relation, many classes of permutation trinomials having the form $(x^q-x+δ)^s+cx$ without restriction on $δ$ over $\mathbb{F}_{q^2}$ are derived from known permutation trinomials having the form $cx-x^s + x^{qs}$.
Some new results on permutation polynomials over finite fields
有限域上的置换多项式是一个活跃的研究领域，在科学和工程的许多领域都有应用。本文给出了四类单名完全置换多项式和一类三名完全置换多项式，其中之一证实了Wu等人（Sci.中国数学，出现。Doi：10.1007/s11425-014-4964-2）。进一步，我们给出了两类三项式置换多项式，并对Blondeau等人提出的关于幂置换多项式微分一致性的猜想（Int. J. Inf. CodingTheory，2010，1，pp. 149-170）。
Recently, there has been a lot of work on constructions of permutation polynomials of the form $(x^{2^m}+x+δ)^{s}+x$ over the finite field $\F_{2^{2m}}$, especially in the case when $s$ is of the form $s=i(2^m-1)+1$ (Niho exponent). In this paper, we further investigate permutation polynomials with this form. Instead of seeking for sporadic constructions of the parameter $i$, we give a general sufficient condition on $i$ such that $(x^{2^m}+x+δ)^{i(2^m-1)+1}+x$ permutes $\F_{2^{2m}}$, that is, $(2^k+1)i \equiv 1 ~\textrm{or}~ 2^k~(\textrm{mod}~ 2^m+1)$, where $1 \leq k \leq m-1$ is any integer…
We give historical remarks related to arXiv:2112.14547 ("A New Method of Construction of Permutation Trinomials with Coefficients 1", by Guo et al.). In particular, we show that the "new" permutation polynomials in that paper are actually well known. In addition we give a simpler derivation of these permutation polynomials than had been given previously, which demonstrates the general method of producing permutation polynomials that was introduced in arXiv:1310.0776.
Shift-Invariant Permutations with Polynomial Composition: Construction and Algebraic Structure
Hubei University（湖北大学）；Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University（湖北大学数学与统计学院应用数学重点实验室）；Key Laboratory of Intelligent Sensing System and Security (Hubei University), Ministry of Education（教育部智能感知系统与安全防护重点实验室（湖北大学））；Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）
