Related papers: A Class of Permutation Trinomials over Finite Fiel…

Permutation polynomials are of particular significance in several areas of applied mathematics, such as Coding theory and Cryptography. Many recent constructions are based on the Akbary-Ghioca-Wang (AGW) criterion. Along this line of…

Let $F_q$ be the finite field with $q$ elements and $F_q[x_1,\ldots, x_n]$ the ring of polynomials in $n$ variables over $F_q$. In this paper we consider permutation polynomials and local permutation polynomials over $F_q[x_1,\ldots, x_n]$,…

In this paper we discuss the permutational property of polynomials of the form $f(L(x))+k(L(x))\cdot M(x)\in \mathbb F_{q^n}[x]$ over the finite field $\mathbb F_{q^n}$, where $L, M\in \mathbb F_q[x]$ are $q$-linearized polynomials. The…

We prove a conjecture by Tu, Zeng, Li, and Helleseth concerning trinomials $f_{\alpha,\beta}(x)= x + \alpha x^{q(q-1)+1} + \beta x^{2(q-1)+1} \in \mathbb{F}_{q^2}[x]$, $\alpha\beta \neq 0$, $q$ even, characterizing all the pairs…

Up to linear transformations, we give a classification of all permutation polynomials of degree $7$ over $\mathbb{F}_{q}$ for any odd prime power $q$, with the help of the SageMath software.

In this work, we completely characterize (i) permutation binomials of the form $x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb{F}_{2^n}[x], n = 2^st, a \in \mathbb{F}_{2^{2t}}^{*}$, and (ii) permutation trinomials of the form…

Let $q$ be a prime power and $n$ and $r$ be positive integers. It is well known that the linearized binomial $L_r(x)=x^{q^r}+ax\in\mathbb{F}_{q^n}[x]$ is a permutation polynomial if and only if $(-1)^{n/d}a^{{(q^n-1)}/{(q^{d}-1)}}\neq 1$…

We focus on the permutation polynomials of the form $L(X)+\Tr_{m}^{3m}(X)^{s}$ over $\F_{q^3}$, where $\F_q$ is the finite field with $q=p^m$ elements, $p$ is a prime number, $m$ is a positive integer, $\Tr_{m}^{3m}$ is the relative trace…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a prime power and $n$ be a positive integer. In this paper, we explore the factorization of $f(x^{n})$ over $\mathbb{F}_q$, where $f(x)$ is an irreducible polynomial…

In a recent paper Zhang et al. constructed 17 families of permutation pentanomials of the form $x^t+x^{r_1(q-1)+t}+x^{r_2(q-1)+t}+x^{r_3(q-1)+t}+x^{r_4(q-1)+t}$ over $\mathbb{F}_{q^2}$ where $q=2^m$. In this paper for 14 of these 17…

Given a polynomial \( H(x) \) over \(\mathbb{F}_{q^n}\), we study permutation polynomials of the form \( x + \gamma \mathrm{Tr}(H(x)) \) over \(\mathbb{F}_{q^n}\). Let \[P_H=\{\gamma\in \mathbb{F}_{q^n} : x+\gamma \mathrm{Tr}(H(x))~\text{is…

In this paper, we further investigate the local criterion and present a class of permutation polynomials and their compositional inverses over $ \mathbb{F}_{q^2}$. Additionally, we demonstrate that linearized polynomial over…

Let $k$ be a positive integer and $S_{2k}={\tt x}+{\tt x}^4+...+{\tt x}^{4^{2k-1}}\in\Bbb F_2[{\tt x}]$. It was recently conjectured that ${\tt x}+S_{2k}^{4^{2k}}+S_{2k}^{4^k+3}$ is a permutation polynomial of $\Bbb F_{4^{3k}}$. In this…

After a brief review of existing results on permutation binomials of finite fields, we introduce the notion of equivalence among permutation binomials (PBs) and describe how to bring a PB to its canonical form under equivalence. We then…

Motivated by recent results on the constructions of permutation polynomials with few terms over the finite field $\mathbb{F}_{2^n}$, where $n$ is a positive even integer, we focus on the construction of permutation trinomials over…

Let $r$ be a prime power and $q=r^m$. For $0\le i\le m-1$, let $f_i\in \mathbb{F}_r[x]$ be $q$-linearized and $a_i\in \mathbb{F}_q$. Assume that $z\in \mathbb{\bar{F}}_r$ satisfies the equation $\sum_{i=0}^{m-1}a_if_i(z)^{r^i}=0$, where…

Let $p$ be a prime, $q=p^n$, and $D \subset \mathbb{F}_q^*$. A celebrated result of McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^*$, and $f:\mathbb{F}_q \to \mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y) \in…

Let $\mu_{q+1}$ denote the set of $(q+1)$-th roots of unity in $\mathbb{F}_{q^2 }$. We construct permutation polynomials over $\mathbb{F}_{q^2}$ by using rational functions of any degree that induce bijections either on $\mu_{q+1}$ or…

Let $q$ be a prime power, $\mathbb F_q$ be the finite field of order $q$ and $\mathbb F_q(x)$ be the field of rational functions over $\mathbb F_q$. In this paper we classify all rational functions $\varphi\in \mathbb F_q(x)$ of degree 3…

Four recursive constructions of permutation polynomials over $\gf(q^2)$ with those over $\gf(q)$ are developed and applied to a few famous classes of permutation polynomials. They produce infinitely many new permutation polynomials over…