Related papers: On Permutation Binomials over Finite Fields

Let $q$ be a prime power, $2\le r\le q$, and $f=a{\tt X}+{\tt X}^{r(q-1)+1}\in\Bbb F_{q^2}[{\tt X}]$, where $a\ne 0$. The conditions on $r,q,a$ that are necessary and sufficient for $f$ to be a permutation polynomial (PP) of ${\Bbb…

Let $q>2$ be a prime power and $f=-{\tt x}+t{\tt x}^q+{\tt x}^{2q-1}$, where $t\in\Bbb F_q^*$. We prove that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following occurs: (i) $q$ is even and…

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…

Some families of linear permutation polynomials of $\mathbb{F}_{q^{ms}}$ with coefficients in $\mathbb{F}_{q^{m}}$ are explicitly described (via conditions on their coefficients) as isomorphic images of classical subgroups of the general…

We classify complete permutation polynomials of type $aX^{\frac{q^n-1}{q-1}+1}$ over the finite field with $q^n$ elements, for $n+1$ a prime and $n^4 < q$. For the case $n+1$ a power of the characteristic we study some known families. We…

Necessary and sufficient conditions on $A,B\in \mathbb{F}_{q^3}^*$ for $f(X)=X^{q^2-q+1}+AX^{q^2}+BX$ being a permutation polynomial of $\mathbb{F}_{q^3}$ are investigated via a connection with algebraic varieties over finite fields.

For each prime p other than 3, and each power q=p^k, we present two large classes of permutation polynomials over F_{q^2} of the form X^r B(X^{q-1}) which have at most five terms, where B(X) is a polynomial with coefficients in {1,-1}. The…

Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form…

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…

This paper considers permutation polynomials over the finite field $F_{q^2}$ in even characteristic by utilizing low-degree permutation rational functions over $F_q$. As a result, we obtain two classes of permutation binomials and six…

Suppose x^m + c*x^n is a permutation polynomial over GF(p), where p>5 is prime, m>n>0, and c is in GF(p)^*. We prove that gcd(m-n,p-1) is not 2 or 4. In the special case that either (p-1)/2 or (p-1)/4 is prime, this was conjectured in a…

For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have constructed permutation polynomials which have order 2 as permutations, and have 3 terms, or 4 terms as polynomials. Explicit formulas for their coefficients are given in…

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation…

For the finite field $\mathbb{F}_{2^{3m}}$, permutation polynomials of the form $(x^{2^m}+x+\delta)^{s}+cx$ are studied. Necessary and sufficient conditions are given for the polynomials to be permutation polynomials. For this, the…

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…

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 $(-1)^{n/d}a^{{(q^n-1)}/{(q^{d}-1)}}\neq 1$…

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are…

Permutation polynomials (PPs) of the form $(x^{q} -x + c)^{\frac{q^2 -1}{3}+1} +x$ over $\mathbb{F}_{q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form…

Let $p>3$ and consider a prime power $q=p^h$. We completely characterize permutation polynomials of $\mathbb{F}_{q^2}$ of the type $f_{a,b}(X) = X(1 + aX^{q(q-1)} + bX^{2(q-1)}) \in \mathbb{F}_{q^2}[X]$. In particular, using connections…

Let F_{q^n} be the field of order q^n, and let Tr be the trace map from F_{q^n} to its q-element subfield. We exhibit nine sequences of polynomials of the form f(x):=x+c*Tr(x^k), with c in F_{q^n}, such that for each polynomial the function…