A conjecture on permutation trinomials over finite fields of characteristic two

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 $\gcd(m,5)=1$

Permutation polynomials, fractional polynomials, and algebraic curves

In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over $\mathbb{F}_3^{2k}$. In addition, new examples and generalizations of some families of permutation polynomials of $\mathbb{F}_{3^k}$ and $\mathbb{F}_{5^k}$ are given. We also study permutation quadrinomials of type $Ax^{q(q-1)+1} + Bx^{2(q-1)+1} + Cx^{q} + x$. Our method is based on the investigation of an algebraic curve associated with a {fractional polynomial} over a finite field

New Permutation Trinomials From Niho Exponents over Finite Fields with Even Characteristic

In this paper, a class of permutation trinomials of Niho type over finite fields with even characteristic is further investigated. New permutation trinomials from Niho exponents are obtained from linear fractional polynomials over finite fields, and it is shown that the presented results are the generalizations of some earlier works

On a Class of Permutation Trinomials in Characteristic 2

Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form $f(X)=X+aX^{q(q-1)+1}+bX^{2(q-1)+1}\in\Bbb F_{q^2}[X]$, where $q$ is even and $a,b\in\Bbb F_{q^2}^*$. They found sufficient conditions on $a,b$ for $f$ to be a permutation polynomial (PP) of $\Bbb F_{q^2}$ and they conjectured that the sufficient conditions are also necessary. The conjecture has been confirmed by Bartoli using the Hasse-Weil bound. In this paper, we give an alternative solution to the question. We also use the Hasse-Weil bound, but in a different way. Moreover, the necessity and sufficiency of the conditions are proved by the same approach

A new class of permutation trinomials constructed from Niho exponents

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.Comment: 17 pages, three table

Constructions of involutions over finite fields

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications such as cryptography and coding theory. As far as we know, there are not many involutions, and there isn't a general way to construct involutions over finite fields. This paper gives a necessary and sufficient condition for the polynomials of the form x^rh(x^s)\in \bF_q[x] to be involutions over the finite field~\bF_q, where $r\geq 1$ and $s\,|\, (q-1)$. By using this criterion we propose a general method to construct involutions of the form $x^rh(x^s)$ over \bF_q from given involutions over the corresponding subgroup of \bF_q^*. Then, many classes of explicit involutions of the form $x^rh(x^s)$ over \bF_q are obtained

A family of permutation trinomials in Fq2\mathbb{F}_{q^2}

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 with algebraic curves over finite fields, we show that the already known sufficient conditions are also necessary

A Classification of Permutation Polynomials through Some Linear Maps

In this paper, we propose linear maps over the space of all polynomials $f(x)$ in $\mathbb{F}_q[x]$ that map $0$ to itself, through their evaluation map. Properties of these linear maps throw up interesting connections with permutation polynomials. We study certain properties of these linear maps. We propose to classify permutation polynomials by identifying the generalized eigenspaces of these maps, where the permutation polynomials reside. As it turns out, several classes of permutation polynomials studied in literature neatly fall into classes defined using these linear maps. We characterize the shapes of permutation polynomials that appear in the various generalized eigenspaces of these linear maps. For the case of $\mathbb{F}_p$, these generalized eigenspaces provide a degree-wise distribution of polynomials (and therefore permutation polynomials) over $\mathbb{F}_p$. We show that for $\mathbb{F}_q$, it is sufficient to consider only a few of these linear maps. The intersection of the generalized eigenspaces of these linear maps contain (permutation) polynomials of certain shapes. In this context, we study a class of permutation polynomials over $\mathbb{F}_{p^2}$. We show that the permutation polynomials in this class are closed under compositional inverses. We also do some enumeration of permutation polynomials of certain shapes.Comment: 18 page

New Constructions of Permutation Polynomials of the Form xrh(xq−1)x^rh\left(x^{q-1}\right) over Fq2\mathbb{F}_{q^2}

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 trinomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$, where $q=2^k$, $h(x)=1+x^s+x^t$ and $r, s, t, k>0$ are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over $\mathbb{F}_{q^2}$ into that of showing the corresponding fractional polynomials permute a smaller set $\mu_{q+1}$, where $\mu_{q+1}:=\{x\in\mathbb{F}_{q^2} : x^{q+1}=1\}$. Motivated by these results, we characterize the permutation polynomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$ such that $h(x)\in\mathbb{F}_q[x]$ is arbitrary and $q$ is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over $\mathbb{F}_{q^2}$ into that of showing permutations over a small subset $S$ of a proper subfield $\mathbb{F}_{q}$, which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$. Moreover, we can explain most of the known permutation trinomials, which are in [6, 13, 14, 16, 20, 29], over finite field with even characteristic.Comment: 29 pages. An early version of this paper was presented at Fq13 in Naples, Ital

Finding compositional inverses of permutations from the AGW criterion

Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of different forms, we propose a general method for finding these inverses of permutation polynomials constructed by the AGW criterion. As a result, we have reduced the problem of finding the compositional inverse of such a permutation polynomial over a finite field to that of finding the inverse of a bijection over a smaller set. We demonstrate our method by interpreting several recent known results, as well as by providing new explicit results on more classes of permutation polynomials in different types. In addition, we give new criteria for these permutation polynomials being involutions. Explicit constructions are also provided for all involutory criteria.Comment: 24 pages. Revision submitte