Gutierrez and Urroz (2023) have proposed a family of local permutation polynomials over finite fields of arbitrary characteristic based on a class of symmetric subgroups without fixed points called $e$-Klenian groups. The polynomials within this family are referred to as $e$-Klenian polynomials. Furthermore, they have shown the existence of companions for the $e$-Klenian polynomials when the characteristic of the finite field is odd. Here, we present three new families of local permutation polynomials over finite fields of even characteristic. We also consider the problem of the existence of companions for the $e$-Klenian polynomials over finite fields of even characteristic. More precisely, we prove that over finite fields of even characteristic, the $0$-Klenian polynomials do not have any companions. However, for $e \geq 1$, we explicitly provide a companion for the $e$-Klenian polynomials. Moreover, we provide a companion for each of the new families of local permutation polynomials that we introduce.
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.
In this paper, we prove that every PP is an AGW-PP. We also extend the result of Wan and Lidl to other permutation polynomials over finite fields and determine their group structure. Moreover, we provide a new general method to find the compositional inverses of all PPs, some new PPs and their compositional inverses are given.
Let $μ_{q+1}$ 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 $μ_{q+1}$ or between $μ_{q+1}$ and $\mathbb{F}_q \cup \{\infty\}$. In particular, we generalize results from Zieve.
Permutation polynomials with coefficients 1 over finite fields attract researchers' interests due to their simple algebraic form. In this paper, we first construct four classes of fractional permutation polynomials over the cyclic subgroup of $ \mathbb{F}_{2^{2m}} $. From these permutation polynomials, three new classes of permutation polynomials with coefficients 1 over $ \mathbb{F}_{2^{2m}} $ are constructed, and three more general new classes of permutation polynomials with coefficients 1 over $ \mathbb{F}_{2^{2m}} $ are constructed using a new method we presented recently. Some known permu…
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 F_{q^n}-->F_{q^n} given by c-->f(c) is a permutation of F_{q^n}. We also computed all permutation polynomials of this form over finite fields of size less than 5000, and found that our examples comprise all examples with n>1 except for some simple cases where the polynomial induces a homomorphism of the additive group of F_{q^n}, along with a few sporadic e…
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 characteriz…
In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation polynomials that are involutions. Our results provide a fast algorithm (only modular operations are involved) to generate many classes of generalized cyclotomic permutation polynomials, their inverses, and involutions.
Given a permutation polynomial of a large finite field, finding its inverse is usually a hard problem. Based on a piecewise interpolation formula, we construct the inverses of cyclotomic mapping permutation polynomials of arbitrary finite fields.
We show that all permutations in $S_n$ can be generated by affine unicritical polynomials. We use the $\operatorname{PGL}$ group structure to compute the cycle structure of permutations with low Carlitz rank. The tree structure of the group generated by shift and inversion maps is used to study the randomness properties of permutation polynomials.
We investigate the permutation property of polynomials of the form $x^{r}(x^{s} -a)^{t}$, and give the expressions of their inverses. In particular, explicit expressions of inverses of permutation polynomials $x(x^3 -a)^2$ and $x(x^2 -a)^3$ on $\mathbb{F}_{7^n}$ are presented. Then, using some known results, we obtain the inverses of all permutation polynomials of degree $6, 7, 8$ over finite fields.
In this paper, a construction of complete permutation polynomials over finite fields of even characteristic proposed by Tu et al. recently is generalized in a recursive manner. Besides, several classes of complete permutation polynomials are derived by computing compositional inverses of known ones.
Motivated by many recent constructions of permutation polynomials over $\mathbb{F}_{q^2}$, we study permutation polynomials over $\mathbb{F}_{q^3}$ in terms of their coefficients. Based on the multivariate method and resultant elimination, we construct several new classes of sparse permutation polynomials over $\mathbb{F}_{q^3}$, $q=p^{k}$, $p\geq3$. Some of them are complete mappings.
Let $q=4$ and $k$ a positive integer. In this short note, we present a class of permutation polynomials over $\Bbb F_{q^{3k}}$. We also present a generalization.
By using the piecewise method, Lagrange interpolation formula and Lucas' theorem, we determine explicit expressions of the inverses of a class of reversed Dickson permutation polynomials and some classes of generalized cyclotomic mapping permutation polynomials over finite fields of characteristic three.
We study degree preserving maps over the set of irreducible polynomials over a finite field. In particular, we show that every permutation of the set of irreducible polynomials of degree $k$ over $\mathbb{F}_q$ is induced by an action from a permutation polynomial of $\mathbb{F}_{q^k}$ with coefficients in $\mathbb{F}_q$. The dynamics of these permutations of irreducible polynomials of degree $k$ over $\mathbb{F}_q$, such as fixed points and cycle lengths, are studied. As an application, we also generate irreducible polynomials of the same degree by an iterative method.
Let $\mathbb{F}_q$ denote the finite field with $q$ elements. The Carlitz rank of a permutation polynomial is a important measure of complexity of the polynomial. In this paper we find the sharp lower bound for the weight of any permutation polynomial with Carlitz rank $2$, improving the bound found by Gómez-Pérez, Ostafe and Topuzoğlu in that case.
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.
Let $p$ be a prime. In this paper, we give a complete classification of self-reciprocal polynomials arising from Fibonacci polynomials over $\mathbb{Z}$ and $\mathbb{Z}_p$, where $p=2$ and $p>5$. We also present some partial results when $p=3, 5$. We also compute the first and second moments of Fibonacci polynomials $f_{n}(x)$ over finite fields, which give necessary conditions for Fibonacci polynomials to be permutation polynomials over finite fields.
Let $G_n$ be the group of permutations on $\mathbb{Z}_n$ that is induced by a Dickson polynomial, where $n$ is a positive integer. In this work, by solving special types of systems of linear congruences, we obtain $G_n$. In addition, we give an a algorithm that gets $|G_n|$.
Permutation polynomials of finite fields have many applications in Coding Theory, Cryptography and Combinatorics. In the first part of this paper we present a new family of local permutation polynomials based on a class of symmetric subgroups without fixed points, the so called e-Klenian groups. In the second part we use the fact that bivariate local permutation polynomials define Latin Squares, to discuss several constructions of Mutually Orthogonal Latin Squares (MOLS) and, in particular, we provide a new family of MOLS on size a prime power.
For the finite field $\mathbb{F}_{2^{3m}}$, permutation polynomials of the form $(x^{2^m}+x+δ)^{s}+cx$ are studied. Necessary and sufficient conditions are given for the polynomials to be permutation polynomials. For this, the structures and properties of the field elements are analyzed.
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 restriction $L, M\in \mathbb F_q[x]$ implies a nice correspondence between the pair $(L, M)$ and the pair $(g, h)$ of conventional $q$-associates over $\mathbb F_q$ of degree at most $n-1$. In particular, by using the AGW criterion, permutational properties of our class of polynomials translates to some arithmetic properties of polynomials over $\mathbb F_q$, l…
Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al. (Sci. China Math., to appear. Doi: 10.1007/s11425-014-4964-2). Furthermore, we give two classes of trinomial permutation polynomials, and make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by B…
