Some families of permutation polynomials over finite fields
Michael E. Zieve
本文给出了多项式x^r*（1+x^v+x^（2 v）+.+）的充要条件。x^（q））^t来置换有限域GF（q）的元素。当（q-1）/gcd（q-1，v）是小素数时，我们的结果给出了特别简单的准则.
We give necessary and sufficient conditions for a polynomial of the form x^r*(1+x^v+x^(2v)+...+x^(kv))^t to permute the elements of the finite field GF(q). Our results yield especially simple criteria in case (q-1)/gcd(q-1,v) is a small prime.
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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…
Complete permutation polynomials induced from complete permutations of subfields
本文提出了几种利用子域的完全置换构造有限域的完全置换多项式的方法。在某些特殊情况下，可以利用有限域上的任意完全置换多项式构造某些扩域的完全置换。所得结果推广了几位作者最近的工作。
Cyclotomy and permutation polynomials of large indices
我们利用割圆法设计了有限域上新的置换多项式类。这使我们能够以算法的方式生成许多类的置换多项式。它们中的许多是大指数的置换多项式。
A general approach to permutation polynomials from quadratic forms
本文研究特征2有限域上的置换多项式，通过建立其与二次型的联系，提出一般刻画方法，并利用特征和的确定明确描述了几类置换多项式。
Specific permutation polynomials over finite fields
本文提出了有限域上新的置换多项式类。
Enumerating Permutation Polynomials over finite fields by degree
证明了一个置换多项式的次数小于$q-2$的置换个数的渐近公式。
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.
Permutation polynomials over finite fields by the local criterion
本文进一步研究了局部判据，提出了一类在 $\mathbb{F}_{q^2}$ 上的置换多项式及其复合逆，并证明了线性化多项式与所有置换多项式在特定条件下均为局部置换多项式。
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.
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.
Some permutation polynomials via linear translators
本文受先前工作启发，扩展线性平移子概念，并据此在有限域上构造新的置换多项式类。
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.
Further Investigation on Cyclotomic Mapping Permutation Polynomials over Finite Fields
本文研究了有限域上分圆映射置换多项式与特定形式置换多项式的关系，提出新的判定条件，并据此构造了若干置换三项式和二项式。
On a Class of Permutation Polynomials and Their Inverses
本文提出一类有限域上的置换多项式，给出其显式形式及逆，并推广至更一般形式。
On Permutation Trinomials and Complete Permutation Polynomials via Fiber Criteria over Finite Fields
本文通过Zieve纤维准则和AGW准则结合，提出了一种构造有限域上完全排列多项式的方法，并通过实例验证了其有效性。
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.
New results on permutation polynomials over finite fields
本文得到了有限域上置换多项式的几个新结果。首先，通过使用线性翻译器，我们构造了形式为$L(x)+\sum_{j=1}^k γ_jh_j(f_j(x))$和$x+\sum_{j=1}^kγ_jf_j(x)$的置换多项式。这些结果推广了Kyureghyan在2011年得到的结果。因此，我们刻画了$L(x)+\sum_{i=1} ^lγ_i {\rm Tr}_{{\bf F}_{q^m}/{\bf F}_{q}}(h_i(x))$形式的置换多项式，这推广了Charpin和Kyureghyan在2009年得到的定理。
Necessary conditions for reversed Dickson polynomials of the second kind to be permutational
本文给出了第二类逆Dickson多项式$E_{n}(1, x)$是$\mathbb{F}_{q}$的置换的几个必要条件.特别地，我们给出明确的评估的总和$\sum_{a\in \mathbb{F}_{q}}E_{n}(1,a)$。
The compositional inverses of three classes of permutation polynomials over finite fields
本文利用局部方法证明了已有文献提出的有限域$\mathbb{F}_{q^3}$上三类置换三项式确为置换多项式，并给出了它们的复合逆。
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 $|G_n|$.
In this paper, we provide a local method to find compositional inverses of all PPs, some new PPs and their compositional inverses are given.
A note on two classes of permutation polynomials over finite fields
有限域上的置换多项式在数学和工程中有着广泛的应用。近年来，屠维岳等给出了偶特征有限域上的几类完全置换多项式.本文在Akbary等人的一个工作的基础上给出了一些性质，利用这些性质，我们给出了Tu等人的推广.的结果任意的特征。
Constructing permutation polynomials over finite fields
本文构造了有限域上几个新的置换多项式。首先，利用线性化多项式，我们构造${\bf F}_{q^{m}}$上形式为$\sum_{i=1}^k(L_{i}(x)+γ_i)h_i(B(x))$的置换多项式，其中$L_i(x)$和$B(x)$是线性化多项式。这推广了Coulter，Henderson和Matthews的一个定理.因此，我们通过构造$x h(λ_{j}(x))$和$xh(μ_{j}(x))$形式的置换多项式推广了Marcos的一个结果，其中$λ_{j}(x)$是$x, x^{q}, ..., x^{q^{m-1}}$和$μ_{j}(x)=\textup{Tr}_{{\bf F}_{q^{m}}/{\bf F}_{q}}(x^{j})$的第$j$阶初等对称多项式.这回答了Zieve在2010年提出的一个开放问题。最后，利用线性变换器，构造了${\bf F}_{q^{m}}$上$L_1(x)+L_{2}(γ)h(f(x))$形式的置换多项式，推广了Kyureghyan的一个结果.
Proof of a Conjecture on Permutation Polynomials over Finite Fields
令$k$为正整数，且$S_{2k}={\tt x}+{\tt x}^4+...+{\tt x}^{4^{2k-1}}\in\Bbb F_2[{\tt x}]$。最近证明了${\tt x}+S_{2k}^{4^{2k}}+S_{2k}^{4^k+3}$是$\Bbb F_{4^{3k}}$的置换多项式。本文证实了这一猜想，并得到了一个推广.
Constructing permutation polynomials using generalized Redei functions
本文确定了有限域上两类由广义Redei函数分母构造的多项式中的所有置换，推广了八个近期结果，并给出更简洁的证明。
Sparse Lifts of q-Linearized Permutation Polynomials with Explicit Inverses
该研究构造了保持复合逆的q线性化置换多项式的稀疏提升，得到二维二项式、三维三项式等的显式逆，为有限域上置换多项式的构造提供了新方法。
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…
A Construction of Permutation Polynomials Using Rédei Function in Even Characteristic
本文受奇特征工作启发，利用AGW准则，在特征2域上通过Rédei函数递归构造置换多项式。
Permutation polynomials of finite fields of even characteristic from character sums
本文利用特征和计算，刻画并构造了偶特征有限域上形如$\mathrm{Tr}(Ax^{q+1})+L(x)$的置换多项式。
Let $\Bbb F_q$ be the finite field with $q$ elements and let $p=\text{char}\,\Bbb F_q$. It was conjectured that for integers $e\ge 2$ and $1\le a\le pe-2$, the polynomial $X^{q-2}+X^{q^2-2}+\cdots+X^{q^a-2}$ is a permutation polynomial of $\Bbb F_{q^e}$ if and only if (i) $a=2$ and $q=2$, or (ii) $a=1$ and $\text{gcd}(q-2,q^e-1)=1$. In the present paper we confirm this conjecture.
Inverses of six classes of permutation polynomials of the form $x+γ\operatorname{Tr}_q^{q^2}(h(x))$ over finite fields of even characteristic
本文研究了形如$x+γ\operatorname{Tr}_q^{q^2}(h(x))$的六类置换多项式在有限域上的逆元问题，基于前人研究，进一步探讨了其逆元结构和性质。
Algebraic Structure of Permutational Polynomials over $\mathbb{F}_{q^n}$
该研究提出有限域 $\mathbb{F}_{q^n}$ 上置换多项式的新代数结构，并据此构造多类新置换多项式，解决了 Charpin 与 Kyureghyan 的公开问题。
The compositional inverses of three classes of permutation polynomials over finite fields
基于P. Yuan的局部方法，本文计算了有限域上三类置换多项式的复合逆，包括特定形式的$ax^q+bx+(x^q-x)^k$、$f(x)$及$A^{m}(x)+L(x)$，并明确了各项参数与映射空间的约束条件。
Let $r$ be a positive integer, $h(X)\in\Bbb F_{q^2}[X]$, and $μ_{q+1}$ be the subgroup of order $q+1$ of $\Bbb F_{q^2}^*$. It is well known that $X^rh(X^{q-1})$ permutes $\Bbb F_{q^2}$ if and only if $\text{gcd}(r,q-1)=1$ and $X^rh(X)^{q-1}$ permutes $μ_{q+1}$. There are many ad hoc constructions of permutation polynomials of $\Bbb F_{q^2}$ of this type such that $h(X)^{q-1}$ induces monomial functions on the cosets of a subgroup of $μ_{q+1}$. We give a general construction that can generate, through an algorithm, {\em all} permutation polynomials of $\Bbb F_{q^2}$ with this property, includin…
In this paper, we use the method developed previously by Hong, Qin and Zhao to obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,k}(1,x)$ of the $(k+1)$-th kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n,k}(1,a)$. Our results extend the known results from the case $0\le k\le 3$ to the general $k\ge 0$ case.
Permutation Polynomials of Degree 6 or 7 over Finite Fields of Characteristic 2
在\cite{D1}中，Dickson列出了任意有限域上5次以下的所有置换多项式，以及奇特征的有限域上6次的所有置换多项式。特征为2的有限域上的6次置换多项式的分类是不完整的。本文完成了特征为2的有限域上6次置换多项式的分类。此外，还对特征为2的有限域上的7次置换多项式进行了分类。
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.
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.
Complete permutation polynomials over finite fields of odd characteristic
本文给出了奇特征有限域上三类完全置换单项式。同时，也给出了这些完全置换多项式的合成逆。
The Slope Polynomial and Collinear Points in Permutations
由于致命错误而退出。
Permutation Polynomials modulo m
本文主要研究了所谓的“置换多项式模$m$“问题，即能诱导环Z_m={0，. m-1}。给出了置换多项式的充要条件，并估计了所有给定次数置换多项式的个数和诱导双射的个数。提出了一种由诱导多项式函数确定所有等价多项式的方法，该方法可用于确定诱导给定双射的所有等价多项式。本论文中还有一些问题没有解决，有待于进一步的研究。注：在完成初稿后，我们注意到本文得到的一些结果可以用其他方法证明（见注2）。在这种情况下，这项工作给出了不同的和独立的相关结果的证明。
A new construction of permutation polynomials over $\mathbb{F}_{q^3}$
本文确定了有限域F_{q^3}上若干多项式族中的所有排列多项式，并给出了新的简单系数排列多项式族，解决了张等人的猜想推广问题。
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 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.
A Class of Binomial Permutation Polynomials
本文给出了一类二项式是置换多项式的一个判别准则。由此得到了许多类二项置换多项式和单项完全置换多项式。这些单项式中的指数是Niho型的。
Collection of Polynomials over Finite Fields Providing Involutary Permutations
本文针对满足特定同余条件的奇素数幂，构造了一类有限域上的置换多项式集合，证明它们均能生成具有特定不动点数量的对合置换，并明确给出了三项式与六项式的数量分布。
More Classes of Complete Permutation Polynomials over $\F_q$
本文利用Zieve给出的置换多项式的一个强有力的判别准则，给出了$\F_{q^r}$上的几类完全置换单项式.此外，我们还提出了一类完全置换多项式，这是最近工作的推广。
Construction of Permutation Polynomials of Certain Specific Cycle Structure over Finite Fields
本文针对奇数阶有限域，构造了具有特定圈结构的置换二项式和置换三项式族，包括具有一个不动点且其余元素构成等长互斥圈的置换，以及具有特定频率的多长度圈置换多项式。
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…
Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal latin squares
本文给出了一种由子域置换得到有限域上置换多项式的一般方法。通过将这种技巧应用于子域上最简单的置换多项式类，我们得到了几个新的置换多项式族。其中一些具有额外的性质，即f（x）和f（x）+x都诱导场的置换，这具有组合后果。我们使用我们的置换多项式来展示相互正交拉丁方的完整集合。此外，我们解决了开放的问题，从最近的一份文件，吴和林，我们更简单的证明更一般的版本的结果在最近的两份文件。
Non-permutation phenomena in trivariate families over $\F_{2^m}$ and resolution of a conjecture
研究三元函数在有限域上的非排列现象，解决关于三元形式的猜想的排列部分。
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.
Classes of permutation polynomials based on cyclotomy and an additive analogue
我提出了一个建设的置换多项式的基础上割圆，添加剂类似物的建设，和推广的添加剂类似物似乎没有乘法类似物。这些构造推广了Marcos在arXiv：0810.2738v1中的结果。
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