Related papers: Further Investigation on Cyclotomic Mapping Permut…

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…

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…

In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds…

Let $f={\tt X}^r(a+{\tt X}^{2(q-1)})\in{\Bbb F}_{q^2}[{\tt X}]$, where $a\in{\Bbb F}_{q^2}^*$ and $r\ge 1$. The parameters $(q,r,a)$ for which $f$ is a permutation polynomial (PP) of ${\Bbb F}_{q^2}$ have been determined in the following…

A representation of finite fields that has proved useful when implementing finite field arithmetic in hardware is based on an isomorphism between subrings and fields. In this paper, we present an unified formulation for multiplication in…

In this paper, by some arithmetic properties of the Pell sequence and some $p$-adic tools, we study certain cyclotomic matrices involving squares over finite fields. For example, let $1=s_1,s_2,\cdots,s_{(q-1)/2}$ be all the nonzero squares…

We construct classes of permutation polynomials over F_{Q^2} by exhibiting classes of low-degree rational functions over F_{Q^2} which induce bijections on the set of (Q+1)-th roots of unity in F_{Q^2}. As a consequence, we prove two…

We construct four new classes of permutation trinomials over the cubic extension of a finite field with even characteristic. Additionally, we explicitly provide the compositional inverse of each class of permutation trinomials in polynomial…

This is a short note that explains a problem on polynomial maps over finite fields for non-experts. The problem is: Do there exist odd polynomial automorphisms over the finite fields with 4,8,16,32,64,... elements? The explanation is very,…

Let $\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\mathbb{F}_{q^3}$ of the family $$ f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\in \mathbb{F}_{q}[X]. $$ Using…

Let $\mathbb{F}$ be a finite field and let $f$ be a linear polynomial in $\mathbb{F}[x]$. We investigate the number of polynomials of degree $d$ which commute with $f$ under composition. In so doing, we rediscover a result of Park, but with…

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 paper, the compositional inverses of a class of linearized permutation polynomials of the form $P(x)=x+x^2+\tr(\frac{x}{a})$ over the finite field $\mathbb{F}_{2^n}$ for an odd positive integer $n$ are explicitly determined.

Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $\mathbb{F}_q$. A general result of our study is that…

For an integer $r$, a prime power $q$, and a polynomial $f$ over a finite field ${\mathbb F}_{q^r}$ of $q^r$ elements, we obtain an upper bound on the frequency of elements in an orbit generated by iterations of $f$ which fall in a proper…

We prove an asymptotic formula for the number of permutation for which the associated permutation polynomial has degree smaller than $q-2$.

Let $q$ be a prime power and $\mathbb F_{q^n}$ be the finite field with $q^n$ elements, where $n>1$. We introduce the class of the linearized polynomials $L(x)$ over $\mathbb F_{q^n}$ such that…

The proof of the theorem concerning to the inverse cyclotomic Discrete Fourier Transform algorithm over finite field is provided.

In this paper we describe a class of codes called {\it permutation codes}. This class of codes is a generalization of cyclic codes and quasi-cyclic codes. We also give some examples of optimal permutation codes over binary, ternary, and…

Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their varied applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear…