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Lucas Reis

Publications and source records attributed to Lucas Reis.

44 records · Page 3Linked to original sources
arXiv2017

Existence of primitive $1$-normal elements in finite fields

An element $α\in \mathbb F_{q^n}$ is \emph{normal} if $\mathcal{B} = \{α, α^q, \ldots, α^{q^{n-1}}\}$ forms a basis of $\mathbb F_{q^n}$ as a vector space over $\mathbb F_{q}$; in this case, $\mathcal{B}$ is a normal basis of $\mathbb F_{q^n}$ over $\mathbb F_{q}$. The notion of $k$-normal elements was introduced in Huczynska et al (2013). Using the same notation as before, $α$ is $k$-normal if $\mathcal{B}$ spans a co-dimension $k$ subspace of $\mathbb F_{q^n}$. It can be shown that $1$-normal elements always exist in $\mathbb F_{q^n}$, and Huczynska et al (2013) show that elements that are simultaneously primitive and $1$-normal exist for $q \geq 3$ and for large enough $n$ when $\gcd(n,q) = 1$ (we note that primitive $1$-normals cannot exist when $n=2$). In this paper, we complete this theorem and show that primitive, $1$-normal elements of $\mathbb F_{q^n}$ over $\mathbb F_{q}$ exist for all prime powers $q$ and all integers $n \geq 3$, thus solving Problem 6.3 from Huczynska, et al (2013).

math.NT↗
arXiv2017

A group action on multivariate polynomials over finite fields

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a power of a prime $p$. Recently, a particular action of the group $\mathrm{GL}_2(\mathbb F_q)$ on irreducible polynomials in $\mathbb F_q[x]$ has been introduced and many questions concerning the invariant polynomials have been discussed. In this paper, we give a natural extension of this action on the polynomial ring $\mathbb F_q[x_1, \ldots, x_n]$ and study the algebraic properties of the invariant elements.

math.RA↗
arXiv2017

A note on primitive $1-$normal elements over finite fields

Let $q$ be a prime power of a prime $p$, $n$ a positive integer and $\mathbb F_{q^n}$ the finite field with $q^n$ elements. The $k-$normal elements over finite fields were introduced and characterized by Huczynska et al (2013). Under the condition that $n$ is not divisible by $p$, they obtained an existence result on primitive $1-$normal elements of $\mathbb F_{q^n}$ over $\mathbb F_q$ for $q>2$. In this note, we extend their result to the excluded case $q=2$.

math.NT↗
arXiv2016

Nilpotent linearized polynomials over finite fields and applications

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 $$L^{(t)}(x):=\underbrace{L(L(\cdots(x)\cdots))}_{t \quad\text{times}}\equiv 0\pmod {x^{q^n}-x}$$ for some $t\ge 2$, called nilpotent linearized polynomials (NLP's). We discuss the existence and construction of NLP's and, as an application, we show how to construct permutations of $\mathbb F_{q^n}$ from these polynomials. For some of those permutations, we can explicitly give the compositional inverse map and the cycle structure. This paper also contains a method for constructing involutions over binary fields with no fixed points, which are useful in block ciphers.

math.NT↗
arXiv2016

The action of $\rm{GL}_2(\mathbb{F}_q)$ on irreducible polynomials over $\F_q$

Let $\F_q$ be the finite field with $q$ elements, $p=\Char \F_q$. The group $\GL_2(\F_q)$ acts naturally in the set of irreducible polynomials over $\F_q$ of degree at least $2$. In this paper we are interested in the characterization and number of the irreducible polynomials that are fixed by the elements of a subgroup $H$ of $\GL_2(\F_q)$. We make a complete characterization of the fixed polynomials in the case when $H$ has only elements of the form $\left(\begin{matrix} 1&b\\ 0&1 \end{matrix}\right)$, corresponding to translations $x\mapsto x+b$ and, as a consequence, the case when $H$ is a $p-$subgroup of $\GL_2(\F_q)$. This paper also contains alternative solutions for the cases when $H$ is generated by an element of the form $\left(\begin{matrix} a&0\\ 0&1 \end{matrix}\right)$, obtained by Garefalakis (2010) and $H=\rm{PGL}$$_2(\F_q)$, obtained by Stichtenoth and Topuzoglu (2011).

math.NT↗
arXiv2015

Factoring polynomials of the form $f(x^n)\in \mathbb{F}_q[x]$

Let $f(x)\in \mathbb{F}_q[x]$ be an irreducible polynomial of degree $m$ and exponent $e$, and $n$ be a positive integer such that $ν_p(q-1)\ge ν_{p}(e)+ν_p(n)$ for all $p$ prime divisor of $n$. We show a fast algorithm to determine the irreducible factors of $f(x^n)$. We also show the irreducible factors in the case when ${\rm rad}(n)$ divides $q-1$ and ${\rm gcd}(m, n)=1$. Finally, using this algorithm we split $x^n-1$ into irreducible factors, in the case when $n=2^mp^t$ and $q$ is a generator of the group $\mathbb{Z}_{p^2}^*$.

math.NT↗