Let $r$, $n$ be positive integers, $k$ be a non-negative integer and $q$ be any prime power such that $r\mid q^n-1.$ An element $α$ of the finite field $\mathbb{F}_{q^n}$ is called an {\it $r$-primitive} element, if its multiplicative order is $(q^n-1)/r$, and it is called a {\it $k$-normal} element over $\mathbb{F}_q$, if the greatest common divisor of the polynomials $m_α(x)=\sum_{i=1}^{n} α^{q^{i-1}}x^{n-i}$ and $x^n-1$ is of degree $k.$ In this article, we define the characteristic function for the set of $k$-normal elements, and with the help of this, we establish a sufficient condition for the existence of an element $α$ in $\mathbb{F}_{q^n}$, such that $α$ and $α^{-1}$ both are simultaneously $r$-primitive and $k$-normal over $\mathbb{F}_q$. Moreover, for $n>6k$, we show that there always exists an $r$-primitive and $k$-normal element $α$ such that $α^{-1}$ is also $r$-primitive and $k$-normal in all but finitely many fields $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, where $q$ and $n$ are such that $r\mid q^n-1$ and there exists a $k$-degree polynomial $g(x)\mid x^n-1$ over $\mathbb{F}_q$. In particular, we discuss the existence of an element $α$ in $\mathbb{F}_{q^n}$ such that $α$ and $α^{-1}$ both are simultaneously $1$-primitive and $1$-normal over $\mathbb{F}_q$.
In this paper, we consider rational functions $f$ with some minor restrictions over the finite field $\mathbb{F}_{q^n},$ where $q=p^k$ for some prime $p$ and positive integer $k$. We establish a sufficient condition for the existence of a pair $(α,f(α))$ of primitive normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}.$ Moreover, for $q=2^k$ and rational functions $f$ with quadratic numerators and denominators, we explicitly find that there are at most $55$ finite fields $\mathbb{F}_{q^n}$ in which such a pair $(α,f(α))$ of primitive normal elements may not exist.
The previous paper [4] proved the existence of primitive polynomials and primitive normal polynomials of degree n with k prescribed coefficients in the finite field GF(q) for all sufficiently large q. This paper presents a loger versions of the result on primitve normal polynomials.
The celebrated Primitive Normal Basis Theorem states that for any $n\ge 2$ and any finite field $\mathbb F_q$, there exists an element $α\in \mathbb F_{q^n}$ that is simultaneously primitive and normal over $\mathbb F_q$. In this paper, we prove some variations of this result, completing the proof of a conjecture proposed by Anderson and Mullen (2014). Our results also imply the existence of elements of $\mathbb F_{q^n}$ with multiplicative order $(q^n-1)/2$ and prescribed trace over $\mathbb F_q$.
Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. We prove that there exists a primitive element of $\mathbb{F}_{q^n}$ that produces a completely normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, provided that $n=p^{\ell}m$ with $(m,p)=1$ and $q>m$.
An element w of the extension E of degree n over the finite field F=GF(q) is called free over F if {w, w^q,...,w^{q^{n-1}}} is a (normal) basis of E/F. The Primitive Normal Basis Theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F, there exists an element w in E such that w is simultaneously primitive (i.e., generates the multiplicative group of E) and free over F. In this paper we prove the following strengthening of this theorem: aside from five specific extensions E/F, there exists an element w in E such that both w and w^{-1} are simulta…
We present computational results which strongly support a conjecture of Morgan and Mullen (1996), which states that for every extension $E/F$ of Galois fields there exists a primitive element of $E$ which is completely normal over $F$.
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 $q=2$.
For $q=3^r$ ($r>0$), denote by $\mathbb{F}_q$ the finite field of order $q$ and for a positive integer $m\geq2$, let $\mathbb{F}_{q^m}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive normal element $α$ such that $f(α)$ is a primitive element, where $f(x)= ax^2+bx+c$, with $a,b,c\in \mathbb{F}_{q^m}$ satisfying $b^2\neq ac$ in $\Fm$ except for at most 9 exceptional pairs $(q,m)$.
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 s…
In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of $k$-normal elements: an element $α\in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_α(x)= αx^{n-1}+α^qx^{n-2}+\ldots +α^{q^{n-2}}x+α^{q^{n-1}}$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$, generalizing the concept of normal elements (normal in the usual sense is $0$-normal). In this paper we discuss the existence of $r$-primitive, $k$-normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, where an element $α\in \mathbb{F}_{q^n}^*$ is $r$-primitive i…
Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in \mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such that the rational function $f_1(x)/f_2(x)$ belongs to a certain set which we define, we present a sufficient condition for the existence of a primitive element $α\in \mathbb{F}_{q^n}$, normal over $\mathbb{F}_q$, such that $f_1(α)/f_2(α)$ is also primitive.
A primitive completely normal element for an extension $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ of Galois fields is a generator of the multiplicative group of $\mathbb{F}_{q^n}$, which simultaneously is normal over every intermediate field of that extension. We are going to prove that such a generator exists when $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ is an 'exceptional' regular extension. In combination with [6] our investigations altogether settle the existence of primitive completely normal bases for any regular extension. An important feature of the class of regular extensions is that they comprise ev…
The so called $k$-normal elements appear in the literature as a generalization of normal elements over finite fields. Recently, questions concerning the construction of $k$-normal elements and the existence of $k$-normal elements that are also primitive have attracted attention from many authors. In this paper we give alternative constructions of $k$-normal elements and, in particular, we obtain a sieve inequality for the existence of primitive, $k$-normal elements. As an application, we show the existence of primitive $k$-normal elements for a significant proportion of $k$'s in many field ext…
Given $m, n, q\in \mathbb{N}$ such that $q$ is a prime power and $m\geq 3$, $a\in \mathbb{F}_q$, we establish a sufficient condition for the existence of primitive pair $(α, f(α))$ in $\mathbb{F}_{q^m}$ such that $α$ is normal over $\mathbb{F}_q$ and $\text{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_q}(α^{-1})=a$, where $f(x)\in \mathbb{F}_{q^m}(x)$ is a rational function of degree sum $n$. Further, when $n=2$ and $q=5^k$ for some $k\in \mathbb{N}$, such a pair definitely exists for all $(q, m)$ apart from at most $20$ choices.
Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements and $r$ be a positive divisor of $q^n-1$. An element $α\in \mathbb{F}_{q^n}^*$ is called $r$-primitive if its multiplicative order is $(q^n-1)/r$. Also, $α\in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_α(x) = αx^{n-1}+ α^q x^{n-2} + \ldots + α^{q^{n-2}}x + α^{q^{n-1}}$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$. These concepts generalize the ideas of primitive and normal elements, respectively. In this paper, we consider non-negative integers $m_1,m_2,k_1,k_2$, po…
We examine linear sums of primitive roots and their inverses in finite fields. In particular, we refine a result by Li and Han, and show that every $p> 13$ has a pair of primitive roots $a$ and $b$ such that $a+ b$ and $a^{-1} + b^{-1}$ are also primitive roots mod $p$.
An element $α\in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if $\mathcal{B}=\{α, α^q, α^{q^2}, \cdots, α^{q^{n-1}}\}$ forms a basis of $\mathbb{F}_{q^n}$ as a vector space over $\mathbb{F}_q$. It is well known that $α\in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if and only if $g_α(x)=αx^{n-1}+α^q x^{n-2}+ \cdots + α^{q^{n-2}}x+α^{q^{n-1}}$ and $x^n-1$ are relatively prime over $\mathbb{F}_{q^n}$, that is, the degree of their greatest common divisor in $\mathbb{F}_{q^n}[x]$ is $0$. Using this equivalence, the notion of $k$-normal elements was introduced in Huczynska et al. ($2013$):…
Let $K$ be a finite extension of $\Q_p$, let $L/K$ be a finite abelian Galois extension of odd degree and let $\bo_L$ be the valuation ring of $L$. We define $A_{L/K}$ to be the unique fractional $\bo_L$-ideal with square equal to the inverse different of $L/K$. For $p$ an odd prime and $L/\Q_p$ contained in certain cyclotomic extensions, Erez has described integral normal bases for $A_{L/\Q_p}$ that are self-dual with respect to the trace form. Assuming $K/\Q_p$ to be unramified we generate odd abelian weakly ramified extensions of $K$ using Lubin-Tate formal groups. We then use Dwork's expon…
The use of permutation polynomials has appeared, along to their compositional inverses, as a good choice in the implementation of cryptographic systems. Hence, there has been a demand for constructions of these polynomials which coefficients belong to a finite field. As a particular case of permutation polynomial, involution is highly desired since its compositional inverse is itself. In this work, we present an effective way of how to construct several linear permutation polynomials over $\mathbb{F}_{q^n}$ as well as their compositional inverses using a decomposition of $\displaystyle{\frac{\…
Let $A$ be an abelian variety defined over a number field $K$. We say that a point $P \in A(\overline{\mathbb{Q}})$ is primitive if there is no $Q \in A(\overline{\mathbb{Q}})$ defined on the field of definition of $P$ over $K$ such that $[N]Q=P$ for some positive integer $N \ge 2$. For any primitive point $P \in A(\overline{\mathbb{Q}})$, positive integer $N$ and point $Q \in A(\overline{\mathbb{Q}})$ such that $[N]Q=P$, we prove an effective lower bound on the degree of the field of definition of $Q$ over $K$ of the form $N^δ$ that depends only on $A,K$ and the degree of the field of definit…
