Pairs of $r$-primitive and $k$-normal elements in finite fields

TL;DR

This paper investigates the existence of elements in finite fields that are simultaneously r-primitive and k-normal, and their images under rational functions, extending the understanding of primitive and normal elements.

Contribution

It provides new sufficient conditions for the existence of pairs of elements with specified primitive and normal properties related through rational functions in finite fields.

Findings

Established conditions for the existence of r-primitive, k-normal elements with prescribed properties.

Analyzed the case with specific parameters r1=2, r2=3, k1=2, k2=1, m1=2, m2=1, for n ≥ 7.

Extended the theory of primitive and normal elements in finite fields.

Abstract

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 $\alpha \in \mathbb{F}_{q^n}^*$ is called $r$-primitive if its multiplicative order is $(q^n-1)/r$. Also, $\alpha \in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_{\alpha}(x) = \alpha x^{n-1}+ \alpha^q x^{n-2} + \ldots + \alpha^{q^{n-2}}x + \alpha^{q^{n-1}}$ 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 $m_1,m_2,k_1,k_2$, positive integers $r_1,r_2$ and rational functions $F(x)=F_1(x)/F_2(x) \in \mathbb{F}_{q^n}(x)$ with $\deg(F_i) \leq m_i$ for $i\in\{ 1,2\}$ satisfying certain conditions and we present sufficient conditions for the existence of $r_1$-primitive $k_1$-normal elements $\alpha \in \mathbb{F}_{q^n}$ over $\mathbb{F}_q$, such that $F(\alpha)$ is an $r_2$-primitive $k_2$-normal element over $\mathbb{F}_q$. Finally as an example we study the case where $r_1=2$, $r_2=3$, $k_1=2$, $k_2=1$, $m_1=2$ and $m_2=1$, with $n \ge 7$.