Existence of primitive k-normal elements for critical values over finite fields

TL;DR

This work tackles the existence of primitive $k$-normal elements over finite fields at the critical value $k=\\tfrac{n}{2}$. It extends translate-and-sieve techniques to the critical regime, establishing general existence conditions and obstructions, and then delivers a complete characterization for the case $n=6$, $k=3$ by combining analytic bounds with sieve refinements and explicit constructions. The key contribution is a precise threshold framework showing when primitive $k$-normal elements must exist, complemented by a constructive approach for the stubborn $n=6$ case, including detailed computational verification. The results advance our understanding of the interplay between normality and primitivity in finite field extensions, with potential implications for cryptographic applications requiring bases that are simultaneously primitive and suitably dependent on conjugates.

Abstract

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. An element $α\in \mathbb{F}_{q^n}$ is called $k$-normal over $\mathbb{F}_q$ if $α$ and its conjugates generate a vector subspace of $\mathbb{F}_{q^n}$ of dimension $n-k$ over $\mathbb{F}_q$. The existence of primitive $k$-normal elements and related properties have been studied throughout the past few years for $k > n/2$. In this paper, we provide general results on the existence of primitive $k$-normal elements for the critical value $k = n/2$, which have not been studied until now, except for $n = 4$. Furthermore, we show the strength of this result by providing a complete characterization of the existence of primitive $3$-normal elements in $\mathbb{F}_{q^6}$ over $\mathbb{F}_q$.

Theorems & Definitions (25)

• Definition 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2: Nied, Lemma 3.59
• Definition 2.3
• Theorem 2.4: knormal, Theorem 3.2
• Definition 2.5
• Proposition 2.6
• proof
• Lemma 2.7: katz, Theorem 1