Complexities of normal bases constructed from Gauss periods

Abstract

Let q be a power of a prime p, and let $r=nk+1$ be a prime such that $r\not \mid q$, where n and k are positive integers. Under a simple condition on q, r and k, a Gauss period of type (n, k) is a normal element of ${\mathbb {F}}_{q}^{n}$ over ${\mathbb {F}}_q$; the complexity of the resulting normal basis of ${\mathbb {F}}_{q}^{n}$ over ${\mathbb {F}}_q$ is denoted by C(n, k; p). Recent works determined C(n, k; p) for $k\le 7$ and all qualified n and q. In this paper, we show that for any given $k>0$, C(n, k; p) is given by an explicit formula except for finitely many primes $r=nk+1$ and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute C(n, k; p) for the exceptional primes $r=nk+1$. Our numerical results cover C(n, k; p) for $k\le 20$ and all qualified n and q.