Subquadratic-Time Algorithms for Normal Bases

Abstract

For any finite Galois field extension $\mathsf{K}/\mathsf{F}$, with Galois group $G = \mathrm{Gal}(\mathsf{K}/\mathsf{F})$, there exists an element $α\in \mathsf{K}$ whose orbit $G\cdotα$ forms an $\mathsf{F}$-basis of $\mathsf{K}$. Such a $α$ is called a normal element and $G\cdotα$ is a normal basis. We introduce a probabilistic algorithm for testing whether a given $α\in \mathsf{K}$ is normal, when $G$ is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether $α$ is normal can be reduced to deciding whether $\sum_{g \in G} g(α)g \in \mathsf{K}[G]$ is invertible; it requires a slightly subquadratic number of operations. Once we know that $α$ is normal, we show how to perform conversions between the power basis of $\mathsf{K}/\mathsf{F}$ and the normal basis with the same asymptotic cost.