Subquadratic-Time Algorithms for Normal Bases

Abstract

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