Subquadratic-Time Algorithms for Normal Bases

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