Quadratic Probabilistic Algorithms for Normal Bases
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It is well known that for any finite Galois extension field K/F, with Galois group G = Gal(K/F), there exists an element α∈ K whose orbit G·α forms an F-basis of K. Such an element α is called normal and G·α is called a normal basis. In this paper we introduce a probabilistic algorithm for finding a normal element when G is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether a random element α∈ K is normal can be reduced to deciding whether ∑_σ∈ Gσ(α)σ∈ K[G] is invertible. In an algebraic model, the cost of our algorithm is quadratic in the size of G for metacyclic G and slightly subquadratic for abelian G.
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