Subquadratic-Time Algorithms for Normal Bases
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For any finite Galois field extension 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 a α is called a normal element and G·α is a normal basis. We introduce a probabilistic algorithm for testing whether a given α∈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 ∑_g ∈ G g(α)g ∈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 𝖪/𝖥 and the normal basis with the same asymptotic cost.
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