Subquadratic-Time Algorithms for Normal Bases

05/05/2020
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by   Mark Giesbrecht, et al.
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For any finite Galois field extension ๐–ช/๐–ฅ, with Galois group G = Gal(๐–ช/๐–ฅ), there exists an element ฮฑโˆˆ๐–ช whose orbit Gยทฮฑ forms an ๐–ฅ-basis of ๐–ช. Such a ฮฑ is called a normal element and Gยทฮฑ is a normal basis. We introduce a probabilistic algorithm for testing whether a given ฮฑโˆˆ๐–ช is normal, when G is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether ฮฑ is normal can be reduced to deciding whether โˆ‘_g โˆˆ G g(ฮฑ)g โˆˆ๐–ช[G] is invertible; it requires a slightly subquadratic number of operations. Once we know that ฮฑ is normal, we show how to perform conversions between the power basis of ๐–ช/๐–ฅ and the normal basis with the same asymptotic cost.

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