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| In [[mathematics]], a '''normal basis''' in [[field theory (mathematics)|field theory]] is a special kind of [[basis (linear algebra)|basis]] for [[Galois extension]]s of finite degree, characterised as forming a single [[orbit (group theory)|orbit]] for the [[Galois group]]. The '''normal basis theorem''' states that any finite Galois extension of fields has a normal basis. In [[algebraic number theory]] the study of the more refined question of the existence of a [[normal integral basis]] is part of [[Galois module]] theory.
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| In the case of [[finite field]]s, this means that each of the basis elements is related to any one of them by applying the [[Frobenius map|Frobenius]] ''p''-th power mapping repeatedly, where ''p'' is the [[characteristic (algebra)|characteristic]] of the field. Let GF(''p''<sup>''m''</sup>) be a field with ''p''<sup>''m''</sup> elements, and β an element of it such that the ''m'' elements
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| :<math>
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| \{ \beta, \beta^p, \beta^{p^2}, \ldots, \beta^{p^{m-1}} \}
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| </math>
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| are linearly independent. Then this set forms a normal basis for GF(''p''<sup>''m''</sup>) over GF(''p'').
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| ==Usage==
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| This basis is frequently used in [[cryptography|cryptographic]] applications that are based on the [[discrete logarithm problem]] such as [[elliptic curve cryptography]]. Hardware implementations of normal basis arithmetic typically have far less power consumption than other bases.
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| When representing elements as a binary string (e.g. in GF(2<sup>3</sup>) the most significant bit represents β<sup>4</sup>, the least significant bit represents β), we can square elements by doing a left circular shift with wraparound (left shifting β<sup>4</sup> would give β<sup>8</sup>, but since we are working in GF(2<sup>3</sup>) this wraps around to β). This makes the normal basis especially attractive for cryptosystems that utilize frequent squaring.
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| ==Free elements==
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| If ''E''/''F'' is a Galois extension with group ''G'' and ''x'' in ''E'' generates a normal basis then ''x'' is '''free''' in ''E''/''F''. If ''x'' has the property that for every subgroup ''H'' of ''G'', with fixed field ''H''°, ''x'' is free for ''E''/''H''°, then ''x'' is said to be '''completely free''' in ''E''/''F''. Every Galois extension has a completely free element.<ref>D. Hachenberger, ''Completely free elements'', in Cohen & Niederreiter (1996) 97-107</ref>
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| ==See also==
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| *[[Dual basis in a field extension]]
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| *[[Polynomial basis]]
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| *[[Zech's logarithms]] for reducing high-order polynomials to those within the field
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| ==References==
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| {{reflist}}
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| * Galois Theory, [[Ian Stewart (mathematician)|Ian Stewart]], CRC Press, 1990 ISBN 978-0-412-34550-0
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| * {{cite book | editor1=S. Cohen | editor2=H. Niederreiter | title=Finite Fields and Applications | publisher=Cambridge University Press | isbn=0-521-56736-X | year=1996 }}
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| [[Category:Linear algebra]]
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| [[Category:Field theory]]
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| [[Category:Abstract algebra]]
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| [[Category:Cryptography]]
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Caravan Park and Camping Floor Manager Albert Dargan from Grande-Riviere, has hobbies and interests for instance playing team sports, diet and tool collecting. Maintains a trip site and has plenty to write about after traveling to Hwaseong Fortress.
my website :: simple diet burn fat