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| == Louis Vuitton Sydney Creasy ==
| | 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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| | \{ \beta, \beta^p, \beta^{p^2}, \ldots, \beta^{p^{m-1}} \} |
| | </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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| == Vibram Five Fingers Australia Srinivasan == | | ==Usage== |
| | 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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| == Vibram Five Fingers Educators can quickly assemble == | | ==Free elements== |
| | 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== |
|
| | *[[Dual basis in a field extension]] |
| <li>[http://www.imodium.es/node/266505 http://www.imodium.es/node/266505]</li>
| | *[[Polynomial basis]] |
|
| | *[[Zech's logarithms]] for reducing high-order polynomials to those within the field |
| <li>[http://www.magiyy.com:30006/forum.php?mod=viewthread&tid=411746 http://www.magiyy.com:30006/forum.php?mod=viewthread&tid=411746]</li>
| | |
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| | ==References== |
| <li>[http://playmond.com/activity/p/47187/ http://playmond.com/activity/p/47187/]</li>
| | {{reflist}} |
|
| | * Galois Theory, [[Ian Stewart (mathematician)|Ian Stewart]], CRC Press, 1990 ISBN 978-0-412-34550-0 |
| <li>[http://bbs.yxdvr.com/bbs/forum.php?mod=viewthread&tid=31253 http://bbs.yxdvr.com/bbs/forum.php?mod=viewthread&tid=31253]</li>
| | * {{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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| <li>[http://dirtyhotshot.com/activity/p/26560/ http://dirtyhotshot.com/activity/p/26560/]</li>
| | [[Category:Linear algebra]] |
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| | [[Category:Field theory]] |
| </ul>
| | [[Category:Abstract algebra]] |
| | [[Category:Cryptography]] |
In mathematics, a normal basis in field theory is a special kind of basis for Galois extensions of finite degree, characterised as forming a single 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.
In the case of finite fields, this means that each of the basis elements is related to any one of them by applying the Frobenius p-th power mapping repeatedly, where p is the characteristic of the field. Let GF(pm) be a field with pm elements, and β an element of it such that the m elements
are linearly independent. Then this set forms a normal basis for GF(pm) over GF(p).
Usage
This basis is frequently used in 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.
When representing elements as a binary string (e.g. in GF(23) the most significant bit represents β4, the least significant bit represents β), we can square elements by doing a left circular shift with wraparound (left shifting β4 would give β8, but since we are working in GF(23) this wraps around to β). This makes the normal basis especially attractive for cryptosystems that utilize frequent squaring.
Free elements
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.[1]
See also
References
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- Galois Theory, Ian Stewart, CRC Press, 1990 ISBN 978-0-412-34550-0
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- ↑ D. Hachenberger, Completely free elements, in Cohen & Niederreiter (1996) 97-107