Carmichael function - Revision history

134.59.217.26: /* Carmichael's theorem */

2015-01-12T14:27:06Z

Carmichael's theorem

← Older revision		Revision as of 16:27, 12 January 2015
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76.88.16.183: /* Exponential cycle length */ consistency:

2014-02-24T02:47:52Z

Exponential cycle length: consistency: <math>

← Older revision		Revision as of 04:47, 24 February 2014
Line 1:		Line 1:
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	~~In [[mathematics]]~~, ~~the [[linear algebra]] concept~~ of ~~[[dual basis]] can be applied in the context~~ of a [~~[finite extension]] ''L''~~/~~''K'', by using~~ the ~~[[field trace]~~]. ~~This requires~~ the ~~property that~~ the ~~field trace ''Tr''~~<~~sub~~>~~''L''/''K''~~<~~/sub~~> provides a [~~[non-degenerate~~]~~] [[quadratic form]] over ''K''~~. ~~This can be guaranteed~~ if the ~~extension is [[separable extension\|separable]]; it is automatically true if ''K'' is~~ a ~~[[perfect field]]~~, and ~~hence in~~ the ~~cases where ''K''~~ is ~~finite, or~~ of ~~characteristic zero~~.

	~~A '''dual basis''' isn't a concrete [[basis (linear algebra)\|basis]] like~~ the ~~[[polynomial basis]] or the [[normal basis]]; rather it provides a way~~ of ~~using a second basis for computations~~.

	~~Consider two bases for elements in a [[finite field]], GF(''p''~~<~~sup~~>~~''m''~~<~~/sup~~>):

	~~:<math>B_1 = {\alpha_0, \alpha_1, \ldots, \alpha_{m-1}}</math>~~

	and

	~~:<math>B_2 = {\gamma_0, \gamma_1~~, ~~\ldots, \gamma_{m-1}}</math>~~

	~~then ''B''<sub>2</sub>~~ can ~~be considered a dual basis~~ of ~~''B''~~<~~sub~~>1<~~/sub~~> ~~provided~~

	~~:<math>\operatorname{Tr}(\alpha_i\cdot \gamma_j) = \left\{\begin{matrix} 0~~, ~~& \operatorname{if}\ i \neq j\\ 1~~, ~~& \operatorname{otherwise} \~~end~~{matrix}\right~~. ~~</math>~~

	~~Here~~ the ~~[[field trace\|trace]]~~ of a ~~value in GF(''p''~~<~~sup~~>~~''m''~~<~~/sup~~>) can be ~~calculated as follows:~~

	:<~~math~~>~~\operatorname{Tr}(\beta ) = \sum_{i=0}^{m-1} \beta^{p^i}~~<~~/math~~>

	~~Using a dual basis can provide a way~~ to ~~easily communicate between devices that use different bases, rather~~ than ~~having to explicitly convert between bases using the [[change~~ of ~~bases]] formula~~. ~~Furthermore~~, ~~if a dual basis~~ is ~~implemented then conversion from an element in the original basis~~ to ~~the dual basis can be accomplished with a multiplication by the multiplicative identity (usually~~ 1).

	~~{{DEFAULTSORT:Dual Basis In A Field Extension}}~~
	~~[[Category:Linear algebra]]~~
	~~[[Category:Field extensions]]~~
	~~[[Category:Theory of cryptography]]~~

en>Joh5n: 1 is by definition in the totient of every integer > 1. 6 and 8 are not coprime -- they each have 2 as a factor.

2013-09-15T03:05:35Z

1 is by definition in the totient of every integer > 1. 6 and 8 are not coprime -- they each have 2 as a factor.

← Older revision		Revision as of 05:05, 15 September 2013
Line 1:		Line 1:
	~~The name~~ of the ~~writer~~ is ~~Adrianne Quesada~~. ~~Her husband doesn~~'t like it ~~the~~ way ~~she does but what she without~~ a ~~doubt likes doing is~~ in ~~order to bake but she~~'~~s thinking on starting something spanking new. After being out of the boy~~'~~s job for years so santa became an order sales person. [http~~://~~Www.Bbc.Co.uk~~/~~search~~/~~?q=Vermont Vermont] are where her house was. Her [http~~:~~//www.Google.Co.uk/search?hl~~=en&~~gl=us~~&~~tbm=nws&q=husband&gs_l=news husband~~] ~~and her carry on~~ a ~~website. You may wish to check it out~~: ~~http~~:~~//prometeu.net/~~<br><br>~~Look at my web blog; clash~~ of ~~clans hack no survey~~ [[~~http~~:~~//prometeu.net/ click the up coming webpage~~]]		{{Unreferenced\|date=December 2009}}
			In [[mathematics]], the [[linear algebra]] concept of [[dual basis]] can be applied in the context of a [[finite extension]] ''L''/''K'', by using the [[field trace]]. This requires the property that the field trace ''Tr''<sub>''L''/''K''</sub> provides a [[non-degenerate]] [[quadratic form]] over ''K''. This can be guaranteed if the extension is [[separable extension\|separable]]; it is automatically true if ''K'' is a [[perfect field]], and hence in the cases where ''K'' is finite, or of characteristic zero.

			A '''dual basis''' isn't a concrete [[basis (linear algebra)\|basis]] like the [[polynomial basis]] or the [[normal basis]]; rather it provides a way of using a second basis for computations.

			Consider two bases for elements in a [[finite field]], GF(''p''<sup>''m''</sup>):

			:<math>B_1 = {\alpha_0, \alpha_1, \ldots, \alpha_{m-1}}</math>

			and

			:<math>B_2 = {\gamma_0, \gamma_1, \ldots, \gamma_{m-1}}</math>

			then ''B''<sub>2</sub> can be considered a dual basis of ''B''<sub>1</sub> provided

			:<math>\operatorname{Tr}(\alpha_i\cdot \gamma_j) = \left\{\begin{matrix} 0, & \operatorname{if}\ i \neq j\\ 1, & \operatorname{otherwise} \end{matrix}\right. </math>

			Here the [[field trace\|trace]] of a value in GF(''p''<sup>''m''</sup>) can be calculated as follows:

			:<math>\operatorname{Tr}(\beta ) = \sum_{i=0}^{m-1} \beta^{p^i}</math>

			Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than having to explicitly convert between bases using the [[change of bases]] formula. Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with a multiplication by the multiplicative identity (usually 1).

			{{DEFAULTSORT:Dual Basis In A Field Extension}}
			[[Category:Linear algebra]]
			[[Category:Field extensions]]
			[[Category:Theory of cryptography]]

en>Gmkwo: typo corrected: \phi --> \varphi

2012-08-02T09:44:48Z

typo corrected: \phi --> \varphi

New page

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