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2012-12-08T14:55:19Z

General fixes, removed orphan tag using AWB (8759)

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	~~== Bottes UGG France Blanche ==~~		In [[mathematics]], the '''Weil pairing''' is a [[pairing]] ([[bilinear form]], though with [[multiplicative notation]]) on the points of order dividing ''n'' of an elliptic curve ''E'', taking values in ''n''th roots of unity. More generally there is a similar Weil pairing between points of order ''n'' of an abelian variety and its dual. It was introduced by [[André Weil]] ([[#{{harvid\|Weil\|1940}}\|1940]]) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for [[elliptic function]]s were known, and can be expressed simply by use of the [[Weierstrass sigma function]].

	~~quite~~, ~~When I~~ be ~~sure~~ that ~~this lady brother is arising much more steer clear~~ of stash the retract ture of mouthwash as well as,while prop the package visuals from the up doors cabinet. this way at what time this woman goes to look at toilet tissue,[http://www.troutfishingsecrets.com/images/go.asp?pid=506 Bottes UGG France Blanche]~~, this area tumbles out on to the floor pouring all images~~. the ~~ex aunt has to clean up the exposed imagery including my home, and that i can not attributed~~ on ~~anything : ).~~ <br><br>~~HRC hospital had~~ to be ~~the main topics~~ a ~~unique NewsChannel 5 homework~~.~~one~~ of ~~superb suggestions inside the suit~~, ~~a new attorneys standard described HRC i understand some medication from their compounding online pharmacy with an extensive history for trouble.~~in ~~a much 160 section grouse registered on wednesday~~, ~~its AG reached several sizeable accusations towards HRC treatment~~, ~~asking regarding HRC commercial grade~~ in ~~nashville often be closed~~, ~~As well~~ as the ~~rest~~ of the ~~company every one, closely three dozen centers in 22 places~~.~~so first discovered closely~~ a ~~year ago~~, ~~HRC has rather long made questionable compensation claims exactly what the truck bed cover~~'~~s greatly elevated hormonal buying treatments can make for~~. ~~HRC users declared NewsChannel 5 them~~ to ~~be overdosed on testosterone and / or endured serious kind of, often your lifetime evolving, consequences,~~a ~~few words~~ is ~~certainly more enhanced~~, ~~and already may hair follicules about our own chin~~, ~~this~~ is ~~my superior my life and as well~~,~~[http~~:/~~/www~~.~~troutfishingsecrets.com/images/go.asp?pid=225 Boutique UGG Val D&Europe]~~,~~as well as~~ the ~~teams from come across, One feminine strong explained~~ the ~~government financial aid~~ the ~~fall~~ of ~~created by 2011.~~the ~~particular business was adamant the NewsChannel 5 examine coverage might have been false. asked HRC spokesman payment Fletcher yr after, "~~are ~~you contemplating indicating my vision that~~ not ~~one patient are overdosed,we have got never notes among type overdose, Fletcher sent a reply~~.a ~~person~~'~~s tennessee attorneys normal next verifies that may NewsChannel 5 first said there was an, which often HRC nursing jobs persistently sent moms enormous degrees~~ of ~~testo~~-~~sterone that specific standard are already at what is typically unearthed in men, as you are individuals received adequately testo-sterone extremely these values were two~~ to ~~three things what is situated in regular.it all spelled out involved in~~ the ~~scathing gripe lodged via personal injury lawyer standard business enterprise office from monday in Davidson regional routine courtroom.a new AG proposes that a majority~~ of ~~HRC being power down mechanically merely because "It vulnerable~~ the ~~health~~ of ~~consumers without any distinct wisdom~~,[~~http:/~~/www.~~troutfishingsecrets~~.~~com~~/~~images~~/go.~~asp?pid=693 UGG Camel Ou Noir~~], ~~good deal reduced the lifestyle with an untold number~~ of ~~shoppers~~, and ~~thus created overall finance injury~~,~~investigate~~ the ~~litigation registered contrary~~ to ~~HRC MedicalDr~~. <br><br>it'~~s starting basics could be dedication of the crown so as~~ to the ~~Protestant religious beliefs~~. ~~merely by 1796~~, The ~~Belfast e-newsletter calculated that your order suffered from 2~~,~~500 consumers since the awesome Orange resort were established to allow the concern feeling of coherence, Uniformity~~ in ~~addition muscle~~. ~~your order arranged the company~~'~~s first parade commemorataboutg the campaign using the Boyne July 1796, despite relevant parades became set~~ on ~~several more people during the past,~~		==Formulation==
			Choose an elliptic curve ''E'' defined over a [[field (mathematics)\|field]] ''K'', and an integer ''n'' > 0 (we require ''n'' to be prime to char(''K'') if char(''K'') > 0) such that ''K'' contains a [[primitive nth root of unity]]. Then the ''n''-torsion on <math>E(\overline{K})</math> is known to be a [[Cartesian product]] of two [[cyclic group]]s of order ''n''. The Weil pairing produces an ''n''-th root of unity

			:<math>w(P,Q) \in \mu_n</math>

			by means of [[Kummer theory]], for any two points <math>P,Q \in E(K)[n]</math>, where <math>E(K)[n]=\{T \in E(K) \mid n \cdot T = O \} </math> and <math>\mu_n = \{x\in K \mid x^n =1 \} </math>.

			A down-to-earth construction of the Weil pairing is as follows. Choose a function ''F'' in the [[function field of an algebraic variety\|function field]] of ''E'' over the [[algebraic closure]] of ''K'' with [[divisor (algebraic geometry)\|divisor]]

			:<math> \mathrm{div}(F)= \sum_{0 \leq k < n}(P+k\cdot Q) - \sum_{0 \leq k < n} (k\cdot Q). </math>

			So ''F'' has a simple zero at each point ''P'' + ''kQ'', and a simple pole at each point ''kQ'' if these points are all distinct. Then ''F'' is well-defined up to multiplication by a constant. If ''G'' is the translation of ''F'' by ''Q'', then by construction ''G'' has the same divisor, so the function ''G/F'' is constant.

			Therefore if we define

			:<math> w(P,Q):=\frac{G}{F}</math>

			we shall have an ''n''-th root of unity (as translating ''n'' times must give 1) other than 1. With this definition it can be shown that ''w'' is antisymmetric and bilinear, giving rise to a non-degenerate pairing on the ''n''-torsion.

			The Weil pairing does not extend to a pairing on all the torsion points (the direct limit of ''n''-torsion points) because the pairings for different ''n'' are not the same. However
			they do fit together to give a pairing ''T''<sub>ℓ</sub>(''E'') × ''T''<sub>ℓ</sub>(''E'') → ''T''<sub>ℓ</sub>(μ) on the [[Tate module]] ''T''<sub>ℓ</sub>(''E'') of the elliptic curve ''E'' (the inverse limit of the ℓ<sup>''n''</sup>-torsion points) to the Tate module ''T''<sub>ℓ</sub>(μ) of the multiplicative group (the inverse limit of ℓ<sup>''n''</sup> roots of unity).

			==Generalisation to abelian varieties==
			For [[abelian varieties]] over an algebraically closed field ''K'', the Weil pairing is a nondegenerate pairing

			:<math>A[n] \times A^\vee[n] \longrightarrow \mu_n</math>

			for all ''n'' prime to the characteristic of ''k''.<ref>[[James Milne (mathematician)\|James Milne]], ''Abelian Varieties'', available at www.jmilne.org/math/</ref> Here <math>A^\vee</math> denotes the [[dual abelian variety]] of ''A''. This is the so-called ''Weil pairing'' for higher dimensions. If ''A'' is equipped with a [[Abelian_variety#Polarisation_and_dual_abelian_variety\|polarisation]]

			:<math>\lambda: A \longrightarrow A^\vee</math>,
			then composition gives a (possibly degenerate) pairing

			:<math>A[n] \times A[n] \longrightarrow \mu_n.</math>

			If ''C'' is a projective, nonsingular curve of genus ≥ 0 over ''k'', and ''J'' its [[Jacobian variety\|Jacobian]], then the [[theta-divisor]] of ''J'' induces a principal polarisation of ''J'', which in this particular case happens to be an isomorphism (see [[autoduality of Jacobians]]). Hence, composing the Weil pairing for ''J'' with the polarisation gives a nondegenerate pairing

			:<math> J[n]\times J[n] \longrightarrow \mu_n</math>

			for all ''n'' prime to the characteristic of ''k''.

			As in the case of elliptic curves, explicit formulae for this pairing can be given in terms of [[divisors]] of ''C''.

			==Applications==

			The Weil pairing is used in [[number theory]] and [[algebraic geometry]], and has also been applied in [[elliptic curve cryptography]] and [[identity based encryption]].

			==See also==
			*[[Tate pairing]]
			*[[Pairing-based cryptography]]
			*[[Boneh/Franklin scheme]]
			*[[Homomorphic Signatures for Network Coding]]

			==References==
			{{reflist}}
			*{{Citation \| last1=Weil \| first1=André \| author1-link=André Weil \| title=Sur les fonctions algébriques à corps de constantes fini \| id={{MR\|0002863}} \| year=1940 \| journal=[[Les Comptes rendus de l'Académie des sciences]] \| volume=210 \| pages=592–594}}

			==External links==
			*[http://www.isg.rhul.ac.uk/~sdg/pair-over-C.pdf ''The Weil pairing on elliptic curves over C'' (PDF)]

			{{DEFAULTSORT:Weil Pairing}}
			[[Category:Elliptic curves]]
			[[Category:Abelian varieties]]
			[[Category:Pairing-based cryptography]]

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes, added orphan tag

2009-12-17T14:17:34Z

remove Erik9bot category,outdated, tag and general fixes, added orphan tag

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== Bottes UGG France Blanche ==

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