Parameter identification problem - Revision history

en>Phleg1: dab

2014-11-17T23:35:39Z

dab

← Older revision		Revision as of 01:35, 18 November 2014
Line 1:		Line 1:
	The '''Schoof–Elkies–Atkin algorithm (SEA)''' is an [[algorithm]] used for finding the [[order (group theory)\|order]] of or calculating the number of points on an [[elliptic curve]] over a [[finite field]]. Its primary application is ~~in [[elliptic curve cryptography]]. The algorithm is an extension of [[Schoof's algorithm]] by [[Noam Elkies]]~~ and ~~[[A. O. L. Atkin]] to significantly improve its efficiency (under heuristic assumptions).~~		Ed is what individuals call me and my spouse doesn't like it at all. To play lacross is some thing I really appreciate doing. Ohio is where his house is and his family members loves it. I am presently a journey agent.<br><br>My web blog ... telephone psychic [[http://kard.dk/?p=24252 click]]

	~~==Details==~~
	~~The Elkies-Atkin extension to [[Schoof~~'s algorithm]] works by restricting the set of primes <math>S = \{l_1, \ldots, l_s\}</math> considered to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime <math>l</math> is called an Elkies prime if the characteristic equation: <math>\phi^2-t~~\phi+ q = 0</math> splits over <math>\mathbb{F}_l</math>, while an Atkin prime is a prime that is not an Elkies prime~~. Atkin showed how to combine information obtained from the Atkin primes with the information obtained from Elkies primes to produce an efficient algorithm, which came to be known as the Schoof–Elkies–Atkin algorithm. The first problem to address is to determine whether a given prime is Elkies or Atkin. In order to do so, we make use of [[Classical modular curve\|modular polynomials]] <math>\Phi_l(X,Y)</math> that parametrize pairs of <math>l</math>-[[isogeny\|isogenous]] elliptic curves in terms of their [[j-invariant]]s (in practice alternative modular polynomials may also be used but for the same purpose).

	~~If the instantiated polynomial <math>\Phi_l(X,j(E))</math> has a root <math>j(E')</math> in <math>\mathbb{F}_q</math> then <math>l</math>~~ is an Elkies prime, and we may compute a polynomial <math>f_l(X)</math> whose roots correspond to points in the kernel of the <math>l</math>-isogeny from <math>E</math> to <math>E'</math>. The polynomial <math>f_l</math> is ~~a divisor of the corresponding [[Division polynomials\|division polynomial]] used in Schoof's algorithm,~~ and it ~~has significantly lower degree, <math>O(l)</math> versus <math>O(l^2)</math>~~. ~~For Elkies primes, this allows one to compute the number of points on <math>E</math> modulo <math>l</math> more efficiently than in Schoof's algorithm.~~
	In the case of an Atkin prime, we can gain some information from the factorization pattern of <math>\Phi_l(X,j(E))</math> in <math>\mathbb{F}_l[X]</math>, which constrains the possibilities for the number of points modulo <math>l</math>, but the asymptotic complexity of the algorithm depends entirely on the Elkies primes. Provided there are sufficiently many small Elkies primes (on average, we expect half the primes <math>l</math> to be Elkies primes), this results in a ~~reduction in the running time~~. ~~The resulting algorithm is probabilistic (of [[Las Vegas algorithm\|Las Vegas]] type), and its expected running time is, heuristically,~~ <~~math~~>~~\tilde{O}(\log^4 q)~~<~~/math~~>~~, making it more efficient in practice than Schoof's algorithm.~~
	~~{{reflist}}~~

	~~==Implementations==~~
	~~Schoof–Elkies–Atkin algorithm is implemented in the [[PARI/GP]] computer algebra system in the GP function ellap.~~

	~~==External links==~~
	* [http://archive.numdam.~~org/ARCHIVE/JTNB/JTNB_1995__7_1/JTNB_1995__7_1_219_0/JTNB_1995__7_1_219_0~~.~~pdf "Counting points on elliptic curves over finite fields"]~~
	* [http://mathworld.wolfram.com/Schoof-Elkies-AtkinAlgorithm.~~html article on [~~[~~Mathworld]]]~~
	* [http://~~www.ams~~.~~org~~/~~mcom/1998-67-223/S0025-5718-98-00962-4/home.html "Remarks on the Schoof-Elkies-Atkin algorithm"]~~
	* [http://www.esat.kuleuven.ac.be/cosic/eurocrypt2000/pdf/fre-sea.pdf "The Schoof-Elkies-Atkin Algorithm in Characteristic 2"]

	~~{{DEFAULTSORT:Schoof-Elkies-Atkin algorithm}}~~
	~~[[Category:Asymmetric-key algorithms]]~~
	~~[[Category:Elliptic curve cryptography]]~~
	~~[[Category:Group theory]]~~
	~~[[Category:Finite fields]]~~
	~~[[Category:Number theory~~]]

132.64.180.71: /* More equations */

2013-10-23T12:15:38Z

More equations

← Older revision		Revision as of 14:15, 23 October 2013
Line 1:		Line 1:
	The ~~author~~ is ~~known as Wilber Pegues~~. ~~Some time ago she chose to reside~~ in ~~Alaska~~ and ~~her parents live close~~ by. ~~To play lacross~~ is ~~some thing he would never give up~~. ~~She functions~~ as a ~~travel agent~~ but ~~quickly she~~'~~ll be~~ on ~~her own~~.<br><br>~~my webpage~~ :: ~~real psychics (~~[http://~~help~~.~~ksu~~.~~edu~~.sa/~~node~~/~~65129 help~~.~~ksu~~.~~edu~~.sa])		The '''Schoof–Elkies–Atkin algorithm (SEA)''' is an [[algorithm]] used for finding the [[order (group theory)\|order]] of or calculating the number of points on an [[elliptic curve]] over a [[finite field]]. Its primary application is in [[elliptic curve cryptography]]. The algorithm is an extension of [[Schoof's algorithm]] by [[Noam Elkies]] and [[A. O. L. Atkin]] to significantly improve its efficiency (under heuristic assumptions).

			==Details==
			The Elkies-Atkin extension to [[Schoof's algorithm]] works by restricting the set of primes <math>S = \{l_1, \ldots, l_s\}</math> considered to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime <math>l</math> is called an Elkies prime if the characteristic equation: <math>\phi^2-t\phi+ q = 0</math> splits over <math>\mathbb{F}_l</math>, while an Atkin prime is a prime that is not an Elkies prime. Atkin showed how to combine information obtained from the Atkin primes with the information obtained from Elkies primes to produce an efficient algorithm, which came to be known as the Schoof–Elkies–Atkin algorithm. The first problem to address is to determine whether a given prime is Elkies or Atkin. In order to do so, we make use of [[Classical modular curve\|modular polynomials]] <math>\Phi_l(X,Y)</math> that parametrize pairs of <math>l</math>-[[isogeny\|isogenous]] elliptic curves in terms of their [[j-invariant]]s (in practice alternative modular polynomials may also be used but for the same purpose).

			If the instantiated polynomial <math>\Phi_l(X,j(E))</math> has a root <math>j(E')</math> in <math>\mathbb{F}_q</math> then <math>l</math> is an Elkies prime, and we may compute a polynomial <math>f_l(X)</math> whose roots correspond to points in the kernel of the <math>l</math>-isogeny from <math>E</math> to <math>E'</math>. The polynomial <math>f_l</math> is a divisor of the corresponding [[Division polynomials\|division polynomial]] used in Schoof's algorithm, and it has significantly lower degree, <math>O(l)</math> versus <math>O(l^2)</math>. For Elkies primes, this allows one to compute the number of points on <math>E</math> modulo <math>l</math> more efficiently than in Schoof's algorithm.
			In the case of an Atkin prime, we can gain some information from the factorization pattern of <math>\Phi_l(X,j(E))</math> in <math>\mathbb{F}_l[X]</math>, which constrains the possibilities for the number of points modulo <math>l</math>, but the asymptotic complexity of the algorithm depends entirely on the Elkies primes. Provided there are sufficiently many small Elkies primes (on average, we expect half the primes <math>l</math> to be Elkies primes), this results in a reduction in the running time. The resulting algorithm is probabilistic (of [[Las Vegas algorithm\|Las Vegas]] type), and its expected running time is, heuristically, <math>\tilde{O}(\log^4 q)</math>, making it more efficient in practice than Schoof's algorithm.
			{{reflist}}

			==Implementations==
			Schoof–Elkies–Atkin algorithm is implemented in the [[PARI/GP]] computer algebra system in the GP function ellap.

			==External links==
			* [http://archive.numdam.org/ARCHIVE/JTNB/JTNB_1995__7_1/JTNB_1995__7_1_219_0/JTNB_1995__7_1_219_0.pdf "Counting points on elliptic curves over finite fields"]
			* [http://mathworld.wolfram.com/Schoof-Elkies-AtkinAlgorithm.html article on [[Mathworld]]]
			* [http://www.ams.org/mcom/1998-67-223/S0025-5718-98-00962-4/home.html "Remarks on the Schoof-Elkies-Atkin algorithm"]
			* [http://www.esat.kuleuven.ac.be/cosic/eurocrypt2000/pdf/fre-sea.pdf "The Schoof-Elkies-Atkin Algorithm in Characteristic 2"]

			{{DEFAULTSORT:Schoof-Elkies-Atkin algorithm}}
			[[Category:Asymmetric-key algorithms]]
			[[Category:Elliptic curve cryptography]]
			[[Category:Group theory]]
			[[Category:Finite fields]]
			[[Category:Number theory]]

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-10T12:53:41Z

ISBNs (Build KE)

New page

The author is known as Wilber Pegues. Some time ago she chose to reside in Alaska and her parents live close by. To play lacross is some thing he would never give up. She functions as a travel agent but quickly she'll be on her own.<br><br>my webpage :: real psychics ([http://help.ksu.edu.sa/node/65129 help.ksu.edu.sa])