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Topic: Lenstra elliptic curve factorization

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	Elliptic curve
		Elliptic curves are non-singular, meaning they don't have cusps or self-intersections, and a binary operation can be defined for their points in a natural geometric fashion, thus turning the set of points into an abelian group.
		Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1.
		Elliptic curves over finite fields are used in some cryptographic applications as well as for integer factorization.
	www.starrepublic.org /encyclopedia/wikipedia/e/el/elliptic_curve.html (833 words)

	Lenstra elliptic curve factorization - Wikipedia, the free encyclopedia
		Technically the Lenstra elliptic curve factorization (aka Elliptic Curve Factorization Method or ECM) like Pollard's p-1 algorithm is classified as a deterministic algorithm as all "random steps" such as the choice of curves used can be derandomized and done in a deterministic way.
		Currently, it is still the best algorithm for factoring out divisors of size not greatly exceeding 20 to 25 digits (64 to 83 bits or so), as its running time is dominated by the size of a factor p rather than on the size of the number n to be factored.
		Lenstra Jr., H. "Factoring integers with elliptic curves." Annals of Mathematics (2) 126 (1987), 649-673.
	en.wikipedia.org /wiki/Lenstra_elliptic_curve_factorization (854 words)

	Elliptic curve -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-02)
		These curves are not (A closed plane curve resulting from the intersection of a circular cone and a plane cutting completely through it) ellipses: see (Click link for more info and facts about elliptic integral) elliptic integral for the origin of the term.
		Elliptic curves can be defined over any (A piece of land cleared of trees and usually enclosed) field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with ((biology) taxonomic group containing one or more species) genus 1 with a given point defined over K
		Elliptic curves over finite fields are used in some (Click link for more info and facts about cryptographic) cryptographic applications as well as for (Click link for more info and facts about integer factorization) integer factorization.
	www.absoluteastronomy.com /encyclopedia/e/el/elliptic_curve.htm (1011 words)

	php-deluxe.net - description: Elliptic-curve
		One finds that elliptic curves correspond to embeddings of the torus into the complex+projective+plane; such embeddings generalize to arbitrary field+(mathematics)s, and so it is said that elliptic curves are non-singular projective+variety algebraic+curves of genus+(mathematics) 1 over a field+(mathematics) ''K'', together with a distinguished point defined over ''K''.
		Elliptic curves are especially important in number+theory, and constitute a major area of current research; for example, they were used in the proof of Fermat%27s+last+theorem.
		Elliptic curves over finite fields are used in some cryptographic applications as well as for integer+factorization.
	www.php-deluxe.net /wiwimod,index.page,Elliptic-curve.htm (1249 words)

	RSA Security - Overview of Elliptic Curve Cryptosystems.
		However, once it is done, the resulting elliptic curve parameters may be used for multiple users within a group (just as in the case of discrete logarithm cryptosystems) and each user has his or her public/private key pair.
		Today, elliptic curve cryptosystems over a field of characteristic 2 are considered to offer implementational advantages but only time will tell whether the situation of the classical discrete logarithm problem is repeated in the case of elliptic curves, with some fields requiring larger system parameters for the same level of security.
		Interest in elliptic curve cryptosystems is fueled by the appeal of basing a cryptosystem on a different hard problem and the fact that currently such a choice appears to lead to smaller system parameters and key sizes for the same level of security.
	www.rsasecurity.com /rsalabs/node.asp?id=2013 (3408 words)

	[No title]
		"Factoring Integers with elliptic curves." Annals of Mathematics, 126 (1987) 649-673.
		These curves provide an easy source of different groups to choose from while performing the factorization, enough in fact that in practice we can typically expect to find one whose order is not divisible by a large prime or prime power.
		It is interesting to note that it seems to used a pre-fixed sequence of elliptic curves and thus, the performance of the package on factoring a sequence of integers may vary extremely widely depending on what order they are input in.
	ftp.gwdg.de /linux/crypt/math/lenstra-factoring-algorithm.txt (770 words)

	14H52: Elliptic Curves
		This is a fascinating area of algebraic geometry dealing with nonsingular curves of genus 1 -- in English, solutions to equations y^2 = x^3 + A x + B. It turns out to have important connections to number theory and in particular to factorization of ordinary integers (and thus to cryptography).
		Elliptic curves also played a role in the recent resolution of the conjecture known as Fermat's Last Theorem.
		Two (unstructured) equations equations in three unknowns lead to an elliptic curve (although integer points are not fully known).
	www.math.niu.edu /~rusin/known-math/index/14H52.html (750 words)

	Quadratic sieve - Wikpedia (Site not responding. Last check: 2007-09-02)
		It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties.
		Now, Lenstra elliptic curve factorization has the same asymptotic running time as QS (in the case where n has exactly two prime factors of equal size), but in practice, QS is faster since it uses single-precision operations instead of the multi-precision operations used by the elliptic curve method.
		This was the largest published factorization by a general-purpose algorithm, until NFS was used to factor RSA-130, completed April 10, 1996.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Quadratic_sieve (1407 words)

	Lenstra elliptic curve factorization - Encyclopedia Glossary Meaning Explanation Lenstra elliptic curve factorization (Site not responding. Last check: 2007-09-02)
		Lenstra elliptic curve factorization - Encyclopedia Glossary Meaning Explanation Lenstra elliptic curve factorization.
		Here you will find more informations about Lenstra elliptic curve factorization.
		The orginal Lenstra elliptic curve factorization article can be editet
	encyclopedia-glossary.com /en/Lenstra-elliptic-curve-factorization.html (562 words)

	math lessons - Elliptic curve
		In mathematics, elliptic curves are defined by certain cubic (the superscript exponent is three, a.k.a.
		Elliptic curves are by definition non-singular, meaning they don't have cusps or self-intersections, and a binary operation can be defined for their points in a natural geometric fashion, thus turning the set of points into an abelian group.
		Specifically, to every elliptic curve E there exists a lattice L and a corresponding Weierstrass elliptic function
	www.mathdaily.com /lessons/Elliptic_curve (847 words)

	The Probability That The Number Of Points On An Elliptic Curve Over A Finite Field Is Prime - Galbraith, McKee ... (Site not responding. Last check: 2007-09-02)
		The paper gives a formula for the probability that a randomly chosen elliptic curve over a finite field has a prime number of points.
		The paper also gives a formula for the probability that a randomly chosen elliptic curve over a finite field has kq points where k is a small number and where q is a prime.
		Given two elliptic curves E 1 and E 2 we wish to construct a non...
	citeseer.ist.psu.edu /239676.html (454 words)

	rpb097 (Site not responding. Last check: 2007-09-02)
		Lenstra's integer factorization algorithm is asymptotically one of the fastest known algorithms, and is ideally suited for parallel computation.
		This Technical Report was a preliminary (longer) version of Brent [102], and was one of the first publications on Hendrik Lenstra's elliptic curve method.
		In equation (1.1), the "(1+o(1))" factor should be inside the exponential.
	web.comlab.ox.ac.uk /oucl/work/richard.brent/pub/pub097.html (151 words)

	[No title]
		It is often thought that breaking the RSA system is as hard as factoring the public modulus $n$ used in the system, but this has never been proved.
		Some factoring algorithms require negligible memory, while others also require $L sup 1/2$.) Therefore our attack on the RSA cryptosystem, although based on very special assumptions, appears to be the most general one that has been proposed so far and is substantially faster than factoring $n$.
		The Lenstra algorithm can be avoided by utilizing somewhat more involved, although probably more efficient methods (cf.
	www.dtc.umn.edu /~odlyzko/doc/arch/rsa.attack.troff (1560 words)

	Global Effort Solves Elliptic Curve Crypto Challenge (Site not responding. Last check: 2007-09-02)
		This computation, coordinated from INRIA (the French National Institute for Research in Computer Science and Control), has confirmed theoretical predictions that a 97-bit code based on elliptic curves is harder to break than a 512-bit code based on factorization such as RSA (Rivest-Shamir-Adleman).
		To encourage research into cryptographic applications of elliptic curves and thereby strengthen the case for ECC (Elliptic Curve Cryptography) in the heated RSA-vs-ECC debate, Canadian company Certicom launched a series of increasingly difficult challenge problems in November 1997 with prizes worth up to $100,000.
		That is twice as much as used for the recent factorization of a 512-bit RSA number, announced by Herman Te Riele at CWI Amsterdam on 26 August.
	astronomy.swin.edu.au /~pbourke/other/ecc2-97/press4k.html (725 words)

	EUforum - Re: Reducing fractions (Site not responding. Last check: 2007-09-02)
		That means that you can safely ignore numbers up to either 47,453,132 or 67,108,864 (check the spec for a 64-bit floating point number--you'll get 1 additional digit over whatever the mantissa size is).
		Of course, you may well find that one large factoriztion takes longer than a few small factorizations, so you may have to be more clever in deciding when factoring is worth it.
		Algoritms you could investigate include: Quadratic Sieve General Number Field Sieve Lenstra Elliptic Curve Factorization Google will give you lots of results for these, but implementing them seems far from simple (and will likely include needing a good matrix library).
	www.listfilter.com /EUforum/m12540.html (253 words)

	Elliptic Curve Factorization Method (Site not responding. Last check: 2007-09-02)
		A factorization method, abbreviated ECM, which computes a large multiple of a point on a random
		Elliptic Curve modulo the number to be factored
		Montgomery, P. ``Speeding the Pollard and Elliptic Curve Methods of Factorization.'' Math.
	mathserver.sdu.edu.cn /mathency/math/e/e084.htm (121 words)

	Global Effort Solves Elliptic Curve Crypto Challenge (Site not responding. Last check: 2007-09-02)
		Encryption systems based on elliptic curves have been known since the mid-1980s, but have only recently been adopted by leading encryption companies such as RSA Security Inc. Certicom issued its "ECC Challenge" in November 1997, specifying a series of challenges of increasing difficulty.
		The aim of the challenge is to encourage research in the field of elliptic curves and their applications in encryption, and to strengthen arguments in favor of using elliptic curve cryptography instead of systems based on integer factorization.
		Nevertheless the computing power used, around 16,000 MIPS/years, was twice as much as that used for the factorization of RSA-155 announced by Herman Te Riele of CWI (Amsterdam) and his colleagues on 26 August 1999.
	astronomy.swin.edu.au /~pbourke/other/ecc2-97/pressinria.html (628 words)

	Math 124: Elementary Number Theory (Site not responding. Last check: 2007-09-02)
		Clay Math Institute page on the Birch and Swinnerton-Dyer conjecture, with the paper by Wiles.
		Lenstra's Annals paper on the elliptic curve factorization method
		Speeding the Pollard and elliptic curve methods of factorization by Peter L. Montgomery
	modular.fas.harvard.edu /edu/Fall2001/124 (93 words)

	List of algorithms (Site not responding. Last check: 2007-09-02)
		Predictive search: binary like search which factors in magnitude of search term versus the high and low values in the search.
		Integer factorization: breaking an integer into its prime factors
		Shor's algorithm: provides exponential speedup for factorizing a number
	www.sciencedaily.com /encyclopedia/list_of_algorithms (813 words)

	CALENDAR for Mathematics 690Z S05
		Some factorization algorithms I: Pollard’s rho/p-1 factorization algorithms
		Factorization algorithms II: Fermat factorization and continued fraction factorization
		[B-P] Brent,.P and Pollard,J.M. The factorization of F_8 Math.
	orion.math.iastate.edu /linglong/calendarS05.htm (120 words)

	[No title] (Site not responding. Last check: 2007-09-02)
		Algorithmic Number Theory II Primality testing, Solovay-Strassen test, Quadratic Reciprocity.
		Algorithmic Number Theory IV Factorization: Pollard's p-1 test, and Lenstra's elliptic curve algorithm.
		Division Algebras These are some expository articles I wrote on division algebras.
	www.csun.edu /~asethura/notes.html (133 words)

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