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Topic: Elliptic Curve DSA

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Elliptic curve cryptography

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	Elliptic curve - Wikipedia, the free encyclopedia
		One finds that elliptic curves correspond to embeddings of the torus into the complex projective plane; such embeddings generalize to arbitrary fields, and so it is said that elliptic curves are non-singular projective algebraic curves of genus 1 over a field 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's last theorem.
		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 with a given point defined over K.
	en.wikipedia.org /wiki/Elliptic_curve (1278 words)

	Elliptic curve cryptography - Wikipedia, the free encyclopedia
		Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the mathematics of elliptic curves.
		There are several slightly different versions of elliptic curve cryptography, all of which rely on the widely believed difficulty of solving the discrete logarithm problem for the group of an elliptic curve over some finite field.
		Given an elliptic curve E, and a field GF(q), we consider the abelian group of rational points E(q) of the form (x, y), where both x and y are in GF(q), and where the group operation "+" is defined on this curve as described in the article elliptic curve.
	www.wikipedia.org /wiki/Elliptic_curve_cryptography (1054 words)

	Elliptic curve -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-19)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, elliptic curves are defined by certain cubic (the superscript exponent is three, a.k.a.
		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
	www.absoluteastronomy.com /encyclopedia/E/El/Elliptic_curve.htm (1011 words)

	Elliptic Curve DSA - Wikipedia, the free encyclopedia
		Elliptic Curve DSA (ECDSA) is a variant of the Digital Signature Algorithm (DSA) which operates on elliptic curve groups.
		In practical terms, this means the DSA is slower than RSA as a signature scheme.
		However, elliptic curve groups are not vulnerable to a number field sieve attack, so they can be securely implemented with smaller key sizes and can be faster than RSA.
	en.wikipedia.org /wiki/ECDSA (141 words)

	Digital Signature Algorithm - Wikipedia, the free encyclopedia
		A minor revision was issued in 1996 as FIPS 186-1 [2], and the standard was expanded further in 2000 as FIPS 186-2 [3].
		DSA is covered by U.S. Patent 5,231,668, filed July 26, 1991, and attributed to David W. Kravitz, a former NSA employee.
		DSA is similar to the ElGamal signature scheme.
	en.wikipedia.org /wiki/Digital_Signature_Algorithm (403 words)

	[No title]
		elliptic integral for the origin of the term.
		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
		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
	en-cyclopedia.com /wiki/Elliptic_curve (762 words)

	An intro to Elliptical Curve Cryptography
		Elliptic Curve Cryptography, as described above, is a relative of discrete logarithm cryptography.
		In elliptic curve cryptosystems, the elliptic curve is used to define the members of the set over which the group is calculated, as well as the operations between them which define how math works in the group.
		This is the elliptic curve discrete logarithm problem — and this is the inverse operation in the cryptosystem — the one you effectively have to perform to get the plaintext back from the ciphertext, given only the public key.
	www.deviceforge.com /articles/AT4234154468.html (5991 words)

	Elliptic curve cryptography FAQ v1.12 22nd December 1997
		The crucial property of an elliptic curve is that we can define a rule for "adding" two points which are on the curve, to obtain a 3rd point which is also on the curve.
		agree on a (non-secret) elliptic curve and a (non-secret) fixed curve point F. Alice chooses a secret random integer Ak which is her secret key, and publishes the curve point AP = Ak*F as her public key.
		Equivalent C++ elliptic curve code, and the code used to calculate the curve parameters is at http://ds.dial.pipex.com/george.barwood/crypto.htm
	www.cryptoman.com /elliptic.htm (2917 words)

	Amazon.com: Books: Elliptic Curve Public Key Cryptosystems (The International Series in Engineering and Computer ... (Site not responding. Last check: 2007-10-19)
		Elliptic curves have been intensively studied in algebraic geometry and number theory.
		Elliptic curve cryptosystems potentially provide equivalent security to the existing public key schemes, but with shorter key lengths.
		Elliptic Curve Public Key Cryptosystems is a valuable reference resource for researchers in academia, government and industry who are concerned with issues of data security.
	www.amazon.com /exec/obidos/tg/detail/-/0792393686?v=glance (480 words)

	Crypto-Gram: November 15, 1999
		In the elliptic curve group, there is no definition of smoothness, and hence in order to break elliptic curve algorithms you have to use older methods: Pollard's rho, for example.
		They say things like: "Elliptic curves as algebraic/geometric entities have been studied extensively for the past 150 years, and from these studies has emerged a rich and deep theory." They conclude that because of this, we can gain good confidence that new algorithmic advances won't be too devastating.
		Elliptic curve cryptography was invented only in 1985, and has only been really studied seriously for a few years.
	www.counterpane.com /crypto-gram-9911.html (5324 words)

	DSA, RSA, ECDSA (FIPS 186-2)
		There are three FIPS-approved algorithms for generating and verifying digital signatures: Digital Signature Algorithm (DSA), RSA (as specified in ANSI X9.31), and Elliptic Curve DSA (ECDSA; as specified in ANSI X9.62).
		The DSA and ECDSA algorithms are specified and approved in FIPS 186-2 (with Change Notice 1 dated October 5, 2001).
		Elliptic curves recommended for Federal Government use are specified in Appendix 6 of FIPS 186-2 with Change Notice 1 dated October 5, 2001.
	csrc.nist.gov /cryptval/dss.htm (582 words)

	math lessons - Digital Signature Algorithm
		It was proposed by the National Institute of Standards and Technology (NIST) in August 1991 for use in their Digital Signature Standard (DSS), specified in FIPS 186, adopted in 1993.
		DSA is covered by, filed July 26, 1991, and attributed to David W. Kravitz, a former NSA employee.
		DSA is similar to Elgamal discrete logarithm cryptosystem signatures.
	www.mathdaily.com /lessons/DSA (227 words)

	Elliptic Curve DSA ECDSA: An Enhanced DSA (ResearchIndex) (Site not responding. Last check: 2007-10-19)
		Abstract: The Elliptic Curve Digital Signature Algorithm #ECDSA# is the elliptic curve analogue of the Digital Signature Algorithm #DSA#, and is under consideration for standardization by the ANSI X9 committee.
		Unlike the normal discrete logarithm problem and the integer factorization problem, the elliptic curve discrete logarithm problem has no subexponentialtime algorithm.
		For this reason, the strength-perkey -bit is substantially greater in an algorithm that uses elliptic curves.
	citeseer.ist.psu.edu /276964.html (237 words)

	Elliptic curve -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-19)
		Elliptic curve -- Facts, Info, and Encyclopedia article
		For further developments see (Click link for more info and facts about arithmetic of abelian varieties) arithmetic of abelian varieties.
		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)

	Jonathan Hammell \| Presentations (Site not responding. Last check: 2007-10-19)
		DSA is a NIST standard for producing digital signatures.
		Elliptic Curves are rapidly becoming an accepted basis for cryptographic algorithms.
		Elliptic Curve DSA is a proposed NIST standard.
	www.student.math.uwaterloo.ca /~jfhammell/talks.html (211 words)

	Elliptic Curve Cryptography wins more converts (Site not responding. Last check: 2007-10-19)
		Certicom's Elliptic Curve Cryptography (ECC) can be used for encrypting information and for generating a digital signature, which is encrypted data attached to a transaction to identify the sender.
		But because ECC manipulates points on a curve instead of huge prime numbers, as many encryption techniques do, ECC-based signature keys are smaller and faster to calculate, require smaller memory and processing requirements, and offer longer battery life and lower messaging costs, Certicom said.
		"Elliptic Curve tends to be able to use much shorter keys for the same level of security, compared to RSA," said Miles Smid, acting chief of NIST's Computer Security Division.
	www.fcw.com /article67443-03-14-99-Print (573 words)

	The Elliptic Curve Digital Signature Algorithm (ECDSA) - Johnson, Menezes (ResearchIndex) (Site not responding. Last check: 2007-10-19)
		Unlike the ordinary discrete logarithm problem and the integer factorization problem, no subexponential-time algorithm is known for the elliptic curve...
		126 Reducing elliptic curve logarithms to logarithms in a finite..
		13 The xedni calculus and the elliptic curve discrete logarithm..
	citeseer.ist.psu.edu /johnson99elliptic.html (1350 words)

	Elliptic Curve Cryptography
		Elliptic curve public-key cryptosystems - an introduction, E. De Win and B. Preneel, State of the Art in Applied Cryptography, Springer-Verlag, LNCS 1528, pp.131-141, 1998.
		Elliptic curve discrete logarithms and the index calculus, Joseph H. Silverman and Joe Suzuki, Proc.
		Elliptic Curve Cryptography on a Palm OS Device, A. Weimerskirch, C. Paar, and S. Chang Shantz, To appear in proc.
	www.securitytechnet.com /crypto/algorithm/ecc.html (3215 words)

	Elliptic Curve Multisignatures (Site not responding. Last check: 2007-10-19)
		More recently, the Elliptic Curve Cryptosystem (ECC), in which the difficulty of breaking the system is based on the difficulty of computing a discrete logarithm over an elliptic curve, has also been considered to become a standard in the IEEE P1363 project.
		The DSA is one of the ElGamal-type signature schemes based on the discrete logarithm problem.
		Since DSA is a special form of the original ElGamal scheme, which utilizes another parameter q, where q is a prime factor of p-1, all 18 ElGamal-type variants can be easily converted into DSA-type variants.
	www.cstp.umkc.edu /~harnl/paper16/paper16.htm (3303 words)

	Elliptic Curve DSA -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-19)
		Elliptic Curve DSA -- Facts, Info, and Encyclopedia article
		To be secure DSA requires that to be secure against a (Click link for more info and facts about Number Field Sieve) Number Field Sieve attack and to be secure against a (Click link for more info and facts about Baby-step giant-step) Baby-step giant-step attack.
		However (Click link for more info and facts about elliptic curve) elliptic curve groups are not vulnerable to a (Click link for more info and facts about Number Field Sieve) Number Field Sieve attack, so they can be securely implemented with smaller key sizes and can be faster than RSA.
	www.absoluteastronomy.com /encyclopedia/e/el/elliptic_curve_dsa2.htm (133 words)

	Elliptic Curve DSA (ECDSA): An Enhanced DSA from Certicom Corp. White Papers at Builder UK
		Elliptic Curve DSA (ECDSA): An Enhanced DSA from Certicom Corp. White Papers at Builder UK Home
		Unlike the ordinary discrete logarithm problem and the integer factorization problem, no subexponential-time algorithm is known for the elliptic curve discrete logarithm problem.
		For this reason, the strength-per-key-bit is substantially greater in an algorithm that uses elliptic curves.
	uk.builder.com /whitepapers/0,39026692,60012627p-39000945q,00.htm (145 words)

	Re: Sun's Elliptic Curve Technology Contribution to the OpenSSL Project
		Supposedly the Nyberg-Rueppel version is patented but US Patent number 5600725 patents are the discrete logarithm versions and not the elliptic curve versions at explicit claim.
		The DSA version isn't patented at all (it was patented, but...) and is unencumbered since the owners lost a suit against the US Government.
		Methods to represent an elliptic curve point using a normal basis and methods for efficient computation using such representations.
	monkey.org /openbsd/archive/misc/0210/msg00625.html (1228 words)

	RFC 3278 (rfc3278) - Use of Elliptic Curve Cryptography (ECC) Algorithms i
		The method is the elliptic curve analog of the Digital Signature Algorithm (DSA) [FIPS 186-2].
		Ephemeral-static ECDH is the the elliptic curve analog of the ephemeral-static Diffie-Hellman key agreement algorithm specified jointly in the documents [CMS, Section 12.3.1.1] and [CMS-DH].
		The receiving agent then retrieves the static and ephemeral EC public keys of the originator, from the originator and ukm fields as described in Section 3.2.1, and its static EC public key identified in the rid field and checks that the domain parameters are the same.
	www.faqs.org /rfcs/rfc3278.html (3021 words)

	Elliptic Curve Cryptography
		Faster Attacks on Elliptic Curve Cryptosystems, Michael Wiener and Robert Zuccherato, IEEE P1363: Research Contributions, 1999.
		Elliptic curve lifting problem and its applications, H. Kim, J. Cheon and S. Hahn, Proc.
		Usage of Optimal Extension Fields for Elliptic Curve Cryptosystems, T. Kobayashi, K. Aoki, F. Hoshino, K. Kobayashi and H. Morita, Contribution to IEEE P1363a, 1999.
	cnscenter.future.co.kr /crypto/algorithm/ecc.html (3215 words)

	IEEE’s Central Coast Section official newsletter
		When Miller and Koblitz proposed public-key systems using a group of points on an elliptic curve, elliptic curve cryptography was born.
		Although its application it to cryptography is recent, mathematicians have studied elliptic curves since the seventeenth century.
		On a high level, the two discrete logarithm problems are the same; however, the ECDLP (elliptic curves over finite fields) appears to be much more difficult to solve than the DLP (finite fields).
	www.ewh.ieee.org /r6/central_coast/NL_2_4.html (2098 words)

	hp labs : research : information theory : elliptic curve cryptography
		In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems.
		Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes.
		This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems.
	www.hpl.hp.com /research/info_theory/ellipbook.html (200 words)

	Amazon.ca: Books: Implementing Elliptic Curve Cryptography (Site not responding. Last check: 2007-10-19)
		This book is the first I have read on elliptic curves that actually attempts to explain just how they are used in cryptography from a practical standpoint.
		It does not attempt to prove the many interesting properties of elliptic curves but instead concentrates on the computer code that one might use to put in place an elliptic curve cryptosystem.
		The code the author admits could be done in many other ways, but the one he chose I think does its job in instructing the reader just how to implement elliptic curves in cryptography.
	www.amazon.ca /exec/obidos/ASIN/1884777694 (620 words)

	CACR: 2002 Conferences
		ECC 2002 is the sixth in a series of annual workshops dedicated to the study of elliptic curve cryptography and related areas.
		The discrete logarithm and elliptic curve discrete logarithm problems.
		It is hoped that the meeting will continue to encourage and stimulate further research on the security and implementation of elliptic curve cryptosystems and related areas, and encourage collaboration between mathematicians, computer scientists and engineers in the academic, industry and government sectors.
	www.cacr.math.uwaterloo.ca /conferences/2002/ecc2002/announcement.html (970 words)

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