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Topic: Elliptic curve

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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 (Site not responding. Last check: 2007-11-05)
		In mathematics, elliptic curves are defined by certain cubic (third degree) equations.
		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.
	www.sciencedaily.com /encyclopedia/elliptic_curve (880 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)

	Elliptic Curves and Elliptic Functions
		This mapping is, in effect, a parameterization of the elliptic curve by points in a "fundamental parallelogram" in the complex plane.
		We're mentioning all this to emphasize that an elliptic curve is just a special case of a much more general class of objects that have been studied quite extensively in the general setting.
		Further, the definition of an elliptic curve requires that there are no repeated roots of the polynomial in x, and this may fail to be true when reducing mod p for some primes.
	cgd.best.vwh.net /home/flt/flt03.htm (3513 words)

	PlanetMath: elliptic curve (Site not responding. Last check: 2007-11-05)
		These pictures are in some sense not representative of most of the elliptic curves that people work with, since many of the interesting cases tend to be of elliptic curves over algebraically closed fields.
		The points on an elliptic curve have a natural group structure, which makes the elliptic curve into an abelian variety.
		This is version 28 of elliptic curve, born on 2001-12-12, modified 2005-05-21.
	planetmath.org /encyclopedia/EllipticCurve.html (561 words)

	Elliptic curves (Site not responding. Last check: 2007-11-05)
		On the conjecture of Birch and Swinnerton-Dyer for elliptic curves, by Cristian D. Gonzalez-Aviles.
		Beppo Levi and the arithmetic of elliptic curves, by Norbert Schappacher.
		On an elliptic analogue of Zagier's conjecture, Jörg Wildeshaus.
	www.fermigier.com /fermigier/elliptic.html.en (746 words)

	Elliptic curve cryptography (in Technology > Encryption @ iusmentis.com)
		Elliptic curve cryptography was invented by Neil Koblitz in 1987 and by Victor Miller in 1986.
		Although no general patent on elliptic curve cryptography appears to exist, there are several patents that may be relevant depending on the implementation (US 5,159,632, US 5,271,061, US 5,463,690 and US 6,141,420).
		The trick with elliptic curve cryptography is that if you have a point F on the curve, all multiples of this point are also on the curve.
	www.iusmentis.com /technology/encryption/elliptic-curves (632 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 (6037 words)

	Q31: What are Elliptic Curve Cryptosystems? (Site not responding. Last check: 2007-11-05)
		The curves used in elliptic curve analogs of discrete logarithm cryptosystems are normally of the form
		It is possible that algorithm development in this area will change the security of elliptic curve discrete logarithm cryptosystems to be equivalent to that of general discrete logarithm cryptosystems; this is an open research problem.
		Elliptic curve analogs of RSA have been proposed, and they are based on the difficulty of factoring, just as RSA is. The elliptic curve analogs do not seem to offer any significant advantage over RSA, as the underlying problem is the same and the key sizes are similar for equivalent levels of security.
	www.x5.net /faqs/crypto/q31.html (236 words)

	Elliptic Curve Cryptography according to Steven Galbraith (Site not responding. Last check: 2007-11-05)
		Elliptic curve cryptography (ECC) was proposed by Victor Miller and Neal Koblitz in the mid 1980s.
		An elliptic curve is the set of solutions (x,y) to an equation of the form y^2 = x^3 + Ax + B, together with an extra point O which is called the point at infinity.
		The first family of elliptic curve discrete logarithm problems which can be solved in subexponential time was the case of supersingular curves.
	www.isg.rhul.ac.uk /~sdg/ecc.html (2173 words)

	Elliptic Curves and Elliptic Functions
		Especially so, because there are many other noteworthy analytic properties of elliptic functions that were discovered in the 19th century by Weiestrass and others and which we haven't even mentioned.
		If the equation of E has rational but non-integral coefficients, we would need to assume none of their denominators are divisible by p, so we might as well assume all coefficients to be integral to begin with (since if the denomimators are prime to p they have inverses mod p).
		Returning to the j-invariant, it is the 1:1 map betweem isomorphism classes of elliptic curves and C*.
	www.mbay.net /%7Ecgd/flt/flt03.htm (3513 words)

	Elliptic Curve Database
		Magma includes John Cremona's database of all elliptic curves over Q of small conductor (up to 10 000 [Note that at the moment the ordering of these curves is only considered canonical up to conductor 7000, with the usual exception for conductor 990.] at V2.8 and above).
		The elliptic curve database uses an internal buffer to cache disk reads; if the buffer is large enough then the entire file can be cached and will not need to be read from the disk more than once.
		The sequence of elliptic curves for conductor N from the database.
	www.math.niu.edu /help/math/magmahelp/text1015.html (849 words)

	Elliptic Curve Data
		The optimal curve can be determined in any individual case, but this would take a long time to do for all remaining cases.
		For all N up to 22000 the optimal Gamma_0(N) curve is the one labelled 1 (except for class 960h when it is the curve labelled 3).
		For each curve of positive rank, generators are given for the Mordell group modulo torsion, in projective coordinates.
	www.maths.nott.ac.uk /personal/jec/ftp/data/INDEX.html (1196 words)

	Cryptomathic Labs - Elliptic curve demo (Site not responding. Last check: 2007-11-05)
		The generation of suitable EC parameters is the major obstacle in an elliptic curve environment.
		Advanced mathematics seems to be required in order to obtain curve E and base point G which are not vulnerable to attacks on the discrete logarithm problem.
		For cryptographic curves, the field size is typically about 200 and the number of points has to satisfy additional criteria to be cryptographically useful.
	www.cryptomathic.com /labs/ellipticcurvedemo.html (440 words)

	Amazon.com: Books: Guide to Elliptic Curve Cryptography (Site not responding. Last check: 2007-11-05)
		Anchored by a comprehensive treatment of the practical aspects of elliptic curve cryptography (ECC), this guide explains the basic mathematics, describes state-of-the-art implementation methods, and presents standardized protocols for public-key encryption, digital signatures, and key establishment.
		Elliptic curves have a rich and beautiful history, having been studied by mathematicians for over a hundred years.
		It does not have any of the juicy ellpitic curve mathematics, but that is okay as this book is directed towards engineers and others who want to learn about how elliptic curve cryptosystems are being deployed.
	www.amazon.com /exec/obidos/tg/detail/-/038795273X?v=glance (1230 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)

	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.
	cnscenter.future.co.kr /crypto/algorithm/ecc.html (3215 words)

	elliptic_curve<bigint> (Site not responding. Last check: 2007-11-05)
		An elliptic curve is a curve of the form
		This class is a specialization of the template class elliptic_curve and is for working with a minimal model of an elliptic curve defined over the rationales.
		This curve provides access to the reduction type at all primes, the conductor etc. This is extracted using Tate's algorithm after the curve has been reduced to a minimal model using Laska-Kraus-Connell reduction.
	www.math.psu.edu /local_doc/LiDIA/node105.html (420 words)

	Science: Mathematics: Number Theory: Elliptic Curves - Open Site
		A projective curve defined over a field F of genus one and with a point rational over F. An elliptic curve can always be written as a non-singular projective plane cubic curve and this is an equivalent definition.
		Let E be an elliptic curve defined over a number field K. The set E(K) of points on E with coordinates in K forms a finitely generated abelian group.
		The rank of the elliptic curve is the rank of the torsion-free part.
	open-site.org /Science/Mathematics/Number_Theory/Elliptic_Curves (302 words)

	Elliptic Curves
		Elliptic Functions and Elliptic Integrals by Viktor Prasolov and Yuri Solovyev (nice introduction to elliptic curves, functions and integrals).
		Surveys on Iwasawa theory of elliptic curves by Ralph Greenberg.
		Connell's Handbook of elliptic curves is an ambitious project and still uncomplete.
	www.rzuser.uni-heidelberg.de /~hb3/elleng.html (928 words)

	Shamus Software Ltd - MIRACL
		Curves generated using these free executables may be used for commercial purposes.
		A quarter of all randomly generated curves can be transformed into this form.The former condition makes it easier to find points on the curve, and the latter make calculations on the curve somewhat faster.
		It is thought that by using such curves the user is safe against cryptanalytic advances, except in a circumstance where the whole premise behind Elliptic Curve cryptography collapses and a sub-exponential solution is found for the most general discrete logarithm problem in the elliptic curve setting.
	indigo.ie /~mscott (2781 words)

	Knapp, A.W.: Elliptic Curves. (MN-40).
		An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group.
		The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem.
		Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match.
	pup.princeton.edu /titles/5272.html (272 words)

	Elliptic Curves and Cryptology
		Elliptic curves and related topics, CRM Proceedings & Lecture Notes 4, American Mathematical Society, 1994.
		Elliptic curves and cryptography: A pseudorandom bit generator and other tools, Ph.D. Thesis, MIT, 1988.
		This program uses 2-descent (via 2-isogeny if possible) to determine the rank of an elliptic curve E over Q, list a set of points which generate E(Q) modulo 2E(Q), and finally search for further points on the curve.
	www.geocities.com /CapeCanaveral/Launchpad/9160/biblio_ell.html (1215 words)

	Next Generation Crypto
		Elliptic Curve Cryptography (ECC) is emerging as an attractive public-key cryptosystem for mobile/wireless environments.
		Elliptic Curve Diffie-Hellman Key Exchange for the SSH Transport Level Protocol, IETF Internet-draft draft-stebila-secsh-ecdh-00.txt specifying the use of Elliptic Curve Cryptography with SSH, Nov.
		Elliptic Curve Cryptography: The Next Generation of Internet Security, a white paper describing how Elliptic Curve Cryptography is an ideal match for the Internet's future security needs.
	research.sun.com /projects/crypto (833 words)

	libecc: C++ Elliptic Curve Library
		A start has been made in determining the relationship between the curve parameter a and the structure of the Abelian groups formed by the points on the elliptic curves.
		You probably thought that I was contacted by the CIA who forced me to stop breaking their backdoors in the FIPS provided elliptic curves, but no that is not anything like it.
		In order to do that, one needs to be able to determine the number of solutions of some given elliptic curve over GF(2^m), and the mathematics involved with that are so difficult that I had to gave up.
	libecc.sourceforge.net (1018 words)

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