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# Topic: Elliptic curve cryptography

###### In the News (Mon 20 May 13)

 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, by Andrew Wiles (assisted by one of his former PhD students, Richard Taylor), 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   (1467 words)

 Elliptic curve cryptography - Wikipedia, the free encyclopedia Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. Elliptic curves are also used in several integer factorization algorithms that have applications in cryptography, such as, for instance, Lenstra elliptic curve factorization, but this use of elliptic curves is not usually referred to as "elliptic curve cryptography." The elliptic curve is defined by the constants a and b used in its defining equation. en.wikipedia.org /wiki/Elliptic_curve_cryptography   (2484 words)

 ipedia.com: Elliptic curve Article   (Site not responding. Last check: 2007-11-06) 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.ipedia.com /elliptic_curve.html   (887 words)

 Elliptic Curve Cryptography according to Steven Galbraith   (Site not responding. Last check: 2007-11-06) 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   (2191 words)

 Elliptic Curve Cryptography   (Site not responding. Last check: 2007-11-06) ECC is quite computationally intensive and if it is not carefully programmed running times can blow out to the order of minutes. ECC is not a mature cipher and as such many competing algorithms exist for different aspects. Elliptic curves are defined by cubic equations similar to those that specify an ellipse, hence the name. www.eng.newcastle.edu.au /~c2104330/ECC.htm   (835 words)

 An intro to Elliptical Curve 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. ECC arranges itself so that when you wish to perform an operation the cryptosystem should make easy — encrypting a message with the public key, decrypting it with the private key — the operation you are performing is point multiplication. 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   (5984 words)

 Elliptic curve cryptography (in Technology > Encryption @ iusmentis.com) Public key cryptography systems are usually based on the assumption that a particular mathematical operation is easy to do, but difficult to undo unless you know some particular secret. Elliptic curve cryptography was invented by Neil Koblitz in 1987 and by Victor Miller in 1986. 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)

 The Case for Elliptic Curve Cryptography   (Site not responding. Last check: 2007-11-06) Since their use in cryptography was discovered in 1985, elliptic curve cryptography has also been an active area of study in academia. Despite the many advantages of elliptic curves and despite the adoption of elliptic curves by many users, many vendors and academics view the intellectual property environment surrounding elliptic curves as a major roadblock to their implementation and use. In the elliptic curve case, there is actually one additional bit that needs to be transmitted in each direction which allows the recovery of both the x and y coordinates of an elliptic curve point. www.nsa.gov /ia/industry/crypto_elliptic_curve.cfm?MenuID=10.2.7   (1818 words)

 [No title]   (Site not responding. Last check: 2007-11-06) Furthermore, this document specifies formats to indicate that an elliptic curve public key is to be restricted for use with an indicated set of elliptic curve cryptography algorithms. Elliptic domain parameters usually include further information such as order of the base point, a number called the cofactor, a value called seed which is used to select the curve, and possibly the base point, verifiably at random. For certificates containing elliptic curve subject public keys, or certificates signed with elliptic curve issuer public keys using ECDSA, it is often necessary to identify the particular ECC algorithms and elliptic curve domain parameters that are used. www.ietf.org /internet-drafts/draft-ietf-pkix-ecc-pkalgs-02.txt   (2721 words)

 CACR: 1999 Conferences ECC '99 is the third in a series of annual workshops dedicated to the study of elliptic curve cryptography. ECC '99 will have a broader scope than ECC '98 and ECC '97, which focussed primarily on the elliptic curve discrete logarithm problem. It is hoped that the meeting will 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/1999/ecc99/ecc99-announce.html   (912 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 wins more converts   (Site not responding. Last check: 2007-11-06) 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. ECC was used in a Treasury Department/MasterCard pilot last summer and has been named as an acceptable option under the General Services Administration's Access Certificates for Electronic Services program, which seeks to establish the conditions for secure online citizen-to-government connectivity. www.fcw.com:8443 /article67443-03-14-99-Print   (573 words)

 CACR: 2004 Conferences ECC 2004 is the eighth in a series of annual workshops dedicated to the study of elliptic curve cryptography and related areas. 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. For the first time the ECC workshop will be held together with a summer school on elliptic curve cryptography. www.cacr.math.uwaterloo.ca /conferences/2004/ecc2004/announcement.html   (822 words)

 CFP: Workshop on Elliptic Curve Cryptography   (Site not responding. Last check: 2007-11-06) THE 9TH WORKSHOP ON ELLIPTIC CURVE CRYPTOGRAPHY (ECC 2005) Technical University of Denmark, Copenhagen September 19-21, 2005 First Announcement March 5, 2005 ECC 2005 is the ninth in a series of annual workshops dedicated to the study of elliptic curve cryptography and related areas. At the same time ECC continues to be the premier conference on elliptic curve cryptography. It is hoped that ECC 2005 will further our mission of encouraging and stimulating research on the security and implementation of elliptic curve cryptosystems and related areas, and encouraging collaboration between mathematicians, computer scientists and engineers in the academic, industry and government sectors. www.ieee-security.org /Calendar/cfps/cfp-ECC2005.html   (386 words)

 libecc: C++ Elliptic Curve Library   (Site not responding. Last check: 2007-11-06) Libecc is an elliptic curve crypto library for C++ developers. 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)

 Guide to Elliptic Curve Cryptography - Springer Darrel Hankerson & Alfred J. Menezes & Scott Vanstone   (Site not responding. Last check: 2007-11-06) 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. Guide to Elliptic Curve Cryptography: Information, updates, and errata for the Guide to Elliptic Curve Cryptography, by Hankerson, Menezes, and Vanstone. Bookpool: Guide to Elliptic Curve Cryptography: 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... www.skattabrain.com /css-books-plain/038795273X.html   (415 words)

 Certicom | ECC |   (Site not responding. Last check: 2007-11-06) Dr. Vanstone is one of the leading experts in the field of Elliptic Curve Cryptography (ECC) research, having authored or co-authored over 250 publications on the topic. The research team is very focused on ECC technology, including studying efficient implementations, methods to enhance the security and developing new protocols. The Certicom ECC Challenge was introduced in 1997 to increase industry understanding and appreciation for the difficulty of the elliptic curve discrete logarithm problem, and to encourage and stimulate further research in the security analysis of elliptic curve cryptosystems. www.certicom.com /research.html   (251 words)

 RFC 3278 (rfc3278) - Use of Elliptic Curve Cryptography (ECC) Algorithms i 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. The use of ECC algorithms and keys within X.509 certificates is specified in [PKI- ALG]. www.faqs.org /rfcs/rfc3278.html   (3021 words)

 Dr. Dobb's | Elliptic-Curve Cryptography | July 22, 2001   (Site not responding. Last check: 2007-11-06) Much cryptography, elliptic curve included, is based on the idea of a mathematical group. Different curves are suitable for different purposes, and because every application has different requirements, no elliptic curve can be optimal for all applications. Fortunately, the vast majority of curves are secure, and it is simple and fast to validate the security of a given curve. www.ddj.com /documents/s=896/ddj9912d/9912d.htm   (3863 words)

 Next Generation Crypto ECC Cipher Suites for TLS, IETF Internet-draft specifying the use of Elliptic Curve Cryptography with SSL. 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   (1016 words)

 Amazon.ca: Implementing Elliptic Curve Cryptography: Books   (Site not responding. Last check: 2007-11-06) It would be difficult to understand without having taken a previous course in cryptography, but if you already have some idea of numbers theory, and you need to get a quick feel of ECC this would be a good place to start. 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. www.amazon.ca /exec/obidos/ASIN/1884777694   (651 words)

 Summer School on "Elliptic Curves in Cryptography" in Bochum The European Network of Excellence (ECRYPT) organizes a summer school on elliptic curve cryptography in Bochum, Germany. The workshop is intended for young researchers and practitioners interested in learning the background of elliptic curve cryptography. The topics of the summer school range from the basic ideas of discrete logarithm based cryptography and finite field arithmetic to side-channel attacks and advanced topics such as point counting. www.ruhr-uni-bochum.de /itsc/tanja/summerschool   (501 words)

 TinyECC: Elliptic Curve Cryptography on TinyOS Note that the code size changes with different parameters due to the changes in the ECC parameters and some internal state. It is efficient to implement elliptic curve operations in projective coordinates instead of affine coordinates. For all NIST and most SECG curves, the underlying field primes p were chosen as pseudo-Mersenne primes to allow for optimized modular reduction[2]. discovery.csc.ncsu.edu /~pning/software/TinyECC   (910 words)

 Manning: Implementing Elliptic Curve Cryptography The main purpose of Implementing Elliptic Curve Cryptography is to help "crypto engineers" implement functioning, state-of-the-art cryptographic algorithms in the minimum time. Implementing Elliptic Curve Cryptography assumes the reader has at least a high school background in algebra, but it explains, in stepwise fashion, what has been considered to be a topic only for graduate-level students. The formulas and images in Elliptic Curve Cryptography make conversion to HTML impractical, so we've converted Chapter 5 to PDF format for you to sample. www.manning.com /rosing   (428 words)

 Elliptic curve cryptography In this paper we introduce elliptic curves and their arithmetic. We show how such mathematical objects are realised in a computer and then go on to describe some of the cryptographic protocols one can implement with elliptic curves. Since elliptic curves are particularly interesting in constrained environments such as smart cards we also describe some of the side-channel defences which have been proposed for elliptic curve systems. www.cs.bris.ac.uk /Publications/pub_info.jsp?id=2000419   (105 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)

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