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Guide to Elliptic Curve Cryptography
, 2004
"... Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves ..."
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Cited by 545 (19 self)
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Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves also figured prominently in the recent proof of Fermat's Last Theorem by Andrew Wiles. Originally pursued for purely aesthetic reasons, elliptic curves have recently been utilized in devising algorithms for factoring integers, primality proving, and in publickey cryptography. In this article, we aim to give the reader an introduction to elliptic curve cryptosystems, and to demonstrate why these systems provide relatively small block sizes, highspeed software and hardware implementations, and offer the highest strengthperkeybit of any known publickey scheme.
Pairingbased Cryptography at High Security Levels
 Proceedings of Cryptography and Coding 2005, volume 3796 of LNCS
, 2005
"... Abstract. In recent years cryptographic protocols based on the Weil and Tate pairings on elliptic curves have attracted much attention. A notable success in this area was the elegant solution by Boneh and Franklin [7] of the problem of efficient identitybased encryption. At the same time, the secur ..."
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Cited by 89 (3 self)
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Abstract. In recent years cryptographic protocols based on the Weil and Tate pairings on elliptic curves have attracted much attention. A notable success in this area was the elegant solution by Boneh and Franklin [7] of the problem of efficient identitybased encryption. At the same time, the security standards for public key cryptosystems are expected to increase, so that in the future they will be capable of providing security equivalent to 128, 192, or 256bit AES keys. In this paper we examine the implications of heightened security needs for pairingbased cryptosystems. We first describe three different reasons why highsecurity users might have concerns about the longterm viability of these systems. However, in our view none of the risks inherent in pairingbased systems are sufficiently serious to warrant pulling them from the shelves. We next discuss two families of elliptic curves E for use in pairingbased cryptosystems. The first has the property that the pairing takes values in the prime field Fp over which the curve is defined; the second family consists of supersingular curves with embedding degree k = 2. Finally, we examine the efficiency of the Weil pairing as opposed to the Tate pairing and compare a range of choices of embedding degree k, including k = 1 and k = 24. Let E be the elliptic curve 1.
Elliptic curves and related sequences
, 2003
"... A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic divisibility sequences or EDSs are an important special case where λ1 = h2 2, λ2 = −h1 h3, the hn are integers ..."
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Cited by 27 (3 self)
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A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic divisibility sequences or EDSs are an important special case where λ1 = h2 2, λ2 = −h1 h3, the hn are integers and hn divides hm whenever n divides m. Somos (4) is the particular Somos 4 sequence whose coefficients λi and initial values are all 1. In this thesis we study the properties of EDSs and Somos 4 sequences reduced modulo a prime power pr. In chapter 2 we collect some results from number theory, and in chapter 3 we give a brief introduction to elliptic curves. In chapter 4 we introduce elliptic divisibility sequences, describe their relationship with elliptic curves, and outline what is known about the properties of an EDS modulo a prime power pr (work by Morgan Ward and Rachel Shipsey). In chapter 5 we extend the EDS “symmetry formulae ” of Ward and Shipsey
Elliptic divisibility sequences and the elliptic curve discrete logarithm problem
, 2008
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ARITHMETIC OF ELLIPTIC CURVES Elliptic Curves
, 2015
"... ellipticcurve groups (over finite fields) in cryptosystems. Use of supersingular curves discarded after the proposal of the Menezes–Okamoto–Vanstone (1993) or Frey–Rück (1994) attack. ECDSA was proposed by Johnson and Menezes (1999) and adopted as a digital signature standard. Use of pairing in ne ..."
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ellipticcurve groups (over finite fields) in cryptosystems. Use of supersingular curves discarded after the proposal of the Menezes–Okamoto–Vanstone (1993) or Frey–Rück (1994) attack. ECDSA was proposed by Johnson and Menezes (1999) and adopted as a digital signature standard. Use of pairing in new protocols Sakai–Ohgishi–Kasahara twoparty key agreement (2000) Boneh–Franklin identitybased encryption (2001) Joux threeparty key agreement (2004) Boneh–Lynn–Shacham short signature scheme (2004) Numerous other applications of pairing after this.
Powered Tate pairing computation
, 2005
"... In this paper, we introduce a powered Tate pairing on a supersingular elliptic curve that has the same shortened loop as the modified Tate pairing using the eta pairing approach by Barreto et al. The main significance of our approach is to remove the condition which the latter should rely on. It ..."
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In this paper, we introduce a powered Tate pairing on a supersingular elliptic curve that has the same shortened loop as the modified Tate pairing using the eta pairing approach by Barreto et al. The main significance of our approach is to remove the condition which the latter should rely on. It implies that our method is simpler and potentially general than the eta pairing approach, although they are equivalent in most practical cases.