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39
Short signatures from the Weil pairing
, 2001
"... Abstract. We introduce a short signature scheme based on the Computational DiffieHellman assumption on certain elliptic and hyperelliptic curves. The signature length is half the size of a DSA signature for a similar level of security. Our short signature scheme is designed for systems where signa ..."
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Cited by 703 (28 self)
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Abstract. We introduce a short signature scheme based on the Computational DiffieHellman assumption on certain elliptic and hyperelliptic curves. The signature length is half the size of a DSA signature for a similar level of security. Our short signature scheme is designed for systems where signatures are typed in by a human or signatures are sent over a lowbandwidth channel. 1
Efficient algorithms for pairingbased cryptosystems
, 2002
"... Abstract. We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing. In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in ..."
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Cited by 353 (25 self)
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Abstract. We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing. In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in larger characteristics. We also propose faster algorithms for scalar multiplication in characteristic 3 and square root extraction over Fpm, the latter technique being also useful in contexts other than that of pairingbased cryptography. 1
Supersingular curves in cryptography
, 2001
"... Frey and Rück gave a method to map the discrete logarithm problem in the divisor class group of a curve over ¢¡ into a finite field discrete logarithm problem in some extension. The discrete logarithm problem in the divisor class group can therefore be solved as long ¥ as is small. In the elliptic ..."
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Cited by 97 (9 self)
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Frey and Rück gave a method to map the discrete logarithm problem in the divisor class group of a curve over ¢¡ into a finite field discrete logarithm problem in some extension. The discrete logarithm problem in the divisor class group can therefore be solved as long ¥ as is small. In the elliptic curve case it is known that for supersingular curves one ¥§¦© ¨ has. In this paper curves of higher genus are studied. Bounds on the possible values ¥ for in the case of supersingular curves are given. Ways to ensure that a curve is not supersingular are also given. 1.
Tate Pairing Implementation for Hyperelliptic Curves y2 xp xþ d
 Advances in Cryptology— Proc. ASIACRYPT ’03
, 2003
"... Abstract. The Weil and Tate pairings have been used recently to build new schemes in cryptography. It is known that the Weil pairing takes longer than twice the running time of the Tate pairing. Hence it is necessary to develop more efficient implementations of the Tate pairing for the practical ap ..."
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Cited by 96 (5 self)
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Abstract. The Weil and Tate pairings have been used recently to build new schemes in cryptography. It is known that the Weil pairing takes longer than twice the running time of the Tate pairing. Hence it is necessary to develop more efficient implementations of the Tate pairing for the practical application of pairing based cryptosystems. In 2002, Barreto et al. and Galbraith et al. provided new algorithms for the fast computation of the Tate pairing in characteristic three. In this paper, we give a closed formula for the Tate pairing on the hyperelliptic curve y2 = xp−x+d in characteristic p. This result improves the implementations in [BKLS02], [GHS02] for the special case p = 3. 1
Efficient arithmetic on Koblitz curves
 Designs, Codes, and Cryptography
, 2000
"... Abstract. It has become increasingly common to implement discretelogarithm based publickey protocols on elliptic curves over finite fields. The basic operation is scalar multiplication: taking a given integer multiple of a given point on the curve. The cost of the protocols depends on that of the ..."
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Cited by 90 (0 self)
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Abstract. It has become increasingly common to implement discretelogarithm based publickey protocols on elliptic curves over finite fields. The basic operation is scalar multiplication: taking a given integer multiple of a given point on the curve. The cost of the protocols depends on that of the elliptic scalar multiplication operation. Koblitz introduced a family of curves which admit especially fast elliptic scalar multiplication. His algorithm was later modified by Meier and Staffelbach. We give an improved version of the algorithm which runs 50 % faster than any previous version. It is based on a new kind of representation of an integer, analogous to certain kinds of binary expansions. We also outline further speedups using precomputation and storage.
Evidence that XTR is more secure than supersingular elliptic curve cryptosystems
 J. Cryptology
, 2001
"... Abstract. We show that finding an efficiently computable injective homomorphism from the XTR subgroup into the group of points over GF(p 2) of a particular type of supersingular elliptic curve is at least as hard as solving the DiffieHellman problem in the XTR subgroup. This provides strong evidenc ..."
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Cited by 90 (5 self)
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Abstract. We show that finding an efficiently computable injective homomorphism from the XTR subgroup into the group of points over GF(p 2) of a particular type of supersingular elliptic curve is at least as hard as solving the DiffieHellman problem in the XTR subgroup. This provides strong evidence for a negative answer to the question posed by S. Vanstone and A. Menezes at the Crypto 2000 Rump Session on the possibility of efficiently inverting the MOV embedding into the XTR subgroup. As a side result we show that the Decision DiffieHellman problem in the group of points on this type of supersingular elliptic curves is efficiently computable, which provides an example of a group where the Decision DiffieHellman problem is simple, while the DiffieHellman and discrete logarithm problem are presumably not. The cryptanalytical tools we use also lead to cryptographic applications of independent interest. These applications are an improvement of Joux’s one round protocol for tripartite DiffieHellman key exchange and a non refutable digital signature scheme that supports escrowable encryption. We also discuss the applicability of our methods to general elliptic curves defined over finite fields. 1
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.
The insecurity of the elliptic curve Digital Signature Algorithm with partially known nonces
 Designs, Codes and Cryptography
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Elliptic curve cryptography: The serpentine course of a paradigm shift
 J. NUMBER THEORY
, 2008
"... Over a period of sixteen years elliptic curve cryptography went from being an approach that many people mistrusted or misunderstood to being a public key technology that enjoys almost unquestioned acceptance. We describe the sometimes surprising twists and turns in this paradigm shift, and compare ..."
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Cited by 21 (5 self)
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Over a period of sixteen years elliptic curve cryptography went from being an approach that many people mistrusted or misunderstood to being a public key technology that enjoys almost unquestioned acceptance. We describe the sometimes surprising twists and turns in this paradigm shift, and compare this story with the commonly accepted Ideal Model of how research and development function in cryptography. We also discuss to what extent the ideas in the literature on “social construction of technology” can contribute to a better understanding of this history.
C.: Itoh–Tsujii inversion in standard basis and its application in cryptography and codes
 Des. Codes Cryptogr
, 2002
"... Abstract. This contribution is concerned with a generalization of Itoh and Tsujii’s algorithm for inversion in extension fields GF (q m). Unlike the original algorithm, the method introduced here uses a standard (or polynomial) basis representation. The inversion method is generalized for standard b ..."
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Cited by 19 (2 self)
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Abstract. This contribution is concerned with a generalization of Itoh and Tsujii’s algorithm for inversion in extension fields GF (q m). Unlike the original algorithm, the method introduced here uses a standard (or polynomial) basis representation. The inversion method is generalized for standard basis representation and relevant complexity expressions are established, consisting of the number of extension field multiplications and exponentiations. As the main contribution, for three important classes of fields we show that the Frobenius map can be explored to perform the exponentiations required for the inversion algorithm efficiently. As an important consequence, Itoh and Tsujii’s inversion method shows almost the same practical complexity for standard basis as for normal basis representation for the field classes considered.