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201
Counting Points on Elliptic Curves Over Finite Fields
, 1995
"... We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large; it is based on Shanks's babystepgiantstep strategy. The second algorithm is very efficient when the endomorphism ring ..."
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Cited by 117 (0 self)
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We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large; it is based on Shanks's babystepgiantstep strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is based on calculations with the torsion points of the elliptic curve [18]. This deterministic polynomial time algorithm was impractical in its original form. We discuss several practical improvements by Atkin and Elkies.
A taxonomy of pairingfriendly elliptic curves
, 2006
"... Elliptic curves with small embedding degree and large primeorder subgroup are key ingredients for implementing pairingbased cryptographic systems. Such “pairingfriendly” curves are rare and thus require specific constructions. In this paper we give a single coherent framework that encompasses all ..."
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Cited by 110 (11 self)
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Elliptic curves with small embedding degree and large primeorder subgroup are key ingredients for implementing pairingbased cryptographic systems. Such “pairingfriendly” curves are rare and thus require specific constructions. In this paper we give a single coherent framework that encompasses all of the constructions of pairingfriendly elliptic curves currently existing in the literature. We also include new constructions of pairingfriendly curves that improve on the previously known constructions for certain embedding degrees. Finally, for all embedding degrees up to 50, we provide recommendations as to which pairingfriendly curves to choose to best satisfy a variety of performance and security requirements.
K.: Signatures for network coding
 In: International Journal on Information and Coding
, 2006
"... Abstract. This paper presents a practical digital signature scheme to be used in conjunction with network coding. This signature scheme seems to be the first example of a homomorphic signature scheme. Furthermore, our idea simultaneously provides authentication and detects malicious nodes that inten ..."
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Cited by 79 (2 self)
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Abstract. This paper presents a practical digital signature scheme to be used in conjunction with network coding. This signature scheme seems to be the first example of a homomorphic signature scheme. Furthermore, our idea simultaneously provides authentication and detects malicious nodes that intentionally corrupt content on the network. 1.
Separating Decision DiffieHellman from DiffieHellman in cryptographic groups
, 2001
"... In many cases, the security of a cryptographic scheme based on DiffieHellman does in fact rely on the hardness of... ..."
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Cited by 74 (0 self)
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In many cases, the security of a cryptographic scheme based on DiffieHellman does in fact rely on the hardness of...
Elliptic Curves Suitable for Pairing Based Cryptography
 Designs, Codes and Cryptography
, 2003
"... We give a method for constructing ordinary elliptic curves over finite prime field Fp with small security parameter k with respect to a prime l dividing the group order #E(Fp) such that p << l² ..."
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Cited by 51 (1 self)
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We give a method for constructing ordinary elliptic curves over finite prime field Fp with small security parameter k with respect to a prime l dividing the group order #E(Fp) such that p << l&sup2;
On the Performance of Signature Schemes based on Elliptic Curves
, 1998
"... . This paper describes a fast software implementation of the elliptic curve version of DSA, as specified in draft standard documents ANSI X9.62 and IEEE P1363. We did the implementations for the fields GF(2 n ), using a standard basis, and GF(p). We discuss various design decisions that have t ..."
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Cited by 46 (2 self)
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. This paper describes a fast software implementation of the elliptic curve version of DSA, as specified in draft standard documents ANSI X9.62 and IEEE P1363. We did the implementations for the fields GF(2 n ), using a standard basis, and GF(p). We discuss various design decisions that have to be made for the operations in the underlying field and the operations on elliptic curve points. In particular, we conclude that it is a good idea to use projective coordinates for GF(p), but not for GF(2 n ). We also extend a number of exponentiation algorithms, that result in considerable speed gains for DSA, to ECDSA, using a signed binary representation. Finally, we present timing results for both types of fields on a PPro200 based PC, for a C/C++ implementation with small assemblylanguage optimizations, and make comparisons to other signature algorithms, such as RSA and DSA. We conclude that for practical sizes of fields and moduli, GF(p) is roughly twice as fast as GF(2 ...
The complexity of class polynomial computation via floating point approximations. ArXiv preprint
, 601
"... Abstract. We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest ..."
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Cited by 45 (7 self)
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Abstract. We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmeticgeometric mean. Under the heuristic assumption, justified by experiments, that the correctness of the result is not perturbed by rounding errors, the algorithm runs in time “p “p ”” 3 2 O Dlog D  M Dlog D  ⊆ O ` Dlog 6+ε D  ´ ⊆ O ` h 2+ε´ for any ε> 0, where D is the CM discriminant, h is the degree of the class polynomial and M(n) is the time needed to multiply two nbit numbers. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials. 1. Motivation and
Parallel Algorithms for Integer Factorisation
"... The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends o ..."
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Cited by 44 (17 self)
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The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorisation algorithms have improved greatly, to the point where it is now easy to factor a 60decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. We describe several algorithms, including the elliptic curve method (ECM), and the multiplepolynomial quadratic sieve (MPQS) algorithm, and discuss their parallel implementation. It turns out that some of the algorithms are very well suited to parallel implementation. Doubling the degree of parallelism (i.e. the amount of hardware devoted to the problem) roughly increases the size of a number which can be factored in a fixed time by 3 decimal digits. Some recent computational results are mentioned – for example, the complete factorisation of the 617decimal digit Fermat number F11 = 2211 + 1 which was accomplished using ECM.
Algorithms for computing isogenies between elliptic curves
 Math. Comp
, 2000
"... Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes sh ..."
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Cited by 41 (7 self)
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Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes’s algorithm. In particular, we describe the use of fast algorithms for performing incremental operations on series. We also insist on the particular case of the characteristic 2. 1.