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Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Speeding Up Pollard's Rho Method For Computing Discrete Logarithms
, 1998
"... . In Pollard's rho method, an iterating function f is used to define a sequence (y i ) by y i+1 = f(y i ) for i = 0; 1; 2; : : : , with some starting value y 0 . In this paper, we define and discuss new iterating functions for computing discrete logarithms with the rho method. We compare their pe ..."
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Cited by 44 (7 self)
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. In Pollard's rho method, an iterating function f is used to define a sequence (y i ) by y i+1 = f(y i ) for i = 0; 1; 2; : : : , with some starting value y 0 . In this paper, we define and discuss new iterating functions for computing discrete logarithms with the rho method. We compare their performances in experiments with elliptic curve groups. Our experiments show that one of our newly defined functions is expected to reduce the number of steps by a factor of approximately 0:8, in comparison with Pollard's originally used function, and we show that this holds independently of the size of the group order. For group orders large enough such that the run time for precomputation can be neglected, this means a realtime speedup of more than 1:2. 1. Introduction Let G be a finite cyclic group, written multiplicatively, and generated by the group element g. Given an element h in G, we wish to find the least nonnegative number x such that g x = h. This problem is the discre...
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 41 (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.
SquareRoot Algorithms For The Discrete Logarithm Problem (a Survey)
 In Public Key Cryptography and Computational Number Theory, Walter de Gruyter
, 2001
"... The best algorithms to compute discrete logarithms in arbitrary groups (of prime order) are the babystep giantstep method, the rho method and the kangaroo method. The first two have (expected) running time O( p n) group operations (n denoting the group order), thereby matching Shoup's lower bounds ..."
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Cited by 27 (0 self)
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The best algorithms to compute discrete logarithms in arbitrary groups (of prime order) are the babystep giantstep method, the rho method and the kangaroo method. The first two have (expected) running time O( p n) group operations (n denoting the group order), thereby matching Shoup's lower bounds. While the babystep giantstep method is deterministic but with large memory requirements, the rho and the kangaroo method are probabilistic but can be implemented very space efficiently, and they can be parallelized with linear speedup. In this paper, we present the state of the art in these methods.
On some computational problems in finite abelian groups
 Mathematics of Computation
, 1997
"... Abstract. We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups. We estimate the computational complexity and storage requirements, and we explicitly determine the Oconstants and Ωconstants. We implemented the algorithms for class ..."
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Cited by 23 (7 self)
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Abstract. We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups. We estimate the computational complexity and storage requirements, and we explicitly determine the Oconstants and Ωconstants. We implemented the algorithms for class groups of imaginary quadratic orders and present a selection of our experimental results. Our algorithms are based on a modification of Shanks ’ babystep giantstep strategy, and have the advantage that their computational complexity and storage requirements are relative to the actual order, discrete logarithm, or size of the group, rather than relative to an upper bound on the group order. 1.
Factorization of the tenth and eleventh Fermat numbers
, 1996
"... . We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a ..."
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Cited by 17 (8 self)
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. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40digit factor of the tenth Fermat number was found after about 140 Mflopyears of computation. We discuss aspects of the practical implementation of ECM, including the use of specialpurpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the nth Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...
On the Complexity and Efficiency of a New Key Exchange System
, 1993
"... In [2] Buchmann and Williams presented a new public key exchange system based on imaginary quadratic fields. While in that paper the system was described theoretically and its security was discussed in some detail nothing much was said about the practical implementation. In this paper we discuss the ..."
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Cited by 14 (6 self)
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In [2] Buchmann and Williams presented a new public key exchange system based on imaginary quadratic fields. While in that paper the system was described theoretically and its security was discussed in some detail nothing much was said about the practical implementation. In this paper we discuss the practical aspects of the new system, its efficiency and implementation. In particular we study the crucial point of the method: ideal reduction. We suggest a refinement of the well known reduction method which has been implemented on a computer. We present extensive running time statistics and a detailed complexity analysis of the methods involved. The implementation of the reduction procedure on chips is subject of future research. 1 The DiffieHellman scheme In their paper "New Directions in Cryptography" [3] Diffie and Hellman introduced in 1976 the idea of public key exchange. By this method it is possible to communicate a secret key for some cryptosystem over a public insecure chann...
Comments on search procedures for primitive roots
 Math.Comp.66
, 1997
"... Abstract. Let p be an odd prime. Assuming the Extended Riemann Hypothesis, we show how to construct O((log p) 4 (log log p) −3) residues modulo p, one of which must be a primitive root, in deterministic polynomial time. Granting some wellknown character sum bounds, the proof is elementary, leading ..."
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Cited by 10 (0 self)
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Abstract. Let p be an odd prime. Assuming the Extended Riemann Hypothesis, we show how to construct O((log p) 4 (log log p) −3) residues modulo p, one of which must be a primitive root, in deterministic polynomial time. Granting some wellknown character sum bounds, the proof is elementary, leading to an explicit algorithm. 1.
Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 Rapport de Recherche 911, INRIA, Octobre
, 1988
"... . We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual impl ..."
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Cited by 9 (7 self)
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. We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number. IMPLEMENTATION DU TEST DE PRIMALITE D' ATKIN, GOLDWASSER, ET KILIAN R'esum'e. Nous d'ecrivons un algorithme de primalit'e, principalement du `a Atkin, qui utilise les propri'et'es des courbes elliptiques sur les corps finis et la th'eorie de la multiplication complexe. En particulier, nous expliquons comment l'utilisation du corps de classe et du corps de genre permet d'acc'el'erer les calculs. Nous esquissons l'impl'ementati...