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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.
Fast Variants of RSA
 CryptoBytes
, 2002
"... We survey four variants of RSA designed to speed up RSA decryption and signing. We only consider variants that are backwards compatible in the sense that a system using one of these variants can interoperate with systems using standard RSA. ..."
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Cited by 17 (1 self)
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We survey four variants of RSA designed to speed up RSA decryption and signing. We only consider variants that are backwards compatible in the sense that a system using one of these variants can interoperate with systems using standard RSA.
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...
Novel Runtime Systems Support for Adaptive Compositional Modeling in PSEs
 Future Gener. Comput. Syst.,21(6):878{895
, 2005
"... Grid infrastructures and computing environments have progressed significantly in the past few years. The vision of truly seamless Grid usage relies on runtime systems support that is cognizant of the operational issues underlying grid computations and, at the same time, is flexible enough to accommo ..."
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Cited by 12 (3 self)
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Grid infrastructures and computing environments have progressed significantly in the past few years. The vision of truly seamless Grid usage relies on runtime systems support that is cognizant of the operational issues underlying grid computations and, at the same time, is flexible enough to accommodate diverse application scenarios. This paper addresses the twin aspects of Grid infrastructure and application support through a novel combination of two computational technologies – Weaves, a sourcelanguage independent parallel runtime compositional framework that operates through reverseanalysis of compiled object files, and runtime recommender systems that aid in dynamic knowledgebased application composition. Domainspecific adaptivity is exploited through a novel compositional system that supports runtime recommendation of code modules and a sophisticated checkpointing and runtime migration solution that can be transparently deployed over Grid infrastructures. A core set of “adaptivity schemas” are provided as templates for adaptive composition of largescale scientific computations. Implementation issues, motivating application contexts, and preliminary results are described. 1
Implementing the Elliptic Curve Method of Factoring in Reconfigurable Hardware
"... A novel portable hardware architecture of the Elliptic Curve Method of factoring, designed and optimized for application in the relation collection step of the Number Field Sieve, is described and analyzed. A comparison with an earlier proofofconcept design by Pelzl, Simka, et al. has been perform ..."
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Cited by 11 (1 self)
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A novel portable hardware architecture of the Elliptic Curve Method of factoring, designed and optimized for application in the relation collection step of the Number Field Sieve, is described and analyzed. A comparison with an earlier proofofconcept design by Pelzl, Simka, et al. has been performed, and a substantial improvement has been demonstrated in terms of both the execution time and the areatime product. The ECM architecture has been ported across five different families of FPGA devices in order to select the family with the best performance to cost ratio. A timing comparison with the highly optimized software implementation, GMPECM, has been performed. Our results indicate that lowcost families of FPGAs, such as Spartan3 and Spartan3E, offer at least an order of magnitude improvement over the same generation of microprocessors in terms of the performance to cost ratio. 1.
ECM using Edwards curves
"... Abstract. This paper introduces GMPEECM, a fast implementation of the ellipticcurve method of factoring integers. GMPEECM is based on, but faster than, the wellknown GMPECM software. The main changes are as follows: (1) use Edwards curves instead of Montgomery curves; (2) use twisted inverted E ..."
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Cited by 3 (1 self)
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Abstract. This paper introduces GMPEECM, a fast implementation of the ellipticcurve method of factoring integers. GMPEECM is based on, but faster than, the wellknown GMPECM software. The main changes are as follows: (1) use Edwards curves instead of Montgomery curves; (2) use twisted inverted Edwards coordinates; (3) use signedslidingwindow addition chains; (4) batch primes to increase the window size; (5) choose curves with small parameters a, d, X1, Y1, Z1; (6) choose curves with larger torsion.
Fast Variants of RSA Abstract
"... We survey three variants of RSA designed to speed up RSA decryption. These variants are backwards compatible in the sense that a system using one of these variants can interoperate with a system using standard RSA. 1 ..."
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We survey three variants of RSA designed to speed up RSA decryption. These variants are backwards compatible in the sense that a system using one of these variants can interoperate with a system using standard RSA. 1
Factorig N = p^r q for Large r
 PROC. OF CRYPTO'99, LNCS 1666
, 1999
"... We present an algorithm for factoring integers of the form N = p^r q for large r. Such integers were previously proposed for various cryptographic applications. When r log p our algorithm runs in polynomial time (in log N ). Hence, we obtain a new class of integers that can be efficiently factored. ..."
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We present an algorithm for factoring integers of the form N = p^r q for large r. Such integers were previously proposed for various cryptographic applications. When r log p our algorithm runs in polynomial time (in log N ). Hence, we obtain a new class of integers that can be efficiently factored. When r p log p the algorithm is asymptotically faster than the Elliptic Curve Method. Our results suggest that integers of the form N = p r q should be used with care. This is especially true when r is large, namely r greater than p log p.
Factoring N = p^r q for Large r (Extended Abstract)
 IN CRYPTOLOGY, CRYPTO 99, LNCS
, 1999
"... We present an algorithm for factoring integers of the form N = p^r q for large r. Such integers were previously proposed for various cryptographic applications. When r ~ log p, our algorithm runs in polynomial time (in log N). Hence, we obtain a new class of integers that can be efficiently factored ..."
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We present an algorithm for factoring integers of the form N = p^r q for large r. Such integers were previously proposed for various cryptographic applications. When r ~ log p, our algorithm runs in polynomial time (in log N). Hence, we obtain a new class of integers that can be efficiently factored. When r ~ sqrt(log p) the algorithm is asymptotically faster than the Elliptic Curve Method. Our results suggest that integers of the form N = p^r q should be used with care. This is especially true when r is large, namely r greater than log p.