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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.
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...
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.
Novel Runtime Systems Support for Adaptive Compositional Modeling in PSEs, Future Generation Computing Systems (Special Issue on "Complex PSEs for Grid Computing
 Future Generation Computing Systems (Special Issue
, 2005
"... Problem solving environments (PSEs) have progressed significantly in the past few years. The vision of truly seamless PSEs relies on runtime systems support that is cognizant of the operational issues underlying scientific computations and, at the same time, is flexible enough to accommodate diverse ..."
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Cited by 12 (3 self)
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Problem solving environments (PSEs) have progressed significantly in the past few years. The vision of truly seamless PSEs relies on runtime systems support that is cognizant of the operational issues underlying scientific computations and, at the same time, is flexible enough to accommodate diverse application scenarios. This paper presents a PSE runtime support solution 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 runtime recommendation of code modules and a sophisticated checkpointing framework for transparent deployment. 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
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.
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.