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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 43 (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.
Theory and applications of the doublebase number system
 IEEE Transactions on Computers
, 1999
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Recent progress and prospects for integer factorisation algorithms
 In Proc. of COCOON 2000
, 2000
"... Abstract. The integer factorisation and discrete logarithm problems are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed difficulty of solving these problems. This paper considers primarily the integer factorisation problem. In ..."
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Cited by 23 (1 self)
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Abstract. The integer factorisation and discrete logarithm problems are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed difficulty of solving these problems. This paper considers primarily the integer factorisation problem. In recent years the limits of the best integer factorisation algorithms have been extended greatly, due in part to Moore’s law and in part to algorithmic improvements. It is now routine to factor 100decimal digit numbers, and feasible to factor numbers of 155 decimal digits (512 bits). We outline several integer factorisation algorithms, consider their suitability for implementation on parallel machines, and give examples of their current capabilities. In particular, we consider the problem of parallel solution of the large, sparse linear systems which arise with the MPQS and NFS methods. 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...
Integer Factorisation on the AP1000
, 1995
"... We compare implementations of two integer factorisation algorithms, the elliptic curve method (ECM) and a variant of the Pollard "rho " method, on three machines (the Fujitsu AP1000, VP2200 and VPP500) with parallel and/or vector architectures. ECM is scalable and well suited for b ..."
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We compare implementations of two integer factorisation algorithms, the elliptic curve method (ECM) and a variant of the Pollard &quot;rho &quot; method, on three machines (the Fujitsu AP1000, VP2200 and VPP500) with parallel and/or vector architectures. ECM is scalable and well suited for both vector and parallel architectures.
Factorizations of a^n ± 1, 13 ≤ a < 100: Update 2
, 1996
"... This Report updates the tables of factorizations of a n \Sigma 1 for 13 a ! 100, previously published as CWI Report NMR9212 (June 1992) and updated in CWI Report NMR9419 (September 1994). A total of 760 new entries in the tables are given here. The factorizations are now complete for n ! 67, an ..."
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This Report updates the tables of factorizations of a n \Sigma 1 for 13 a ! 100, previously published as CWI Report NMR9212 (June 1992) and updated in CWI Report NMR9419 (September 1994). A total of 760 new entries in the tables are given here. The factorizations are now complete for n ! 67, and there are no composite cofactors smaller than 10 94 . 1991 Mathematics Subject Classification. Primary 11A25; Secondary 1104 Key words and phrases. Factor tables, ECM, MPQS, SNFS To appear as Report NMR96??, Centrum voor Wiskunde en Informatica, Amsterdam, March 1996. Copyright c fl 1996, the authors. Only the front matter is given here. For the tables, see rpb134u2.txt . rpb134u2 typeset using L a T E X 1 Introduction For many years there has been an interest in the prime factors of numbers of the form a n \Sigma 1, where a is a small integer (the base) and n is a positive exponent. Such numbers often arise. For example, if a is prime then there is a finite field F with a n ...