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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 Fermat Number
 MATH. COMP
, 1999
"... We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factor ..."
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Cited by 22 (10 self)
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We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 ; : : : ; F 11 .
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...
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 ...
Factorizations of a^n±1, 13 ≤ a < 100: Update 2
"... This Report updates the tables of factorizations of a 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 th ..."
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This Report updates the tables of factorizations of a 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 .