The factorization of the ninth Fermat number

Lenstra, Arjen K.; Lenstra, H.W.; Manasse, Mark S.; Pollard, J. M.

doi:10.1090/s0025-5718-1993-1182953-4

Cited by 71 publications

(35 citation statements)

References 37 publications

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“…If a 60 Mhz SuperSparc is counted as a 60-Mips machine, then our computation took about 240 Mips-years. This is comparable to the 340 Mips-years estimated for sieving to factor F 9 by SNFS [44]. (SNFS has since been improved, so the 340 Mips-years could now be reduced by an order of magnitude, see [32, §10].…”

Section: Factorization Of F 10supporting

confidence: 53%

“…Because the F n grow rapidly in size, a method which factors F n may be inadequate for F n+1 . Historical details and references can be found in [21,35,36,44,74], and some recent results are given in [17,26,27,34].…”

Section: Introduction and Historical Summarymentioning

confidence: 99%

“…The difficulty was finally overcome by the invention of the (special) number field sieve (SNFS), based on a new idea of Pollard [43,61]. In 1990, Lenstra, Lenstra, Manasse and Pollard, with the assistance of many collaborators and approximately 700 workstations scattered around the world, completely factored F 9 by SNFS [44,45]. The factorization is In §8 we show that it would have been possible (though more expensive) to complete the factorization of F 9 by ECM.…”

Section: Introduction and Historical Summarymentioning

confidence: 99%

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Factorization of the tenth Fermat number

Brent

1999

Math. Comp.

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Abstract. 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 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 , . . . , F 11 .

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Section: Factorization Of F 10supporting

confidence: 53%

Section: Introduction and Historical Summarymentioning

confidence: 99%

Section: Introduction and Historical Summarymentioning

confidence: 99%

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Factorization of the tenth Fermat number

Brent

1999

Math. Comp.

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“…Test whether n is a power of a prime (see [22,Section 2]) or is divisible by a prime that is less than or equal to y. In either case, Output the prime and stop.…”

Section: Analytic Interludementioning

confidence: 99%

Factoring integers with the number field sieve

Buhler

Lenstra

Pomerance

1993

Lecture Notes in Mathematics

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146

171

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Abstract. In 1990, the ninth Fermat number was factored into primes by means of a new algorithm, the "number field sieve", which was invented by John Pollard. The present paper is devoted to the description and analysis of a more general version of the number field sieve. It should be possible to use this algorithm to factor arbitrary integers into prime factors, not just integers of a special form like the ninth Fermat number. Under reasonable heuristic assumptions, the analysis predicts that the time needed by the general number field sieve to factor n is exp((c+o(l))(logn)1/3(loglogn)2/3) (for n-*oo), where c=(64/9)l/3 = 1.9223. This is asymptotically faster than all other known factoring algorithms, such äs the quadratic sieve and the elliptic curve method. There does not yet exist an Implementation of the number field sieve for general integers, so that a practical comparison cannot yet be made.

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“…Consider the 155-decimal digit number Using these and related ideas, Lenstra et al [31] factored F 9 in June 1990, obtaining…”

Section: Snfs Examplesmentioning

confidence: 99%

Recent Progress and Prospects for Integer Factorisation Algorithms

Brent

2000

Lecture Notes in Computer Science

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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 100-decimal 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.

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The factorization of the ninth Fermat number

Cited by 71 publications

References 37 publications

Factorization of the tenth Fermat number

Factorization of the tenth Fermat number

Factoring integers with the number field sieve

Recent Progress and Prospects for Integer Factorisation Algorithms

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