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Factoring integers with the number field sieve

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Book cover The development of the number field sieve

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1554))

Abstract

In 1990, the ninth Fermat number was factored into primes by means of a new algorithm, the “number field sieve”, which was proposed 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(1))(logn)1/3(loglogn)2/3) (for n → ∞), where c=(64/9)1/3=1.9223. This is asymptotically faster than all other known factoring algorithms, such as the quadratic sieve and the elliptic curve method.

The authors wish to thank Dan Bernstein, Arjeh Cohen, Michael Filaseta, Andrew Granville, Arjen Lenstra, Victor Miller, Robert Rumely, and Robert Silverman for their helpful suggestions. The authors were supported by NSF under Grants No. DMS 90-12989, No. DMS 90-02939, and No. DMS 90-02538, respectively. The second and third authors are grateful to the Institute for Advanced Study (Princeton), where part of the work on which this paper is based was done.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Reed College, 97202, Portland, OR, USA

    J. P. Buhler

  2. Department of Mathematics, University of California, 94720, Berkeley, CA, USA

    H. W. Lenstra Jr.

  3. Department of Mathematics, University of Georgia, 30602, Athens, GA, USA

    Carl Pomerance

Authors
  1. J. P. Buhler
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  2. H. W. Lenstra Jr.
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  3. Carl Pomerance
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Arjen K. Lenstra Hendrik W. Lenstra Jr.

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© 1993 Springer-Verlag

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Buhler, J.P., Lenstra, H.W., Pomerance, C. (1993). Factoring integers with the number field sieve. In: Lenstra, A.K., Lenstra, H.W. (eds) The development of the number field sieve. Lecture Notes in Mathematics, vol 1554. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091539

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  • DOI: https://doi.org/10.1007/BFb0091539

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57013-4

  • Online ISBN: 978-3-540-47892-8

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