The number field sieve

  • A. K. Lenstra
  • H. W. LenstraJr.
  • M. S. Manasse
  • J. M. Pollard
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1554)


The number field sieve is an algorithm to factor integers of the form res for small positive r and |s|. The algorithm depends on arithmetic in an algebraic number field. We describe the algorithm, discuss several aspects of its implementation, and present some of the factorizations obtained. A heuristic run time analysis indicates that the number field sieve is asymptotically substantially faster than any other known factoring method, for the integers that it applies to. The number field sieve can be modified to handle arbitrary integers. This variant is slower, but asymptotically it is still expected to beat all older factoring methods.


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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • A. K. Lenstra
    • 1
  • H. W. LenstraJr.
    • 2
  • M. S. Manasse
    • 3
  • J. M. Pollard
    • 4
  1. 1.MorristownUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.DEC SRCPalo AltoUSA
  4. 4.Tidmarsh CottageReadingEngland

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