The development of the number field sieve

Volume 1554 of the series Lecture Notes in Mathematics pp 50-94


Factoring integers with the number field sieve

  • J. P. BuhlerAffiliated withDepartment of Mathematics, Reed College
  • , H. W. LenstraJr.Affiliated withDepartment of Mathematics, University of California
  • , Carl PomeranceAffiliated withDepartment of Mathematics, University of Georgia

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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.