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.

1991 Mathematics Subject Classification

Primary 11Y05, 11Y40

Key words and phrases

Factoring algorithm algebraic number fields