Date: 04 Oct 2006

Factoring integers with the number field sieve

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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.

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.