Factoring integers with the number field sieve
- Cite this paper as:
- 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
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.
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