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Published online by Cambridge University Press: 30 May 2025
We describe the main ideas underlying integer factorization using the number field sieve.
The number field sieve is a factoring algorithm that tries to factor a hard composite number by exploiting factorizations of smooth numbers in a well-chosen algebraic number field. It is similar in nature to the quadratic sieve algorithm, but the underlying number theory is less elementary, and the actual implementation involves a fair amount of optimization of the various parameters.
The key idea of the algorithm, the use of smooth numbers in number rings different from , was proposed in 1988 by Pollard. Many people have contributed theoretical and practical improvements since then. An excellent reference for many of the details left out in this paper is [Lenstra and Lenstra 1993]. It contains complete bibliography of the early years of the number field sieve, as well as original contributions by most of the main developers of the algorithm.
Among the successes of the algorithm are the 2005 factorization of the 663-bit RSA challenge number into a product of two primes of 100 decimal digits each, and the factorization in 2006 of the 275-digit Cunningham number.
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