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# Special number field sieve

The special number field sieve (SNFS) is a special-purpose factorization algorithm. The general number field sieve (GNFS) was derived from it.

The special number field sieve is efficient for integers of the form re ± s, where r and s are small. In particular, it is ideal for factoring Mersenne numbers.

Its running time, in asymptotic notation, is conjectured to be:

$\displaystyle{ \Theta\left(\exp\left( \left(\frac{32}{9}n\right)^{\frac{1}{3}} (\log n)^{\frac{2}{3}} \right)\right). }$

The SNFS has been used extensively by NFSNET (a volunteer distributed computing effort) and others to factorise numbers of the Cunningham project.

The first step is the polynomial selection.