Special number field sieve

The special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. When the term "number field sieve" is used without qualification, it refers to GNFS.

The special number field sieve is efficient for integers of the form r^e ± s, where r and s are small. As such, it is ideal for factoring Fermat numbers.

Its running time is

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

The SNFS has been used extensively by NFSNET and others to factorise numbers of the Cunningham project.