In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer 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 (for instance Mersenne numbers). Heuristically, its complexity for factoring an integer n {\displaystyle n} is of the form:
exp ( ( 1 + o ( 1 ) ) ( 32 9 log n ) 1 / 3 ( log log n ) 2 / 3 ) = L n [ 1 / 3 , ( 32 / 9 ) 1 / 3 ] {\displaystyle \exp \left(\left(1+o(1)\right)\left({\tfrac {32}{9}}\log n\right)^{1/3}\left(\log \log n\right)^{2/3}\right)=L_{n}\left[1/3,(32/9)^{1/3}\right]}
in O and L-notations. The SNFS has been used extensively by NFSNet (a volunteer distributed computing effort), NFS@Home and others to factorise numbers of the Cunningham project; for some time the records for integer factorization have been numbers factored by SNFS.
Overview of method The SNFS is based on an idea similar to the much simpler rational sieve; in particular, readers may find it helpful to read about the rational sieve first, before tackling the SNFS. The SNFS works as follows. Let n be the integer we want to factor. As in the rational sieve, the SNFS can be broken into two steps:
First, find a large number of multiplicative relations among a factor base of elements of Z/nZ, such that the number of multiplicative relations is larger than the number of elements in the factor base. Second, multiply together subsets of these relations in such a way that all the exponents are even, resulting in congruences of the form a2≡b2 (mod n). These in turn immediately lead to factorizations of n: n=gcd(a+b,n)×gcd(a-b,n). If done right, it is almost certain that at least one such factorization will be nontrivial. The second step is identical to the case of the rational sieve, and is a straightforward linear algebra problem. The first step, however, is done in a different, more efficient way than the rational sieve, by utilizing number fields.
Details of method Let n be the integer we want to factor. We pick an irreducible polynomial f with integer coefficients, and an integer m such that f(m)≡0 (mod n) (we will explain how they are chosen in the next section). Let α be a root of f; we can then form the ring Z[α]. There is a unique ring homomorphism φ from Z[α] to Z/nZ that maps α to m. For simplicity, we'll assume that Z[α] is a unique factorization domain; the algorithm can be modified to work when it isn't, but then there are some additional complications. Next, we set up two parallel factor bases, one in Z[α] and one in Z. The one in Z[α] consists of all the prime ideals in Z[α] whose norm is bounded by a chosen value N max {\displaystyle N_{\max }} . The factor base in Z, as in the rational sieve case, consists of all prime integers up to some other bound. We then search for relatively prime pairs of integers (a,b) such that:
a+bm is smooth with respect to the factor base in Z (i.e., it is a product of elements in the factor base). a+bα is smooth with respect to the factor base in Z[α]; given how we chose the factor base, this is equivalent to the norm of a+bα being divisible only by primes less than N max {\displaystyle N_{\max }} . These pairs are found through a sieving process, analogous to the Sieve of Eratosthenes; this motivates the name "Number Field Sieve". For each such pair, we can apply the ring homomorphism φ to the factorization of a+bα, and we can apply the canonical ring homomorphism from Z to Z/nZ to the factorization of a+bm. Setting these equal gives a multiplicative relation among elements of a bigger factor base in Z/nZ, and if we find enough pairs we can proceed to combine the relations and factor n, as described above.
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