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Its running time, in asymptotic notation, is conjectured to be: ight)^ (log log n)^ ight) ight). The SNFS has been used extensively by NFSNET (a volunteer distributed computing effort) and others to factorise numbers of the Cunningham project.
Overview of method 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: 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 a monic 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 complex root of f; we can then form the ring '''Z'''α. There is a unique ring homomorphism φ from Z''α'' to '''Z'''/n'''Z''' 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 primes in Z''α'' up to some bound, along with a set of units of Z''α'' sufficient to generate the group of units for that set. Meanwhile, 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: 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. Choice of parameters Let n be the number we want to factor. First, we choose the degree d for the polynomial f. The best value is conjectured to be roughly d=(3 (log n)/(log log n))1/3. Next, we choose the coefficients of f. Recall that, for the algorithm to be effective, n must be of the form re-s, where s here may be positive or negative. We find the smallest integer k with , let t=s×rkd-e, and then define f(x)=xd-e. Finally, we choose m=rk. Limitations of algorithm This algorithm, as mentioned above, is only efficient for numbers of the form re±s, for r and s relatively small. The reason for this is as follows: For arbitrary integers, we can still choose a different f and proceed as above, but in general f will have such large coefficients that the number field Z''α'' will be too complicated to perform the required computations in. The difference between SNFS and GNFS is that the latter has slightly different methods of computation that can work in these complicated number fields. | ||||||||
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