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The first step, however, is done in a different, more efficient way than the rational sieve, by utilizing number fields.
This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve.
The second step is identical to the case of the rational sieve, and is a straightforward linear algebra problem.
In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors.
The factor base in Z, as in the rational sieve case, consists of all prime integers up to some other bound.
As in the rational sieve, the SNFS can be broken into two steps:
The principle of the number field sieve (both special and general) can be understood as an improvement to the simpler rational sieve or quadratic sieve.
The rational sieve, like the general number field sieve, cannot factor numbers of the form p, where p is a prime and m is an integer.
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.
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