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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 r e ± s , where r and s are small (for instance Mersenne numbers ).
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. When using such algorithms to factor a large number n , it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order n 1/2 .
Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of June 2019 [update] , 2 1,193 − 1 is the record-holder, [ 27 ] having been factored with a variant of the special number field sieve that allows the factorization ...
Number field sieve (NFS) is an integer factorization method, it can be: General number field sieve (GNFS): Number field sieve for any integer Special number field sieve (SNFS): Number field sieve for integers of a certain special form
For example, 15 is a composite number because 15 = 3 · 5, but 7 is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4) .
The primary improvement that quadratic sieve makes over Fermat's factorization method is that instead of simply finding a square in the sequence of , it finds a subset of elements of this sequence whose product is a square, and it does this in a highly efficient manner.
Read out all the entries in the sieve region with a large enough value. For the number field sieve application, it is necessary for two polynomials both to have smooth values; this is handled by running the inner loop over both polynomials, whilst the special-q can be taken from either side.
This is a special case of an algebraic function field. ... The main reason why the Number Field Sieve and the Function Field Sieve are faster is that these ...