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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 .
Since typically B 1 > 2, d n are even numbers. The distribution of prime numbers is such that the d n will all be relatively small. It is suggested that d n ≤ ln 2 B 2. Hence, the values of H 2, H 4, H 6, … (mod n) can be stored in a table, and H q n be computed from H q n−1 ⋅H d n, saving the need for exponentiations.
If the pseudorandom number = occurring in the Pollard ρ algorithm were an actual random number, it would follow that success would be achieved half the time, by the birthday paradox in () (/) iterations. It is believed that the same analysis applies as well to the actual rho algorithm, but this is a heuristic claim, and rigorous analysis of ...
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
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.
The index calculus algorithm is much easier to state than the Function Field Sieve and the Number Field Sieve since it does not involve any advanced algebraic structures. It is asymptotically slower with a complexity of L p [ 1 / 2 , 2 ] {\displaystyle L_{p}[1/2,{\sqrt {2}}]} .
John M. Pollard (born 1941) is a British mathematician who has invented algorithms for the factorization of large numbers and for the calculation of discrete logarithms.. His factorization algorithms include the rho, p − 1, and the first version of the special number field sieve, which has since been improved by others.