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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 ).
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
The second-fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. The Lenstra elliptic-curve factorization is named after Hendrik Lenstra. Practically speaking, ECM is considered a special-purpose factoring algorithm, as it is most suitable for finding small factors.
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 ...
In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithms for primality testing and integer factorization, finding solutions to diophantine equations, and explicit methods in arithmetic geometry. [1]
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.