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In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10 100. Heuristically, its complexity for factoring an integer n (consisting of ⌊log 2 n ⌋ + 1 bits) is of the form
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 computation was performed with the Number Field Sieve algorithm, using the open source CADO-NFS software. The team dedicated the computation to Peter Montgomery, an American mathematician known for his contributions to computational number theory and cryptography who died on February 18, 2020, and had contributed to factoring RSA-768. [40]
There are published algorithms that are faster than O((1 + ε) b) for all positive ε, that is, sub-exponential. As of 2022, the algorithm with best theoretical asymptotic running time is the general number field sieve (GNFS), first published in 1993, [6] running on a b-bit number n in time:
The algorithm implicitly involves the ideal structure of the number field of the polynomial; ... For the number field sieve application, it is necessary for two ...
The sieve of Eratosthenes can be expressed in pseudocode, as follows: [8] [9] algorithm Sieve of Eratosthenes is input: an integer n > 1. output: all prime numbers from 2 through n. let A be an array of Boolean values, indexed by integers 2 to n, initially all set to true.
They generated a prime susceptible to the special number field sieve, using the specialized algorithm on a comparatively small subgroup (160-bits). While this is a small subgroup, it was the standardized subgroup size used with the 1024-bit digital signature algorithm (DSA).