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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
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:
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 Topics referred to by the same term
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).
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Integer factorization is the process of determining which prime numbers divide a given positive integer.Doing this quickly has applications in cryptography.The difficulty depends on both the size and form of the number and its prime factors; it is currently very difficult to factorize large semiprimes (and, indeed, most numbers that have no small factors).