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The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of n. Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve.
For current computers, GNFS is the best published algorithm for large n (more than about 400 bits). For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time. Shor's algorithm takes only O(b 3) time and O(b) space on b-bit number inputs.
The factorization was found using the general number field sieve algorithm and an estimated 8000 MIPS-years of computing time. The polynomials were 119377138320*x^5 ...
The discrete logarithm algorithm and the factoring algorithm are instances of the period-finding algorithm, and all three are instances of the hidden subgroup problem. On a quantum computer, to factor an integer N {\displaystyle N} , Shor's algorithm runs in polynomial time , meaning the time taken is polynomial in log N {\displaystyle \log ...
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
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning ...
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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).