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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.
The following is pseudocode which combines Atkin's algorithms 3.1, 3.2, and 3.3 [1] by using a combined set s of all the numbers modulo 60 excluding those which are multiples of the prime numbers 2, 3, and 5, as per the algorithms, for a straightforward version of the algorithm that supports optional bit-packing of the wheel; although not specifically mentioned in the referenced paper, this ...
Magma contains asymptotically fast algorithms for all fundamental integer and polynomial operations, such as the Schönhage–Strassen algorithm for fast multiplication of integers and polynomials. Integer factorization algorithms include the Elliptic Curve Method , the Quadratic sieve and the Number field sieve .
Some examples of those algorithms are the elliptic curve method and the quadratic sieve. Another such algorithm is the class group relations method proposed by Schnorr, [11] Seysen, [12] and Lenstra, [13] which they proved only assuming the unproved generalized Riemann hypothesis.
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.
The primary improvement that quadratic sieve makes over Fermat's factorization method is that instead of simply finding a square in the sequence of , it finds a subset of elements of this sequence whose product is a square, and it does this in a highly efficient manner.
The sieve methods discussed in this article are not closely related to the integer factorization sieve methods such as the quadratic sieve and the general number field sieve. Those factorization methods use the idea of the sieve of Eratosthenes to determine efficiently which members of a list of numbers can be completely factored into small primes.
The two major areas of functionality currently implemented in FLINT are polynomial arithmetic over the integers and a quadratic sieve. The library is designed to be compiled with the GNU Multi-Precision Library (GMP) and is released under the GNU General Public License .