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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 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 this is a quadratic irrational, the continued fraction must be periodic (unless n is square, in which case the factorization is obvious). It has a time complexity of O ( e 2 log n log log n ) = L n [ 1 / 2 , 2 ] {\displaystyle O\left(e^{\sqrt {2\log n\log \log n}}\right)=L_{n}\left[1/2,{\sqrt {2}}\right]} , in the O and L ...
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.
Integer factorization algorithms include the Elliptic Curve Method, the Quadratic sieve and the Number field sieve. Algebraic number theory; Magma includes the KANT computer algebra system for comprehensive computations in algebraic number fields. A special type also allows one to compute in the algebraic closure of a field. Module theory and ...
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 ).
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.
Wheel factorization with n = 2 × 3 × 5 = 30.No primes will occur in the yellow areas. Wheel factorization is a method for generating a sequence of natural numbers by repeated additions, as determined by a number of the first few primes, so that the generated numbers are coprime with these primes, by construction.