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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 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 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 main problem with the page-segmented sieve of Atkin is the difficulty in implementing the prime-square-free culling sequences due to the span between culls rapidly growing far beyond the page buffer span; the time expended for this operation in Bernstein's implementation rapidly grows to many times the time expended in the actual quadratic ...
Quadratic irrational; Integer square root; Algebraic number. Pisot–Vijayaraghavan number; Salem number; Transcendental number. e (mathematical constant) pi, list of topics related to pi; Squaring the circle; Proof that e is irrational; Lindemann–Weierstrass theorem; Hilbert's seventh problem; Gelfond–Schneider theorem; Erdős–Borwein ...
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 quadratic sieve is an optimization of Dixon's method. It selects values of x close to the square root of N such that x 2 modulo N is small, thereby largely increasing the chance of obtaining a smooth number.
The Fast Library for Number Theory (FLINT) is a C library for number theory applications. The two major areas of functionality currently implemented in FLINT are polynomial arithmetic over the integers and a quadratic sieve.