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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 .
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 ...
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 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 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 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.
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 ...