Search results
Results from the WOW.Com Content Network
Given a general algorithm for integer factorization, any integer can be factored into its constituent prime factors by repeated application of this algorithm. The situation is more complicated with special-purpose factorization algorithms, whose benefits may not be realized as well or even at all with the factors produced during decomposition.
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10 100. Heuristically, its complexity for factoring an integer n (consisting of ⌊log 2 n ⌋ + 1 bits) is of the form
Moreover, this factorization is unique up to the order of the factors and the signs of the factors. There are efficient algorithms for computing this factorization, which are implemented in most computer algebra systems. See Factorization of polynomials. Unfortunately, these algorithms are too complicated to use for paper-and-pencil computations.
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method.
Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm .
A simplified version of the LLL factorization algorithm is as follows: calculate a complex (or p-adic) root α of the polynomial () to high precision, then use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm to find an approximate linear relation between 1, α, α 2, α 3, . . . with integer coefficients, which might be an ...
Like the algorithms of the preceding section, Victor Shoup's algorithm is an equal-degree factorization algorithm. [4] Unlike them, it is a deterministic algorithm. However, it is less efficient, in practice, than the algorithms of preceding section. For Shoup's algorithm, the input is restricted to polynomials over prime fields F p.
In computational number theory, Williams's p + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms.It was invented by Hugh C. Williams in 1982.