# Category: Integer factorization algorithms

Integer factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called p
Fermat's factorization method
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: That difference is algebraically factorable as ; if neit
Lenstra elliptic-curve factorization
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. F
RSA numbers
In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of eac
Fast Library for Number Theory
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 integer
Euler's factorization method
Euler's factorization method is a technique for factoring a number by writing it as a sum of two squares in two different ways. For example the number can be written as or as and Euler's method gives
Special number field sieve
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 sp
Trial division
Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer n, the integer to be factore
Factor base
In computational number theory, a factor base is a small set of prime numbers commonly used as a mathematical tool in algorithms involving extensive sieving for potential factors of a given integer.
Pollard's rho algorithm
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and its expected running time is proportional to the s
Pollard's p − 1 algorithm
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
Integer factorization records
Integer factorization is the process of determining which prime numbers divide a given positive integer. Doing this quickly has applications in cryptography. The difficulty depends on both the size an
Williams's p + 1 algorithm
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
Congruence of squares
In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms.
Continued fraction factorization
In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning that it is suitable for factoring any integer n,
Rational sieve
In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is less efficient than the general
General number field sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity for factoring an integer
Lattice sieving
Lattice sieving is a technique for finding smooth values of a bivariate polynomial over a large region. It is almost exclusively used in conjunction with the number field sieve. The original idea of t
Quantum algorithm for linear systems of equations
The quantum algorithm for linear systems of equations, also called HHL algorithm, designed by Aram Harrow, Avinatan Hassidim, and Seth Lloyd, is a quantum algorithm published in 2008 for solving linea
Sum of squares function
In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n as the sum of k squares, where representations that diff
Dixon's factorization method
In number theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method
Algebraic-group factorisation algorithm
Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure is the direct sum of the 'reduced groups' obt
Shanks's square forms factorization
Shanks's square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method. The success of Fermat's method depends on finding