Integer factorization

Integer factorization decomposes an integer greater than one into prime factors. The fundamental theorem of arithmetic guarantees that this decomposition is unique apart from factor order.

Trial division: An integer factorization method that tests successive divisors up to the square root of the target. It is the simplest method and gives a baseline for measuring faster algorithms.
Prime number: An integer greater than one whose only positive divisors are one and itself. Prime numbers are the indivisible factors that form every integer's canonical decomposition.
RSA cryptosystem: A public-key cryptosystem whose security is based on the difficulty of recovering secret information from a product of large primes. Factoring the public modulus can reveal the prime factors used to construct an RSA key.
Integer multiplication: The arithmetic operation that combines integers to produce their product. Multiplication is easy to perform, while recovering its prime inputs can be computationally difficult.
Pollard's rho algorithm: A randomized algorithm for finding a nontrivial factor of an integer using repeated values of a modular sequence. It efficiently finds small factors without testing every possible divisor.
Composite number: An integer greater than one that has a positive divisor other than one and itself. A composite number is precisely the kind of input that can be split into nontrivial factors.
RSA integer factorization challenge: A series of published composite integers intended to benchmark the practical difficulty of integer factorization. Its solved instances recorded concrete advances in factoring large, specially chosen integers.
Discrete logarithm: The problem of finding an exponent that produces a given group element as a power of a specified element. It is a distinct hard problem used in cryptography and also has a quantum algorithm.
Quadratic sieve: An integer factorization algorithm that finds a congruence of squares modulo the target number. It was a leading general-purpose method before the number field sieve.
Divisibility: The relation in which one integer divides another with no remainder. Factorization depends on identifying integers that divide the target exactly.

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