Pollard's rho algorithm

We may search the entire range of possible logarithms by setting and , although in this case Pollard's rho algorithm is more efficient.

(this section discusses only Pollard's rho algorithm).

Pollard's rho algorithm is a special-purpose integer factorization algorithm.

Alternatively one can use Pollard's rho algorithm for logarithms, which has about the same running time as the baby-step giant-step algorithm, but only a small memory requirement.

Pollard's rho algorithm for logarithms is an example for an algorithm using a birthday attack for the computation of discrete logarithms.

In some applications, and in particular in Pollard's rho algorithm for integer factorization, the algorithm has much more limited access to S and to ƒ.

Several number-theoretic algorithms are based on cycle detection, including Pollard's rho algorithm for integer factorization and his related kangaroo algorithm for the discrete logarithm problem.

There are other generic algorithms for computing the order of a group element that are more space efficient, such as Pollard's rho algorithm and the Pollard kangaroo method.

The factors p and q of n should both be very large, around the same order of magnitude as the square root of n; this makes trial division and Pollard's rho algorithm impractical.

The equality test action may involve some nontrivial computation: in Pollard's rho algorithm, it is implemented by testing whether the difference between two stored values has a nontrivial gcd with the number to be factored.

In order to do so quickly, they typically use a hash table or similar data structure for storing the previously-computed values, and therefore are not pointer algorithms: in particular, they usually cannot be applied to Pollard's rho algorithm.

In Pollard's rho algorithm, for instance, 'S' is the set of integers modulo an unknown prime factor of the number to be factorized, so even the size of 'S' is unknown to the algorithm.

Much like the name of another of Pollard's discrete logarithm algorithms, Pollard's rho algorithm, this name refers to the similarity between a visualisation of the algorithm and the Greek letter lambda ().

Similarly, there are simple randomized in-place algorithms for primality testing such as the Miller-Rabin primality test, and there are also simple in-place randomized factoring algorithms such as Pollard's rho algorithm.

Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 for solving the discrete logarithm problem analogous to Pollard's rho algorithm for solving the Integer factorization problem.

Cycle detection, the problem of following a path in a functional graph to find a cycle in it, has applications in cryptography and computational number theory, as part of Pollard's rho algorithm for integer factorization and as a method for finding collisions in cryptographic hash functions.