Application
The algorithm is very fast for numbers with small factors. For example, on a 3 GHz workstation, the original rho algorithm found the factor 274177 of the sixth Fermat number (18446744073709551617) in 26 milliseconds; the Richard Brent variant found the same factor in 5 milliseconds. However, for a semiprime of the same size (10023859281455311421), the same workstation using the original rho algorithm took 109 milliseconds to find a factor; the Richard Brent variant took 31 milliseconds.
For f, we choose a polynomial with integer coefficients. The most common ones are of the form:
The rho algorithm's most remarkable success has been the factorization of the eighth Fermat number by Pollard and Brent. They used Brent's variant of the algorithm, which found a previously unknown prime factor. The complete factorization of F8 took, in total, 2 hours on a UNIVAC 1100/42.
Read more about this topic: Pollard's Rho Algorithm
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