Pollard rho Factorization Method

A prime factorization algorithm also known as Pollard Monte Carlo factorization method. There are two aspects to the Pollard  factorization method. The first is the idea of iterating a formula until it falls into a cycle. Let , where  is the number to be factored and  and  are its unknown prime factors. Iterating the formula

(1)

or almost any polynomial formula (an exception being ) for any initial value  will produce a sequence of number that eventually fall into a cycle. The expected time until the s become cyclic and the expected length of the cycle are both proportional to .

However, since  with  and  relatively prime, the Chinese remainder theorem guarantees that each value of  (mod ) corresponds uniquely to the pair of values (), ). Furthermore, the sequence of s follows exactly the same formula modulo  and , i.e.,

			(2)
			(3)

Therefore, the sequence (mod ) will fall into a much shorter cycle of length on the order of . It can be directly verified that two values  and  have the same value (mod ), by computing

(4)

which is equal to .

The second part of Pollard's method concerns detection of the fact that a sequence has become periodic. Pollard's suggestion was to use the idea attributed to Floyd of comparing  to  for all . A different method is used in Brent's factorization method.

Under worst conditions, the Pollard  algorithm can be very slow.

REFERENCES:

Brent, R. "An Improved Monte Carlo Factorization Algorithm." Nordisk Tidskrift for Informationsbehandlung (BIT) 20, 176-184, 1980.

Brent, R. P. "Some Integer Factorization Algorithms Using Elliptic Curves." Austral. Comp. Sci. Comm. 8, 149-163, 1986.

Bressoud, D. M. Factorization and Primality Testing. New York: Springer-Verlag, pp. 61-67, 1989.

Eldershaw, C. and Brent, R. P. "Factorization of Large Integers on Some Vector and Parallel Computers."

Montgomery, P. L. "Speeding the Pollard and Elliptic Curve Methods of Factorization." Math. Comput. 48, 243-264, 1987.

Pollard, J. M. "A Monte Carlo Method for Factorization." Nordisk Tidskrift for Informationsbehandlung (BIT) 15, 331-334, 1975.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 83 and 102-103, 1991.