Pollard's rho algorithm

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Pollard's rho algorithm is a special-purpose integer factorization algorithm. It was invented by John Pollard in 1975. It is particularly effective at splitting composite numbers with small factors.

Contents

1 Core ideas
2 The algorithm
3 Richard Brent's variant
4 In practice
5 Example factorization
6 References

Core ideas

The rho algorithm is based on Floyd's cycle-finding algorithm and on the observation that (as in the birthday paradox) two numbers x and y are congruent modulo p with probability 0.5 after [1.177\sqrt] numbers have been randomly chosen. If p is a factor of n, the integer we are aiming to factor, then [1 < \gcd \left( |x-y|,n \right) \le n] since p divides both [\left|x-y\right|] and n.

The rho algorithm therefore uses a function modulo n as a generator of a pseudo-random sequence. It runs one sequence twice as "fast" as the other; i.e. for every iteration made by one copy of the sequence, the other copy makes two iterations. Let x be the current state of one sequence and y be the current state of the other. The GCD of |x − y| and n is taken at each step. If this GCD ever comes to n, then the algorithm terminates with failure, since this means x = y and therefore, by Floyd's cycle-finding algorithm, the sequence has cycled and continuing any further would only be repeating previous work.

The algorithm

Inputs: n, the integer to be factored; and f(x), a pseudo-random function modulo n

Output: a non-trivial factor of n, or failure.

x ← 2, y ← 2; d ← 1
While d = 1:
# x ← f(x)
# y ← f(f(y))
# d ← GCD(|x − y|, n)
# If 1 < d < n, then return d.
# If d = n, return failure.

Note that this algorithm will return failure for all prime n, but it can also fail for composite n. In that case, use a different f(x) and try again.

Richard Brent's variant

In 1980, Richard Brent published a faster variant of the rho algorithm. He used the same core ideas as Pollard, but he used a different method of cycle detection that was faster than Floyd's original algorithm.

Brent's algorithm is as follows:

Input: n, the integer to be factored; x₀, such that 0 ≤ x₀ ≤ n; m such that m > 0; and f(x), a pseudo-random function modulo n.

Output: a non-trivial factor of n, or failure.

y ← x₀, r ← 1, q ← 1.
Do:
# x ← y
# For i = 1 To r:
## y ← f(y)
#k ← 0
# Do:
## ys ← y
## For i = 1 To min(m, r − k):
### y ← f(y), q ← (q × |x − y|) mod n
## g ← GCD(q, n), k ← k + i
# Until (k ≥ r or g > 1)
# r ← 2r
Until g > 1
If g = n then
# Do:
## ys ← f(ys), g ← GCD(|x − ys|, n)
# Until g > 1
If g = n then return failure, else return g

In practice

The algorithm is very fast for numbers with small factors. For example, on a 733 MHz workstation, an implementation of the rho algorithm, without any optimizations, found the factor 274177 of the sixth Fermat number in about half a second. The sixth Fermat number is 18446744073709551617 (20 decimal digits). However, for a semiprime of the same size, the same workstation took around 9 seconds to find a factor of 10023859281455311421 (the product of 2 10-digit primes).

For f, we choose a polynomial with integer coefficients. The most common ones are of the form:

[f(x)=x^2+c\hboxn,\,c\neq0,-2.]

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.

Example factorization

Let n = 8051 and f(x) = x² + 1 mod 8051.

i x_i y_i GCD(|x_i − y_i|, 8051)

1 5 26 1

2 26 7474 1

3 677 871 97

97 is a non-trivial factor of 8051. Other values of c may give the cofactor (83) of 97 instead of 97.

References

Richard P. Brent. An Improved Monte Carlo Factorization Algorithm, BIT 20, 1980, pp.176-184, http://web.comlab.ox.ac.uk/oucl/work/richard.brent/pd/rpb051i.pdf
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 31.9: Integer factorization, pp.896–901 (this section discusses only Pollard's rho algorithm).

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