Pollard's rho algorithm

This article is about the integer factorization algorithm. For the discrete logarithm algorithm, see Pollard's rho algorithm for logarithms.

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.

Core ideas

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

1. x ← 2, y ← 2; d ← 1
2. While d = 1:
   1. x ← f(x)
   2. y ← f(f(y))
   3. d ← GCD(|x − y|, n)
3. If d = n, return failure.
4. Else, return d.

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.

Speeding up the Algorithm

In 1980, Richard Brent published a faster variant of the rho algorithm. He used the same core ideas as Pollard but a different method of cycle detection, replacing Floyd's cycle finding algorithm with the related Brent's cycle finding method.

A further improvement was made by Pollard and Brent. They observed that if $\gcd(a,n)>1$, then also $\gcd(ab,n)>1$ for any positive integer $b$. In particular, instead of computing $\gcd(|x-y|,n)$ at every step, it suffices to define $z$ as the product of $100$ consecutive $|x-y|$ terms modulo n, and then compute a single $\gcd(z,n)$. A major speed up results as 100 $gcd$ steps are replaced with $99$ multiplications modulo $n$ and a single $gcd$. Occasionally it may cause the algorithm to fail by introducing repeated factor, for instance when $n$ is a square. But it then suffices to go back to the previous gcd term, where $\gcd(z,n)=1$, and use the regular Rho algorithm from there.

In practice

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:

$f(x)=x^{2}+c{\hbox{ mod }}n,\,c\neq 0,-2.$

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	xi	yi	GCD(|xi − yi|, 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) instead of 97.

Complexity

The algorithm offers a trade-off between its running time and the probability that it finds a factor. If n is a product of two distinct primes of equal length, running the algorithm for O(n^1/4 polylog(n)) steps yields a factor with probability roughly half. (Note that this is a heuristic claim, and rigorous analysis of the algorithm remains open.)

References

• J.M. Pollard. "A Monte Carlo method for factorization", BIT Numerical Mathematics 15(3), 1975, pp. 331-334.
• Richard P. Brent. An Improved Monte Carlo Factorization Algorithm, BIT 20, 1980, pp.176-184, http://wwwmaths.anu.edu.au/~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 0-262-03293-7. Section 31.9: Integer factorization, pp.896–901 (this section discusses only Pollard's rho algorithm).

External links

• Java Implementation