Dixon's Factorization Method

In order to find integers  and  such that

(1)

(a modified form of Fermat's factorization method), in which case there is a 50% chance that  is a factor of , choose a random integer , compute

(2)

and try to factor . If  is not easily factorable (up to some small trial divisor ), try another . In practice, the trial s are usually taken to be , with , 2, ..., which allows the quadratic sieve factorization method to be used. Continue finding and factoring s until  are found, where  is the prime counting function. Now for each , write

(3)

and form the exponent vector

(4)

Now, if  are even for any , then  is a square number and we have found a solution to (◇). If not, look for a linear combination  such that the elements are all even, i.e.,

(5)

(6)

Since this must be solved only mod 2, the problem can be simplified by replacing the s with

(7)

Gaussian elimination can then be used to solve

(8)

for , where  is a vector equal to  (mod 2). Once  is known, then we have

(9)

where the products are taken over all  for which . Both sides are perfect squares, so we have a 50% chance that this yields a nontrivial factor of . If it does not, then we proceed to a different  and repeat the procedure. There is no guarantee that this method will yield a factor, but in practice it produces factors faster than any method using trial divisors. It is especially amenable to parallel processing, since each processor can work on a different value of 

REFERENCES:

Bressoud, D. M. Factorization and Primality Testing. New York: Springer-Verlag, pp. 102-104, 1989.

Dixon, J. D. "Asymptotically Fast Factorization of Integers." Math. Comput. 36, 255-260, 1981.

Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Number Theory." In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (Ed. J. van Leeuwen). New York: Elsevier, pp. 673-715, 1990.

Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math. Soc. 43, 1473-1485, 1996.