Dixon's Factorization Method - Basic Idea

Basic Idea

Dixon's method is based on finding a congruence of squares modulo the integer N which we intend to factor. Fermat's factorization algorithm finds such a congruence by selecting random or pseudo-random x values and hoping that the integer x2 mod N is a perfect square (in the integers):

For example, if N = 84923, we notice (by starting at 292, the first number greater than √N and counting up) that 5052 mod 84923 is 256, the square of 16. So (505 − 16)(505 + 16) = 0 mod 84923. Computing the greatest common divisor of 505 − 16 and N using Euclid's algorithm gives us 163, which is a factor of N.

In practice, selecting random x values will take an impractically long time to find a congruence of squares, since there are only √N squares less than N.

Dixon's method replaces the condition "is the square of an integer" with the much weaker one "has only small prime factors"; for example, there are 292 squares smaller than 84923; 662 numbers smaller than 84923 whose prime factors are only 2,3,5 or 7; and 4767 whose prime factors are all less than 30. (Such numbers are called B-smooth with respect to some bound B.)

If we have lots of numbers whose squares can be factorized as for a fixed set of small primes, linear algebra modulo 2 on the matrix will give us a subset of the whose squares combine to a product of small primes to an even power — that is, a subset of the whose squares multiply to the square of a (hopefully different) number mod N.