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.
Read more about this topic: Dixon's Factorization Method
Famous quotes containing the words basic and/or idea:
“It is not an exaggeration to say that play is as basic to your child’s total development as good food, cleanliness, and rest.”
—Joanne E. Oppenheim (20th century)
“... the idea of a classless society is ... a disastrous mirage which cannot be maintained without tyranny of the few over the many. It is even more pernicious culturally than politically, not because the monolithic state forces the party line upon its intellectuals and artists, but because it has no social patterns to reflect.”
—Agnes E. Meyer (1887–1970)