Editing Congruence of squares (section)

==Using a factor base==
A technique pioneered by [[Dixon's factorization method]] and improved by [[continued fraction factorization]], the [[quadratic sieve]], and the [[general number field sieve]], is to construct a congruence of squares using a [[factor base]].

Instead of looking for one pair <math>\textstyle x^2 \equiv y^2 \pmod n</math> directly, we find many "relations" <math>\textstyle x^2 \equiv y \pmod n</math> where the ''y'' have only small [[prime number|prime]] factors (they are [[smooth numbers]]), and multiply some of them together to get a [[square (algebra)|square]] on the right-hand side.

The set of small primes which all the ''y'' factor into is called the factor base.<!--The factor base generally includes -1 as well, a detail omitted here-->  Construct a [[logical matrix]] where each row describes one ''y'', each column corresponds to one prime in the factor base, and the entry is the parity (even or odd) of the number of times that factor occurs in ''y''.  Our goal is to select a subset of rows whose sum is the all-zero row.  This corresponds to a set of ''y'' values whose product is a square number, i.e. one whose factorization has only even exponents.  The products of ''x'' and ''y'' values together form a congruence of squares.

This is a classic [[system of linear equations]] problem, and can be efficiently solved using [[Gaussian elimination]] as soon as the number of rows exceeds the number of columns.<!--Not "equals"; that would only guarantee the trivial solution "choose no rows".-->  Some additional rows are often included to ensure that several solutions exist in the nullspace of our matrix, in case the first solution produces a trivial congruence.

A great advantage of this technique is that the search for relations is [[embarrassingly parallel]]; a large number of computers can be set to work searching different ranges of ''x'' values and trying to factor the resultant ''y''s.  Only the found relations need to be reported to a central computer, and there is no particular hurry to do so.  The searching computers do not even have to be trusted; a reported relation can be verified with minimal effort.

There are numerous elaborations on this technique.  For example, in addition to relations where ''y'' factors completely in the factor base, the "large prime" variant also collects "partial relations" where ''y'' factors completely except for one larger factor.  A second partial relation with the same larger factor can be multiplied by the first to produce a "complete relation".