The p(x) values are factorized (this is easy since we are certain they factorize completely over the factor base) and the exponents of the prime factors are converted into an exponent vector mod 2. For example, if the factor base is {2, 3, 5, 7} and the p(x) value is
30870, we have:

30870 = 2¹.3².5¹.7³

This gives an exponent vector of:

If we can find some way to add these exponent vectors together (equivalent to multiplying the corresponding relations together) to produce the zero vector (mod 2), then we can get a congruence of squares. Thus we can put the exponent vectors together into a matrix, and formulate an equation:

This can be converted into a matrix equation:

This matrix equation is then solved (using, for example, Gaussian elimination) to find the vector c. Then:

where the products are taken over all k for which C_k= 1. At least one of the C_kmust be one. Because of the way we have solved for c, the right-hand side of the above congruence is a square. We then have a congruence of squares.