I once mentioned the same idea to Peter Montgomery, he replied that a certain Bob Silverman

mentioned it almost ten years earlier. He also told me that this idea is not implemented in the CWI siever, however multiple polynomial NFS for factoring

*one* number is. Maybe the code could be adapted to the many-numbers-at-once idea.

He also mentioned another nice trick to use symmetry in the algebraic polynomial: for even polynomials, F(a,b) is smooth iff F(-a,b) is, so the algebraic side only needs to sieve nonnegative values. For symmetric polynomials (same coefficients left-to-right as right-to-left), F(a,b) is smooth iff F(b,a) is, though this symmetry seems harder to exploit. He said these tricks are not implemented in the CWI siever, either.

Alex