Saving computation in ECM
Suppose we have run 2900 curves with B1 = 3e6 and we now want to run 5500 curves with B1 = 11e6. If we continue on from the old residues then we've already done about 15% of the work. This saving is superficially significant. (15% = (3e6 * 2900) / (11e6 * 5500)).
Of course, the snag here is conditional probability: since we know that 2900 curves have failed to yield a factor, continuing them to B1 = 11e6 has a lower probability of success than running 2900 'fresh' curves to B1 = 11e6.
There are some good papers out there which show us how to reselect parameters after failure, but none (that I've found) suggest that we keep the residues from previous trials. Has anyone looked into working out such an algorithm?
Dave
