While we're on the subject, can someone explain how to derive the group
order for factors found using ECM? I've been carrying out ECM on an old PC
for almost a year now, and I'd like to be able to derive, and factorise,
the group orders for the factors that I've found.

I've been making an effort to understand the maths, and I'm getting there
slowly, but I've found nothing yet that explains how to derive the group
orders. If my understanding is correct, you would need to know the
equations used by mprime to derive the co-ordinates of the starting point
for each curve.

Anyway, if someone could explain how to derive the group order, or point
me in the right direction, I'd be very grateful.


> If the sigma is the same, then a curve with B1=250000 will find any
> factor that a curve with B1=50000 finds.
> When you run 700 random curves at B1=250000, you might theoretically
> miss a factor that someone else finds with B1=50000, if he gets a lucky
> sigma so that the group order is very smooth. But in general, using the
> same number of curves, the higher bound should find all the factors that
> the lower bound can find.
> But dont be tempted into running only a few curves at very high bounds.
> The strength of ECM is that you can try curves with different group
> orders until a sufficiently smooth one comes along. So "skipping" bound
> levels is usually not a good idea unless you have reason to believe the
> the number unter attack has only large factors which call for a higher
> bound.

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