I wrote:
> > From: Brian J. Beesley [mailto:[EMAIL PROTECTED]]
>
> > If P-1 does find a factor which is compound, then running P-1 again
> > with smaller limits will eventually recover a smaller factor. These
> > extra runs will obviously take less time than the original
>
> Indeed, and with care one can usually choose the bounds so only one more
run
> is necessary. Factoring c-1 (where c is the composite factor found) and
> judiciously chosing which primes to omit is the method. This
factorization
> is extremely easy, because of the way in which c was discovered. In
> practice, discarding the prime factor of c found in stage 2 is all that's
> usually needed. If c was found after stage 1, and so there is no large
> prime, discarding half (rounded up if the number is odd) of the powers of
2
> usually does the trick.
>
> Of course, all these computations are performed on c, and not the original
> integer.
>
> It's a pity that a similar procedure isn't known for ECM, or at least not
> known to me.
On reflection, this last sentence is silly, and arose only because I hadn't
thought about the matter properly.
As long as the coefficients of the curve and the starting point are
recorded, we can re-run exactly the same computation, with the small primes
curtailed as in the p-1 case, on the same curve and the number c. It's
because my software doesn't normally output the curve and starting point
used that the idea hadn't occurred to me.
Paul
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