Progressive ECM bounds

[R.D. Silverman]

Originally Posted by alpertron

So, as I suspected, the contents of the worker's computers were not backed up.

No. They *were* backed up while I was working there. The backups
were erased as well.

[Zeta-Flux]

Dear R. D. Silverman,

I can easily download a pdf version of the paper off the web. If you'd like a copy, you can private message me your email address, and I'll send it to you. Then you can distribute it according to copyright laws to those who ask you for a copy.

-Pace

[garo]

I think the issue at hand is not a scanned pdf which is easy to obtain but the latex source.

Anyway, we are getting way off topic which is to help George decide the right formulae for deciding ECM limits.

Bob, and I'll go out on a limb here, while your paper does solve the problem completely (I've read it, several times, carefully), it is not trivial to convert it into an implementation practical for a server.

[R.D. Silverman]

Originally Posted by garo

I think the issue at hand is not a scanned pdf which is easy to obtain but the latex source.

Anyway, we are getting way off topic which is to help George decide the right formulae for deciding ECM limits.

Bob, and I'll go out on a limb here, while your paper does solve the problem completely (I've read it, several times, carefully), it is not trivial to convert it into an implementation practical for a server.

I agree. It is not easy to implement. But the theory presented was
quite new at the time. The idea (summarized for everyone else) is
as follows:

We know the theoretical distribution for the size of prime factors
of a larger number. It is given by Dickman's Function. We now run
ECM and it fails. How do we use the information gained from the
failure to reselect parameters for the next run? The answer is to use
Bayes' Theorem. Dickman's function gives use a prior distribution.
The ECM failure is a sample. We combines these with Baye's theorem
to get a posterior, then take the expected value of the posterior as the
size of the new target prime.

BTW, the numbers in the paper assumed B2 = 100 B1, because that was
what was available at that time; Noone had yet implemented a
convolution based step 2 for ECM.

And, as has been pointed out, there is a mistake in the formula
given for rho_2, the second Dickman function. However, the computations
were done using an independently written piece of software (Sam Wagstaff
wrote it) and we validated that software against known tables published
by Knuth....