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-   -   B1 and # curves for ECM (https://www.mersenneforum.org/showthread.php?t=17091)

Walter Nissen 2012-08-21 04:48

B1 and # curves for ECM
 
In the table in the readme , we find :[code]digits optimal B1 expected curves
N(B1,B2,D)
default poly
20 11e3 74
25 5e4 214
30 25e4 430
35 1e6 904
40 3e6 2350
45 11e6 4480
50 43e6 7553
55 11e7 17769
60 26e7 42017
65 85e7 69408

Table 1: optimal B1 and expected number of curves to find a
factor of D digits with GMP-ECM.[/code]Why do the second derivatives change sign ?
Does this come from the mathematics or is it from the details of
the computer technology used to compute the estimates , such as
available RAM , cache sizes at various levels , disk access times ,
etc. ?

fivemack 2012-08-21 07:06

Which second derivatives are you talking about? If you mean 'why are plots of log(B1) against number of digits concave', that is a corollary to the expected runtime of ECM being sub-exponential in the number of digits of the factor.

These figures from gmp-ecm come from the mathematics rather than from computer technology - you get slightly different ones if you optimise measured runtime rather than number of curves

Dubslow 2012-08-21 07:44

[QUOTE=fivemack;308743]Which second derivatives are you talking about? If you mean 'why are plots of log(B1) against number of digits concave', that is a corollary to the expected runtime of ECM being sub-exponential in the number of digits of the factor.

These figures from gmp-ecm come from the mathematics rather than from computer technology - you get slightly different ones if you optimise measured runtime rather than number of curves[/QUOTE]

No, he means this: Take, e.g., B1:

11e3
5e4 -- ratio to previous 4.5
25e4 -- ratio to previous 5.0
1e6 -- ratio to previous 4.0
3e6 -- ratio to previous 3.0
11e6 -- ratio to previous 3.67
43e6 -- ratio to previous 3.91

If you plot those ratios, the second derivative (which should be zero) swings from positive to negative and back, without every really settling at zero.

The same thing happens with necessary curves -- the ratio changes for each digit jump.

I've actually had this same question myself more than once (which is probably why I understood him).

pinhodecarlos 2012-08-21 09:20

ECM - [U][B]e[/B]lliptic[/U] [B]c[/B]urve factorization [B]m[/B]ethod

axn 2012-08-21 10:00

[QUOTE=pinhodecarlos;308751]ECM - [U][B]e[/B]lliptic[/U] [B]c[/B]urve factorization [B]m[/B]ethod[/QUOTE]

:huh:?

Walter Nissen 2012-08-26 00:07

cites ?
 
[QUOTE=pinhodecarlos;308751]ECM - [U][B]e[/B]lliptic[/U] [B]c[/B]urve factorization [B]m[/B]ethod[/QUOTE]Perhaps part of your post is an answer with a
quite serious intent .
If so , what parts of what papers do I need to read to view this
ellipticity for myself ?

Extending the readme table using
[url]http://www.loria.fr/~zimmerma/records/ecm/params.html[/url]
and computing some ratios and differences gives :[code]digits B1 ratio curves diff ratio
20 11000 74
25 50000 4.55 214 140 2.89
30 250000 5. 430 216 2.01
35 1000000 4. 904 474 2.1
40 3000000 3. 2350 1446 2.6
45 11000000 3.67 4480 2130 1.91
50 43000000 3.91 7553 3073 1.69
55 110000000 2.56 17769 10216 2.35
60 260000000 2.36 42017 24248 2.36
65 850000000 3.26 69408 27391 1.65
70 2900000000 3.4 102212 32804 1.47
75 7600000000 2.62 188056 85844 1.84
80 25000000000 3.29 265557 77501 1.41[/code]

Splicing the tables together introduces a minor inconsistency .
32804 would be 32741 , if earlier 69471 were used .

rcv 2012-08-26 16:11

[QUOTE=Walter Nissen;309231]... what parts of what papers do I need to read ...?[/QUOTE]
Walter: I think the journal article you seek is "A Practical Analysis of the Elliptic Curve Factoring Algorithm", by Robert D. Silverman and Samuel S. Wagstaff, Jr., Mathematics of Computation, v61, n203, July 1993, pp 445-462.

According to Table 3, a round B1 value for 40 digits might have been better chosen as about 4000000. As the paper points out, the curves are very flat where we are working. "Changes of +/- 10% in B1 can result in less than a 1% change in the global response surface..." So, at 40 digits, B1 is a little lower than perhaps it should be. But we do more curves to compensate.

Disclaimer: The above referenced paper is, I believe, the primary basis by which curve choices are made. I don't necessarily agree with those choices. My own (unpublished) analysis, using some different assumptions, shows that we may be doing too many curves at each level. But that's a topic for another day.

Walter Nissen 2012-08-27 14:59

Thanks much for your perceptive response .

Computing a couple more ratios ( of ratios ) here :
In the B1 ratio column , the ratio of the largest ratio
to the smallest ratio is more than 2.1 .
In the # of curves ratio column , the ratio of the largest ratio
to the smallest ratio is more than 2 .

Their Figure 2. , albeit logarithmic , doesn't suggest such wide
variation .
There must be a lot of detail that doesn't appear in the paper ,
but I don't see any function in the paper which would produce
changes in the sign of the second derivatives =
the roller coasters .

Walter Nissen 2012-09-01 23:07

[url]http://www.mersenneforum.org/showthread.php?t=11615&page=11[/url]
contains some informative messages from akruppa .

The depth of my ignorance is such that I can't cite an example with
specific digit sizes .
So , I'll just make some up , which may be suitable , or not .
I wouldn't expect optimal B1's to have the same effectiveness in
finding p38's as p39's .
Nor p39's as p40's .
But it's not immediately apparent to me why the change in
effectiveness from p38 to p39 might change in a different direction
compared to that from p39 to p40 .

Would we expect the same if instead of focusing on digit sizes of
20 , 25 , 30 , etc. , attention were paid to 20.5 , 25.5 , 30.5 ,
etc. ?
Or 24.6 , 29.6 , 34.6 , etc. ?

rcv 2012-09-02 05:43

@Walter: If my comments didn't help, then will you please make clear exactly what second derivatives are changing sign. Use specific numbers from published tables and *show us* the second derivatives that are changing.

@mod: Is it really necessary to randomly change the title of every thread in the forums? Perhaps it is my weak mind, but I don't read everything, and I can't keep track of the topics I'm following when thread titles change. ty

Dubslow 2012-09-02 06:33

[QUOTE=rcv;310021]@Walter: If my comments didn't help, then will you please make clear exactly what second derivatives are changing sign. Use specific numbers from published tables and *show us* the second derivatives that are changing.[/QUOTE]
I'm pretty sure this is what he meant. He didn't say it wasn't :razz:
[QUOTE=Dubslow;308746]No, he means this: Take, e.g., B1:

11e3
5e4 -- ratio to previous 4.5
25e4 -- ratio to previous 5.0
1e6 -- ratio to previous 4.0
3e6 -- ratio to previous 3.0
11e6 -- ratio to previous 3.67
43e6 -- ratio to previous 3.91

If you plot those ratios, the second derivative (which should be zero) swings from positive to negative and back, without every really settling at zero.

The same thing happens with necessary curves -- the ratio changes for each digit jump.

I've actually had this same question myself more than once (which is probably why I understood him).[/QUOTE]


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