mersenneforum.org B1 and # curves for ECM
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 2012-08-21, 04:48 #1 Walter Nissen     Nov 2006 Terra 4E16 Posts 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. 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. ?
 2012-08-21, 07:06 #2 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 142628 Posts 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
2012-08-21, 07:44   #3
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3×2,399 Posts

Quote:
 Originally Posted by fivemack 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
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).

 2012-08-21, 09:20 #4 pinhodecarlos     "Carlos Pinho" Oct 2011 Milton Keynes, UK 22×7×132 Posts ECM - elliptic curve factorization method
2012-08-21, 10:00   #5
axn

Jun 2003

5×23×41 Posts

Quote:
 Originally Posted by pinhodecarlos ECM - elliptic curve factorization method
?

2012-08-26, 00:07   #6
Walter Nissen

Nov 2006
Terra

1168 Posts
cites ?

Quote:
 Originally Posted by pinhodecarlos ECM - elliptic curve factorization method
quite serious intent .
If so , what parts of what papers do I need to read to view this
ellipticity for myself ?

http://www.loria.fr/~zimmerma/records/ecm/params.html
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
Splicing the tables together introduces a minor inconsistency .
32804 would be 32741 , if earlier 69471 were used .

Last fiddled with by Walter Nissen on 2012-08-26 at 00:14 Reason: restored missing column label

2012-08-26, 16:11   #7
rcv

Dec 2011

100011112 Posts

Quote:
 Originally Posted by Walter Nissen ... what parts of what papers do I need to read ...?
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.

 2012-08-27, 14:59 #8 Walter Nissen     Nov 2006 Terra 2·3·13 Posts 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 .
 2012-09-01, 23:07 #9 Walter Nissen     Nov 2006 Terra 2×3×13 Posts http://www.mersenneforum.org/showthr...=11615&page=11 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. ?
 2012-09-02, 05:43 #10 rcv   Dec 2011 2178 Posts @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
2012-09-02, 06:33   #11
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3×2,399 Posts

Quote:
 Originally Posted by rcv @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.
I'm pretty sure this is what he meant. He didn't say it wasn't
Quote:
 Originally Posted by Dubslow 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).

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