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Old 2007-12-11, 13:12   #2
akruppa
 
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"Nancy"
Aug 2002
Alexandria

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I get these figures for the expected number of curves, all without Brent-Suyama extension:

B1=1e6
B2: Curves to find p35: Ratio:
1e6 14317 0.12
1e7 3388 0.49
1e8 1674 1
1e9 993 1.68
1e10 639 2.62
1e11 430 3.89
1e12 297 5.62

B1=1e7
B2: Curves to find p45: Ratio:
1e7 94363 0.12
1e8 22594 0.5
1e9 11286 1
1e10 6779 1.66
1e11 4421 2.55
1e12 3015 3.74
1e13 2113 5.34

B1=1e8
B2: Curves to find p55: Ratio:
1e8 410456 0.12
1e9 101581 0.51
1e10 51549 1
1e11 31434 1.64
1e12 20823 2.47
1e13 14436 3.57
1e14 10291 5.00

The closest function I got to these ratios is
0.12 + 0.88 * (log_10(B2 / B1) / 2) ^ 1.5, which produces for log_10(B2 / B1) = 0 ... 6
0: 0.1200000000000000000000000000
1: 0.4311269837220809107363715193
2: 1.000000000000000000000000000
3: 1.736663230236897544810207489
4: 2.609015869776647285890972155
5: 3.598505426185217265198782899
6: 4.692614131981836054912458342


The values for 1 and 6 are somewhat too low, but such B2/B1 ratios will probably not be used much. For 3 it's somewhat too high. If you want really accurate estimates, you could call Pari/GP as an external program and let the rho.gp script compute probabilities. Or I'll write a small wrapper around the C code in GMP-ECM.

Alex

Last fiddled with by akruppa on 2007-12-11 at 21:39 Reason: Should be 0.12 + 0.88 * ... I guess
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