Thread: ECM odds
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Old 2015-05-25, 17:12   #3
R.D. Silverman
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Nov 2003

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Originally Posted by wblipp View Post
The question is more complicated that it sounds. If a factor of a particular size exists, the probability of finding that factor is the same for every curve of the set. But that includes re-finding factors already found by earlier curves in the set. And you seldom know that a factor of a particular size exists, so working through a set curves without finding any such factor makes it more likely that a factor of that size does not exist. These are not particularly complicated probability ideas, but there are several things to keep track of. We could help you work through them if you want.

But the answer you probably want can be figured out in a simpler manner. Recommended change points happen because the probability of finding a factor per unit of computing power has just become higher for the bigger curve. So right at the change over the probabilities are the same. (In fact, they are nearly the same for a broad range around the change point, but that's not necessary for this argument). So if the new curve takes "k" times more computing power, it is "k" times more likely to find a factor. If it were not so, the recommended change point would be early or later - at the point where it IS true.

So if you only want the relative odds - that's how to do it. If you want to know the actual values of these probabilities, we need to dig deeper into the issues of the first paragraph.
The reply was almost totally devoid of information. It is hand-waving nonsense.

The OP's question may be answered by reading my joint paper with Sam Wagstaff Jr.:
A Practical Analysis of ECM, Math. Comp.
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