Talk:Lenstra elliptic-curve factorization

message: Can anyone provide pseudocode and an example for this topic to illustrate the method? Four steps are given but they are too abstract to be useful IMO. Compare with the excellent topic on Miller-Rabin primality test, for example.
14:08, 16 October 2008 (UTC)
replies: text: I don't like pseudocode, but did try to make the steps more concrete. Will also add an example soon. --GaborPete (talk) 08:42, 1 April 2009 (UTC)
replies: text:  OK, I have now put in an example. --GaborPete (talk) 04:56, 3 April 2009 (UTC)

text: Also, I changed the "over Z/nZ" wording to "(mod n)". One reason is that the old version sounded too much as if Z/nZ was a field. Another reason is that my undergrads, for example, know what (mod n) is, but don't know "over Z/nZ". --GaborPete (talk) 08:42, 1 April 2009 (UTC)

text: Frankly, both ECM and QS/NFS are heuristic algorithms. There's no proof that they're either deterministic (in the sense of always returning a result) or probabilistic (in the sense of having a known bound on their error rate). For example, there's no reason to believe, by merely examining a description of NFS, that it will not always yield a trivial difference of squares; the possibility exists that there is still some input for which it will produce this behavior, just as the Fermat primality test always fails on Carmichael numbers. I know less about ECM, but I'm pretty sure it also relies on unproven assumptions but works well in practice. The best thing to do is just label them heuristic algorithms, and avoid other labels. It would be valid to say that the description of ECM assumes the presence of a source of random bits (e.g. whenever it says "select a random..."), which is not an assumption made by QS/NFS; perhaps this is the distinction you're getting at. Dcoetzee 09:13, 30 January 2009 (UTC)
thread_title:  Pseudocode ? 

message: I know an online site called Alpertron that uses this method to find primes and it's the fastest online method I know to use, but sometimes it doesn't succeed well. This means that it sometimes says that a number is composite but doesn't find any factors within 325 curves. (Why 325?? It's because it requires a lot of patience, and after curve #325, it requires a lot of patience per curve, even more than it does until curve #325.) Any faster online site?? Georgia guy (talk) 21:40, 13 January 2012 (UTC)
thread_title:  Faster online site?? 

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message: I think that the Hasse theorem gives only an upper bound to the order of the subgroup. We have to consider the subgroup that belongs to the starting point. This a divisor of the number of points on the curve. The order of the subgroup can be much smaller as the prime p. 
It means, the algorithm is in fact even faster.
95.222.84.67 (talk) 20:08, 6 December 2021 (UTC)
thread_title:  Order of the subgroup of number of points on the curve 

message: The algorithm is currently described as three steps. But Step 2 is mainly background material about the group law on an elliptic curve, right? And then there's some ill-specified stuff about u and v. And then Step 3 is a legitimate algorithmic step, involving v not at all, except that decisions must be made based on v.
A competent programmer, who is trained in number theory, could not implement the algorithm as stated. It should be stated much more clearly. Mgnbar (talk) 01:22, 19 September 2019 (UTC)
thread_title:  Statement of algorithm