Difficult: So we can just pick any elliptic curve and it will probably factor the number? How can we be sure within a good probability that we have a curve that will work? It is interesting that we can just keep picking curves and one may work though, not like the p-1 method. I also did not understand how the singular curves were worked with.
Reflective: So the p-1 and brute force algorithms are just special cases of the elliptic curve algorithm? Never thought of it that way before, of course. Or is it just that there are special cases of the elliptic curve algorithm that are the same as the p-1 and brute force algorithms? The moniker "smooth" for numbers with only small prime factors caught my eye as I have been working with "smooth" functions on manifolds a lot recently. Is there some relation in this terminology? I couldn't think of one.
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