1. I thought I understood elliptic curves until I read this section. I understood the first few paragraphs and that's about it. In the first example, why are we trying to compute 10!P? It's probably a bad sign that I don't even see how this relates to the goal of factoring n. Also, I don't quite understand the significance of smoothness in this factorization method. Singular curves were also confusing.
2. I like the concept of utilizing differences in p and q in order to factor n=pq. Assuming I follow what it is saying, the very last statement in this section was interesting. Does it mean that the p-1 method and trial division are a part of the algorithm? Or that performing the algorithm is equivalent to doing both methods?
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