Difficult: Why do they define addition of points on the curve as the reflection of the third point where the secant line hits the curve? It seems like it would be more natural to just define the sum of two points on the curve to be the point where the secant line hits the curve. Is there some sort of computational benefit to adding in the reflection that I just do not see? The formulas seem to follow easily from the definition given, but it would seem to me just as easy to come up with formulas for the nonreflected point.
Reflective: Putting a group structure on points of a curve is particularly interesting to me in light of the research I have done in topology. This structure does not seem to be quite as natural a structure as what I am used to, so it is more interesting to think of what it all means. For example, I spent some time thinking about the curve with two components (x(x-1)(x+1)) and what the definition of addition does with that group. It looks like the points on the circle would generate the full group, as you can obtain any point on the line from two points on the circle (including the point at infinity). Possibly they do not generate though, since I see no way you could get a point on the circle from one or two points on the circle.
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