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Elliptic curves are "generalized modular arithmetic" only in the sense that they involve a different kind of addition law (the set of points on a nice EC is a group).

Also, you can look at the points on an elliptic curve where you allow the coordinates to be real, complex, rational, or even the integers mod p (any field will do), so the last choice gives you a closer link with modular arithmetic. It's best to treat ECs as their own weird, wonderful beasts!

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