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There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman

When I first learned about Diffie-Hellman and especially elliptic curve Diffie-Hellman, I had one rather obvious question: Why elliptic curves? Why use this strange group that seems rather arbitrar…

We probably want something with finite objects, because computers aren’t very good at dealing with things that don’t fit into their little memories and disks and registers and stuff. We will need a tiny amount of handwaving, but I think it is not too hard to see that taking tuples of elements of finite fields as a base might be a good idea. They result in secure constructions, but point counting is hard, and you get these “special divisors” which make constant time programming a nightmare, and get nothing in return.

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