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 arbitrary, with its third intersection of a line and then reflected? Why not use, say, the Monster Group? Surely a monster is better equipped to guard your secrets than some curve thing named after, but legally distinct from, a completely different curve thing! I wouldn’t have a satisfying answer to this question until way into my graduate studies, and the answer makes a perfect blog post for the “Cryptography (Incomprehensible)” category of this blog, so here we go. Groups First, as a recap, what do we mean by Diffie-Hellman? Well, we need a group , and element of this group with some order , so is the smallest positive integer with where is the neutral element of the group. The we take our private key and compute our public key as . We can now compute a shared secret with some other public key a ... Read full article.