A curve by any other name

Winter Break Of Code Day 7 Yesterday was a day of meetings. Discussion and debate flourished. Conversations ranged over all parts of every project. Words spoken aloud may have outnumbered lines of code shipped. The entire team was fully engaged and people nearly had to be dragged out of the house for an afternoon hike to the top of the ridge. Despite our intense collective focus on conceptual progress, when Trevor agreed to present an overview of elliptic curve cryptography, the entire team, veterans and “li’l Whisperers” alike, fell silent and gathered ‘round the whiteboard. We learned about fields and curves and groups, of basepoints and cofactors, secrets and signatures. Questions abounded and Trevor delivered the answers, one after another, albeit with enough handwaving that I thought he might lift off and fly himself back to the mainland. However, there was one question that even he could not answer: Why are they called elliptic curves? From Weierstrass to Montgomery to Edwards formats, these geometric objects that form the essential mathematical underpinnings of many modern crypto systems are not defined by ellipses, nor do they resemble ellipses. Not even for very stretchy definitions of an ellipse. There is no immediately obvious connection. So why do we call them that? As usual, to understand the universe, we must first bake an apple pie from scratch. Or at least pretend to. Read more...