June 21, 2021


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A curve by any other name

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.

A curve by any other name

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.