Well, this was thoroughly unenlightening and way too full of romantic fluff.
Don’t get me wrong, I love me some mathematical drama, but the article doesn’t actually explain anything except “they come from elliptic integrals”.
This is a much better explanation. Roughly, it goes a little something like this:
You are trying to find the arclength of an ellipse.
This leads to complicated integrals, called elliptic integrals, which do not have elementary antiderivatives
Elliptic integrals look analogous to, shall we say, “circular” integrals, for example the antiderivative of arcsin is the integral of the square root of a quadratic function.
Just like with “circular” integrals the most natural function to study is sine, not arcsine, with elliptic integrals we also look at the inverse function, which we call an elliptic function.
These elliptic functions are doubly periodic, just like circular functions are singly periodically.
This double periodicity uniquely characterises them: any doubly periodic function must be an elliptic function.
Turns out that elliptic functions satisfy a differential equation [y']^2 = cubic(y).
This differential equation can be used to parametrise elliptic functions with curves of the form y2 = cubic.
Well, this was thoroughly unenlightening and way too full of romantic fluff.
Don’t get me wrong, I love me some mathematical drama, but the article doesn’t actually explain anything except “they come from elliptic integrals”.
This is a much better explanation. Roughly, it goes a little something like this:
And that’s why they’re called elliptic curves.
Shouldn’t this also be tagged as “security”?
It could. I elected not to because despite the application of curves to crypto, the post is really only about the history.