[curves] Climbing the elliptic learning curve (was: Re: Finalizing XEdDSA)

[Ron Garret]

On Nov 1, 2016, at 2:40 PM, Trevor Perrin <trevp at trevp.net> wrote:

> It would be be great if there were better surveys on modern ECC and
> engineering issues.  If someone wanted to suggest a reading list /
> bibliography that would be a nice contribution (but also a bunch of
> work).

I decided it would be a useful exercise for me to undertake to write such a survey (even if I couldn’t actually finish it), and right away I ran into a snag.  I was trying to reconcile all the different forms of elliptic curve formulas commonly found in the literature, and found the following promising-looking lead on mathworld:

http://mathworld.wolfram.com/EllipticCurve.html

Ax^3 + Bx^2y + Cxy^2 + Dy^3 + Ex^2 + Fxy + Gy^2 + hHx + Iy + J = 0

This is consistent (AFAICT) with the definition given in section 4.4.2.a of Cohen and Frey.  But then there are Edwards curves, which have a x^2y^2 term in them.  How do those fit in?

In fact, as I started thinking about this I realized that Edwards curves are really weird because they’re quartic and not cubic (aren’t they?) and all elliptic curves are supposed to be cubic (aren’t they?)  How can a fourth-order polynomial be birationally equivalent to a third-order polynomial?

I tried taking a look at some of the proofs that Edwards curves are birationally equivalent to Montgomery curves but they went way over my head.  Is there a more elementary way of understanding this?

Thanks,
rg