[curves] Climbing the elliptic learning curve (was: Re: Finalizing XEdDSA)
Ron Garret
ron at flownet.com
Thu Nov 3 12:30:49 PDT 2016
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
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