I think we're all aware of the "classical" Weierstrass (short?) elliptic curve equation: $y^2\equiv x^3 + ax +b \pmod p$. Well known examples of these curves include the NIST's and Brainpool ones.
Now there's also the "Montgomery" representation: $by^2\equiv x^3 + ax^2 + x \pmod p$ where the famous Curve25519 is an example for.
To get things even more complicated there are "Edwards" and "Twisted Edwards" curves (of which I don't even have the general equations). A well-known sample would be Ed25519 I think.
- Now what are the equations of these curves ([twisted] edwards only)?
- Can they be converted into each other?
- What are the computational advantages over using a given curve equation over another?
- Are there any security implications (including side-channels) in choosing a specific equation?