Anyone have a good explanation for why elliptic curves have a 'natural' group law? I've seen the definition of the group law in R before, where you draw a line through two points, find the third point, and mirror-image. I feel like there's something deeper going on though.
As far as I've seen, the group law is what makes elliptic curves special. Are they the _only_ flavour of curve that has a nice geometric group law? (let's say aside from really simple cases like lines through the origin, where you can just port over the additive group from R)
aleph_minus_one 22 minutes ago [-]
>
Anyone have a good explanation for why elliptic curves have a 'natural' group law? [...]
As far as I've seen, the group law is what makes elliptic curves special. Are they the _only_ flavour of curve that has a nice geometric group law?
I asked the same question to a professor who works in topics related to algebraic geometry. His answer was very simple: it's because elliptic curves form Abelian varieties
Being an algebraic group means that the group law on the variety can be defined by regular functions.
Basically, he told to read good textbooks about abelian varieties if one is interested in this topic.
> Are they the _only_ flavour of curve that has a nice geometric group law?
The Jacobian of a hyperelliptic curve (which generalize elliptic curves) also forms an abelian variety. Its use in cryptography is named "hyperelliptic curve cryptography":
Nice explanation of elliptic curves especially the emphasis on how the underlying field changes what the curve actually is. The transition from intuitive equations to the formal definition (smooth, projective genus one) is very well done and the Curve1174 example helps clarify why not all elliptic curves look like Weierstrass forms
commandersaki 3 hours ago [-]
Dr Cook has been smashing out some excellent very digestible math content lately.
Edit: Just realised this was posted in 2019.
Momade 3 hours ago [-]
Ola
Rendered at 11:14:21 GMT+0000 (Coordinated Universal Time) with Vercel.
As far as I've seen, the group law is what makes elliptic curves special. Are they the _only_ flavour of curve that has a nice geometric group law? (let's say aside from really simple cases like lines through the origin, where you can just port over the additive group from R)
I asked the same question to a professor who works in topics related to algebraic geometry. His answer was very simple: it's because elliptic curves form Abelian varieties
> https://en.wikipedia.org/wiki/Abelian_variety
i.e. a projective variety that is also an algebraic group
> https://en.wikipedia.org/wiki/Algebraic_group
Being an algebraic group means that the group law on the variety can be defined by regular functions.
Basically, he told to read good textbooks about abelian varieties if one is interested in this topic.
> Are they the _only_ flavour of curve that has a nice geometric group law?
The Jacobian of a hyperelliptic curve (which generalize elliptic curves) also forms an abelian variety. Its use in cryptography is named "hyperelliptic curve cryptography":
> https://en.wikipedia.org/wiki/Hyperelliptic_curve_cryptograp...
(y-a)(y-b) = (x-c)(x-d)(x-k)
By varying terms on both sides or making a term as a constant, you get generalizations for conics etc.
https://en.wikipedia.org/wiki/EdDSA
Edit: Just realised this was posted in 2019.