alt Hacker NewsWhat is an elliptic curve? (2019)
Comments
I prefer a more generic form:
(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.
If folks have ever seen “ed25519” - say when generating an ssh key, and wondered what it meant and how that tiny thing could still be secure
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
Dr Cook has been smashing out some excellent very digestible math content lately.
Edit: Just realised this was posted in 2019.
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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)