5 ms·

What is an elliptic curve? (2019)

139 points by tzury 10mo ago

zkmon 10mo ago
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.
commandersaki 10mo ago
Dr Cook has been smashing out some excellent very digestible math content lately. Edit: Just realised this was posted in 2019.
- astrobiased 10mo ago
  John is not only smart and knowledgeable, but an incredibly great person to know in general. I worked with him on a project briefly back in 2012 and he stood out as a champion for science, coding, and education. His posts clearly reflect him well.
Momade 10mo ago
[flagged]
Rakshath_1 10mo ago
[flagged]
jasonjmcghee 10mo ago
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 https://en.wikipedia.org/wiki/EdDSA https://en.wikipedia.org/wiki/EdDSA
soVeryTired 10mo ago
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 10mo 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 > https://en.wikipedia.org/wiki/Abelian_variety 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 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_cryptography https://en.wikipedia.org/wiki/Hyperelliptic_curve_cryptograp...
  - ogogmad 10mo ago
    The following is one of my favourite submissions to HN: https://news.ycombinator.com/item?id=44083753 https://news.ycombinator.com/item?id=44083753
- daynthelife 10mo ago
  I find a lot of motivation from topology. If you plot a smooth degree d curve over the complex numbers, it forms a surface of degree g=(d-1)(d-2)/2. In the case of a cubic, we get genus 1, i.e. a torus. Now tori admit a very natural group action, namely addition in (R/Z)^2. And sure enough, if you pick the right homeomorphism, this corresponds to the group action given by the elliptical curve. Of course, the homeomorphism to (R/Z)^2 does not respect the geometry (it is not conformal). If we want the map to preserve angles, we need our fundamental domain to be a parallelogram instead of a rigid square. The shape of the parallelogram depends on the coefficients of the cubic, and the isomorphism is uniquely defined up to choice of a base point O (mapping to the identity element; for elliptic curves, this is normally taken to be the point at infinity). You still get a group law on the parallelogram from vector addition in the same way, and this pulls back to the precise group action on the elliptic curve. The real magic is that the resulting group law is algebraic, meaning that a*b can be written as an algebraic function of a and b. This means you can do the same arithmetic over any field, not just the complex numbers, and still get a group action.

139 points by tzury 10mo ago

zkmon 10mo ago

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.

commandersaki 10mo ago

Dr Cook has been smashing out some excellent very digestible math content lately. Edit: Just realised this was posted in 2019.

astrobiased 10mo ago
John is not only smart and knowledgeable, but an incredibly great person to know in general. I worked with him on a project briefly back in 2012 and he stood out as a champion for science, coding, and education. His posts clearly reflect him well.

Momade 10mo ago

[flagged]

Rakshath_1 10mo ago

[flagged]

jasonjmcghee 10mo ago

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 https://en.wikipedia.org/wiki/EdDSA https://en.wikipedia.org/wiki/EdDSA

soVeryTired 10mo ago

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 10mo 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 > https://en.wikipedia.org/wiki/Abelian_variety 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 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_cryptography https://en.wikipedia.org/wiki/Hyperelliptic_curve_cryptograp...
- ogogmad 10mo ago
  The following is one of my favourite submissions to HN: https://news.ycombinator.com/item?id=44083753 https://news.ycombinator.com/item?id=44083753
daynthelife 10mo ago
I find a lot of motivation from topology. If you plot a smooth degree d curve over the complex numbers, it forms a surface of degree g=(d-1)(d-2)/2. In the case of a cubic, we get genus 1, i.e. a torus. Now tori admit a very natural group action, namely addition in (R/Z)^2. And sure enough, if you pick the right homeomorphism, this corresponds to the group action given by the elliptical curve. Of course, the homeomorphism to (R/Z)^2 does not respect the geometry (it is not conformal). If we want the map to preserve angles, we need our fundamental domain to be a parallelogram instead of a rigid square. The shape of the parallelogram depends on the coefficients of the cubic, and the isomorphism is uniquely defined up to choice of a base point O (mapping to the identity element; for elliptic curves, this is normally taken to be the point at infinity). You still get a group law on the parallelogram from vector addition in the same way, and this pulls back to the precise group action on the elliptic curve. The real magic is that the resulting group law is algebraic, meaning that a*b can be written as an algebraic function of a and b. This means you can do the same arithmetic over any field, not just the complex numbers, and still get a group action.