The animated elliptic curve
curves.ulfheim.net
The animated elliptic curve
1–10 of 62 posts
Re: The animated elliptic curve
#2Author here, let me know if there are any questions or comments!
Re: The animated elliptic curve
#3Author here, let me know if there are any questions or comments!
Very nice work! Useful and informative. I’ll spread it at work.
Re: The animated elliptic curve
#4In the past 5 years I've seen maybe a dozen "how elliptic curves work", but this is the first to actually illustrate how they work on a small field. I think that's key to understanding, seeing it in a small enough field that you can literally see all of the points. Nicely done.
If you want to keep going, as an advanced beginner I'd like to see:
Arbitrary bigint math - how do you do Exp/Sqrt with arbitrary sized ints? (I'm familiar with two crypto libs that do this, and MPIs & branches confuse me).
Second: why is C25519 faster than SecP256R1 or BrainPool? maybe some insight there? (Isn't Ed25519 the signature name? and X25519 the ECDH name?)
Greedy of me, but thanks!
Re: The animated elliptic curve
#5Author here, let me know if there are any questions or comments!
This is really great! I was fortunate enough to do my master's under a professor who included these visual representations in his lectures. I loved enough to write my thesis on the next step generalization, hyperelliptic curves.
You might be interested in the fact that a variant of this visual representation still works: https://www.juricho.me/files/masterarbeit-hyperelliptic_curv...
Re: The animated elliptic curve
#6In the past 5 years I've seen maybe a dozen "how elliptic curves work", but this is the first to actually illustrate how they work on a small field. I think that's key to understanding, seeing it in a small enough field that you can literally see all of the points. Nicely done. If you want to keep going, as an advanced beginner I'd like to see: Arbitrary bigint math - how do you do Exp/Sqrt with arbitrary sized ints?…
> this is the first to actually illustrate how they work on a small field
That means a lot, thanks! That was my design goal with the page: figuring out the best way to get the idea across _without_ the user having to read a lot of text and stare at the wall until they got it.
I've been casting around for the next idea to do a visualization of, adding yours to the list.
(as for a partial answer your second question: the answer is going to be that Montgomery curves like Curve25519 have a method [Montgomery ladder] to quickly and timing-safely calculate only the x values of point multiplication. Faster _and_ more likely to be implemented securely than NIST curves, by design. Unfortunately I don't know the details of BrainPool, yet?)
Yes, Curve25519 is the curve itself (and associate params), Ed25519 is the signature system implemented on top of C25519, and X25519 is the ECDHE mechanism implemented on top of C25519. This page talks mostly about the C and a little of the X, and doesn't go into Ed.
Re: The animated elliptic curve
#7Author here, let me know if there are any questions or comments!
Great article and great visuals! One very minor missing detail is how the base point P is picked.
Re: The animated elliptic curve
#8My broken record take on this stuff is that whatever value there is in visualizing elliptic curves, there's more intuition to build just by playing with the curve formulae directly (it's pretty basic math) and seeing how the operations work in code. I don't have the math background to back this up but my understanding of the group rules is that they're --- at least for the application we're using them in --- a little arbitrary.
Re: The animated elliptic curve
#9Author here, let me know if there are any questions or comments!
Great article and great visuals! One very minor missing detail is how the base point P is picked.
Ah, I tried to trim the page down as much as possible, but there's a million tangents like this I could have gone down.
Each point that you pick is going to have a different number of times it can be added to itself before it lands on a point that has the same x-value but different y-value, and then the "point addition" operation draws a vertical line and the point goes to infinity. The number of times you can add a point to itself before it happens and the cycle resets is called the point's "order".
Most of the points on the graph will repeat themselves after less than a dozen times. The one I picked repeats itself after 72 points, which is great because that's every point on the curve. I chose it by writing a little program that tried each point and returned the best one.
Compare that to a "real" curve like Curve25519: it has the base point at x=9 and can repeat itself over 2^252 times before repeating. The author of that curve used a different technique to find the point's order (obviously he didn't try adding the point to itself a trillion^6 times) but the idea's the same.
Re: The animated elliptic curve
#10My broken record take on this stuff is that whatever value there is in visualizing elliptic curves, there's more intuition to build just by playing with the curve formulae directly (it's pretty basic math) and seeing how the operations work in code. I don't have the math background to back this up but my understanding of the group rules is that they're --- at least for the application we're using them in --- a little…
That's true for people whose verbal intelligence exceeds their spatial intelligence but might not be true for everybody. As for the importance of understanding eliptic curves... perhaps people would be best served by thinking of them as a black box discrete logarithm problem.