What does the secp256k1 elliptic curve really look like?

[Fresh_Milk]

My brain says  math but my imagination says Christmas tree.

I know the question sounds strange, but if there’s one place where I can ask it, it’s here — definitely not at a family dinner. I’ve been wondering how people visualize the secp256k1 elliptic curve used in Bitcoin.

I’ve searched all over the internet and found all kinds of representations:

– some show a wobbly, twisted curve,
– others show a circle or parallel straight lines,
– others show a cloud of points inside a rectangle,
– and some even show galaxies… which I kind of like.

Basically, nobody shows the same thing, and I’m asking myself: what does it actually look like?
Is the word “curve” even appropriate???

Personally, in my imagination — with zero artistic talent — it looks like a Christmas tree.
The point G is the star at the top, and as you go down, there are more and more branches: a perfectly symmetric tree. 🎄⇅

I’m curious: what do you see?

[pooya87]

I’ve searched all over the internet and found all kinds of representations:

The bitcoin wiki has a page on it that also shows the curve.
https://en.bitcoin.it/wiki/Secp256k1

Also here is a fun site you can use to play around with such stuff. Here is the curve Bitcoin uses https://www.desmos.com/calculator/xcnlmiauec

Here is another more complete one that allows you to perform and visualize point addition as well: https://www.desmos.com/calculator/ialhd71we3 (fix the values for b and a to get secp256k1 curve like I did in previous link)

[stwenhao]

I’ve been wondering how people visualize the secp256k1 elliptic curve used in Bitcoin.

There is a repository for that: https://github.com/vjudeu/curves1000/tree/master/png

Everything up to 1000 is covered, for example for p = 967: https://github.com/vjudeu/curves1000/blob/master/png/967.png

what does it actually look like?

You have (x,y) points, where each coordinate is in range from 1 to p - 1. And you have N points. Which means, that instead of 967 x 967 square with 906 white dots, you have 115792089237316195423570985008687907853269984665640564039457584007908834671663 x 115792089237316195423570985008687907853269984665640564039457584007908834671663 square with 115792089237316195423570985008687907852837564279074904382605163141518161494336 white dots inside.

Technically, you can represent it in many different ways, but in general, you have just (x,y) points in a huge space. Which is why around 2^128 operations are needed to break it.

I think the easiest way to understand it, is to start with smaller examples, and to increase numbers gradually, until you get it. And again, there is even another topic about it: https://bitcointalk.org/index.php?topic=5459153.0

[Fresh_Milk]

 Merci Thank you  for your feedback and for the interesting links.

So the curve is indeed the one I keep seeing everywhere visually, but i be honnest, it becomes mentally ::)difficult to represent it once we go beyond a certain number of scalar multiplications with respect to point G, which I call a ‘branch’ in the representation of my symmetrical Christmas-tree

[odolvlobo]

I imagine that it looks something like this:

[hmbdofficial]

My brain says   math but my imagination says Christmas tree.

I know the question sounds strange, but if there’s one place where I can ask it, it’s here — definitely not at a family dinner. I’ve been wondering how people visualize the secp256k1 elliptic curve used in Bitcoin.

I’ve searched all over the internet and found all kinds of representations:

– some show a wobbly, twisted curve,
– others show a circle or parallel straight lines,
– others show a cloud of points inside a rectangle,
– and some even show galaxies… which I kind of like.

Basically, nobody shows the same thing, and I’m asking myself: what does it actually look like?
Is the word “curve” even appropriate???

Personally, in my imagination — with zero artistic talent — it looks like a Christmas tree.
The point G is the star at the top, and as you go down, there are more and more branches: a perfectly symmetric tree. 🎄⇅

I’m curious: what do you see?

I use to imagine the elliptical curve multiplication in many ways before I came across a video on YouTube that gave me the convincing representation. The G point start at the top by drawing  a tangent from the point to intercept the curve.  and take the point it intercepts to the opposite point that’s like taking the inverse of that point. the starting point will be G1 and the other point from the tangent is G2 if you take another tangent and repeat same process you get more points the G points are the multiplications thats how I understand it. let me share the link with everyone here
https://youtu.be/GT6AuURD8Y0?si=5gHN3xEeVTr5MnDc

[ABCbits]

--snip--
I use to imagine the elliptical curve multiplication in many ways before I came across a video on YouTube that gave me the convincing representation. The G point start at the top by drawing  a tangent from the point to intercept the curve.  and take the point it intercepts to the opposite point that’s like taking the inverse of that point. the starting point will be G1 and the other point from the tangent is G2 if you take another tangent and repeat same process you get more points the G points are the multiplications thats how I understand it. let me share the link with everyone here
https://youtu.be/GT6AuURD8Y0?si=5gHN3xEeVTr5MnDc

That video is 1 hour long. So for other who just want to see visualization, check these page (from same author).
https://learnmeabitcoin.com/technical/cryptography/elliptic-curve/
https://learnmeabitcoin.com/technical/cryptography/elliptic-curve/ecdsa/

[NotATether]

It looks like this:



It's literally just a bunch of random points, like would be output by an RNG.

Source: Bitcoin Wiki

[satscraper]

It's literally just a bunch of random points, like would be output by an RNG.

Source: Bitcoin Wiki

Not precisely. It’s more accurate to say “a bunch of pseudo‑random points.” We use pseudo because all those points satisfy the equation 𝑦^2 ≡ 𝑥^3 + 7  (mod 𝑝), which means they are deterministically defined ^{(whereas true random numbers do not follow any deterministic rule/equation)}. We use random because, statistically, these points are distributed uniformly across the field.

Source, you are refereed to says "its graph will in reality look like random scattered points" , and this is correct.

You just missed "look like".