user1878: I'm reading up on ECC curves and on many of them I see an illustration that looks like this

What does the comparable curve in Bitcoin look like, or are all curves generally the same?

user49450: Since the underlying field for the elliptic curve for secp256k1 is Fp where p=2256-232-977, and since p is very close to a power of 2,
it makes sense to try to graph the elliptic curve y2=x3+7 over the field of 2-adic numbers or related rings. Since we only have a finite amount of room to post here, let us simply graph the elliptic curve over a ring of the form Z2n.
The ring of 2-adic integers is the inverse limit of the rings of the form Z2n. Therefore, the rings Z2n can be thought of as finite approximations for the ring of 2-adic integers. Now, we want to graph the elliptic curve in such a way so that two points which are near each other with respect to the 2-adic metric are also near each other on the graphical representation of the elliptic curve.
Let f:Z2n->{0,...,2n-1} be the function where
f(a020+...+an-12n-1)=
an-120+...+a02n-1
whenever a0,...,an-1 are all elements of the set {0,1}. In other words, f simply reverses the bits in the binary representation of an element of
Z2n. Then the white pixels in the following graph are precisely at the points with coordinates (f(x),f(y)) where y2=x3+7 mod 2n where n=9. This picture is an approximation of the image of the elliptic curve over the ring of 2-adic integers.

Since the field of complex numbers is isomorphic to any ultraproduct of the algebraic closures of finite fields Fp by a non-principal ultrafilter on the set of all prime numbers, one should think of the finite fields as an approximation to the set of all complex numbers. Furthermore, the field of p-adic numbers embeds into the field of complex numbers, so one may think of the finite fields as objects that approximate a field that contains the field of p-adic numbers as a sub-field. Therefore, it is appropriate to use the graphs of the elliptic curve y2=x3+7 over the reals, complex numbers, or even the p-adic numbers as a visualization for the fields used in elliptic curve cryptography. One just needs to realize that one is working in a different field for visualization purposes.

user30396: you can check  the Bitcoin doc https://en.bitcoin.it/wiki/Secp256k1 ,
there you will find some technical details about the secp256k1 used in bitcoin.
Below an illustration of the secp256k1's elliptic curve y2 = x3 + 7 over the real numbers (plot using www.desmos.com/calculator/ialhd71we3)

in the context of a finite field Zp, which greatly changes the ECC appearance but not its underlying equation or special properties. the picture below represents the same equation in a finite field F17 (the x and y values are integers between 0 and 17).

and here over F59 :

You will find an online opensource tool here https://cdn.rawgit.com/andreacorbellini/ecc/920b29a/interactive/modk-add.html
which will help you to plot the graphe and to make addition or scalar multiplication on a EC. E.g plot over F97 with P+Q.

another good article to read about ECDSA in Bitcoin is  https://github.com/bellaj/Bitcoin_Ethereum_docs/blob/6bffb47afae6a2a70903a26d215484cf8ff03859/ecdsa_bitcoin.pdf

secp256k1 was almost never used before Bitcoin became popular, but it
  is now gaining in popularity due to its several nice properties. Most
  commonly-used curves have a random structure, but secp256k1 was
  constructed in a special non-random way which allows for especially
  efficient computation. As a result, it is often more than 30% faster
  than other curves if the implementation is sufficiently optimized.
  Also, unlike the popular NIST curves, secp256k1's constants were
  selected in a predictable way, which significantly reduces the
  possibility that the curve's creator inserted any sort of backdoor
  into the curve.


user208: I'm afraid you won't like the answer.
These curves - including the secp256k1 curve, y2 = x3 + 7 - 'look' nice when evaluated in typical fields (like the real numbers), but secp256k1 is defined over the finite field Z2256-232-977, which means the X and Y coordinates are 256-bit integers modulo a large number. Curves using such coordinates do not have any concept of continuous lines.
I've tried to plot this curve over a similar but much smaller field, Z28+1. Coordinates extend from -128 to 128.

Note that even though it may not make sense geometrically anymore, it still has all properties you need. A line (which means, a group of points with equation ay + bx + c = 0) that 'intersects' 2 points of the curve, will intersect a third. Tangent again has no geometric interpretation anymore, but you can still symbolically compute a derivative of the equation in a given point, which will have the property of intersecting the curve in a second point.
To show you what you'd get if this were over the real numbers, here is a plot of the same curve equation for that case. Once with coordinates -128 through 128, once with -8 through 8.



1878: <p>I'm reading up on ECC curves and on many of them I see an illustration that looks like this</p>

<p><img src="https://i.stack.imgur.com/hAaxS.png" alt="enter image description here"></p>

<p>What does the comparable curve in Bitcoin look like, or are all curves generally the same?</p>
 49450: <p>Since the underlying field for the elliptic curve for secp256k1 is F<sub>p</sub> where p=2<sup>256</sup>-2<sup>32</sup>-977, and since p is very close to a power of 2,
it makes sense to try to graph the elliptic curve y<sup>2</sup>=x<sup>3</sup>+7 over the field of 2-adic numbers or related rings. Since we only have a finite amount of room to post here, let us simply graph the elliptic curve over a ring of the form Z<sub>2<sup>n</sup></sub>.
The ring of 2-adic integers is the inverse limit of the rings of the form Z<sub>2<sup>n</sup></sub>. Therefore, the rings Z<sub>2<sup>n</sup></sub> can be thought of as finite approximations for the ring of 2-adic integers. Now, we want to graph the elliptic curve in such a way so that two points which are near each other with respect to the 2-adic metric are also near each other on the graphical representation of the elliptic curve.</p>
<p>Let f:Z<sub>2<sup>n</sup></sub>-&gt;{0,...,2<sup>n</sup>-1} be the function where
f(a<sub>0</sub>2<sup>0</sup>+...+a<sub>n-1</sub>2<sup>n-1</sup>)=
a<sub>n-1</sub>2<sup>0</sup>+...+a<sub>0</sub>2<sup>n-1</sup>
whenever a<sub>0</sub>,...,a<sub>n-1</sub> are all elements of the set {0,1}. In other words, f simply reverses the bits in the binary representation of an element of
Z<sub>2<sup>n</sup></sub>. Then the white pixels in the following graph are precisely at the points with coordinates (f(x),f(y)) where y<sup>2</sup>=x<sup>3</sup>+7 mod 2<sup>n</sup> where n=9. This picture is an approximation of the image of the elliptic curve over the ring of 2-adic integers.</p>
<p><a href="https://i.stack.imgur.com/Ko2nV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ko2nV.png" alt="enter image description here" /></a></p>
<p>Since the field of complex numbers is isomorphic to any ultraproduct of the algebraic closures of finite fields F<sub>p</sub> by a non-principal ultrafilter on the set of all prime numbers, one should think of the finite fields as an approximation to the set of all complex numbers. Furthermore, the field of p-adic numbers embeds into the field of complex numbers, so one may think of the finite fields as objects that approximate a field that contains the field of p-adic numbers as a sub-field. Therefore, it is appropriate to use the graphs of the elliptic curve y<sup>2</sup>=x<sup>3</sup>+7 over the reals, complex numbers, or even the p-adic numbers as a visualization for the fields used in elliptic curve cryptography. One just needs to realize that one is working in a different field for visualization purposes.</p>
 30396: <p>you can check  the Bitcoin doc <a href="https://en.bitcoin.it/wiki/Secp256k1" rel="noreferrer">https://en.bitcoin.it/wiki/Secp256k1</a> ,
there you will find some technical details about the secp256k1 used in bitcoin.</p>

<p>Below an illustration of the secp256k1's elliptic curve y2 = x3 + 7 over the real numbers (plot using <a href="http://www.desmos.com/calculator/ialhd71we3" rel="noreferrer">www.desmos.com/calculator/ialhd71we3</a>)</p>

<p><a href="https://i.stack.imgur.com/JGSXb.png" rel="noreferrer"><img src="https://i.stack.imgur.com/JGSXb.png" alt="secp256k1 plot using www.desmos.com/calculator/ialhd71we3"></a></p>

<p>in the context of a finite field Zp, which greatly changes the ECC appearance but not its underlying equation or special properties. the picture below represents the same equation in a finite field F17 (the x and y values are integers between 0 and 17).</p>

<p><a href="https://i.stack.imgur.com/SvjBj.png" rel="noreferrer"><img src="https://i.stack.imgur.com/SvjBj.png" alt=" Elliptic curve over F17"></a></p>

<p>and here over F59 :</p>

<p><a href="https://i.stack.imgur.com/HugPF.png" rel="noreferrer"><img src="https://i.stack.imgur.com/HugPF.png" alt="enter image description here"></a></p>

<p>You will find an online opensource tool here <a href="https://cdn.rawgit.com/andreacorbellini/ecc/920b29a/interactive/modk-add.html" rel="noreferrer">https://cdn.rawgit.com/andreacorbellini/ecc/920b29a/interactive/modk-add.html</a>
which will help you to plot the graphe and to make addition or scalar multiplication on a EC. E.g plot over F97 with P+Q.</p>

<p><a href="https://i.stack.imgur.com/I3NpQ.png" rel="noreferrer"><img src="https://i.stack.imgur.com/I3NpQ.png" alt="enter image description here"></a></p>

<p>another good article to read about ECDSA in Bitcoin is  <a href="https://github.com/bellaj/Bitcoin_Ethereum_docs/blob/6bffb47afae6a2a70903a26d215484cf8ff03859/ecdsa_bitcoin.pdf" rel="noreferrer">https://github.com/bellaj/Bitcoin_Ethereum_docs/blob/6bffb47afae6a2a70903a26d215484cf8ff03859/ecdsa_bitcoin.pdf</a></p>

<blockquote>
  <p>secp256k1 was almost never used before Bitcoin became popular, but it
  is now gaining in popularity due to its several nice properties. Most
  commonly-used curves have a random structure, but secp256k1 was
  constructed in a special non-random way which allows for especially
  efficient computation. As a result, it is often more than 30% faster
  than other curves if the implementation is sufficiently optimized.
  Also, unlike the popular NIST curves, secp256k1's constants were
  selected in a predictable way, which significantly reduces the
  possibility that the curve's creator inserted any sort of backdoor
  into the curve.</p>
</blockquote>
 208: <p>I'm afraid you won't like the answer.</p>
<p>These curves - including the <code>secp256k1</code> curve, <em>y<sup>2</sup> = x<sup>3</sup> + 7</em> - 'look' nice when evaluated in typical fields (like the real numbers), but secp256k1 is defined over the finite field <em>Z<sub>2<sup>256</sup>-2<sup>32</sup>-977</sub></em>, which means the X and Y coordinates are 256-bit integers modulo a large number. Curves using such coordinates do not have any concept of continuous lines.</p>
<p>I've tried to plot this curve over a similar but much smaller field, Z<sub>2<sup>8</sup>+1</sub>. Coordinates extend from -128 to 128.</p>
<p><img src="https://i.stack.imgur.com/Iil7a.png" alt="y² = x³ + 7 over Z257, -128 through 128" /></p>
<p>Note that even though it may not make sense geometrically anymore, it still has all properties you need. A line (which means, a group of points with equation <em>ay + bx + c = 0</em>) that 'intersects' 2 points of the curve, will intersect a third. Tangent again has no geometric interpretation anymore, but you can still symbolically compute a derivative of the equation in a given point, which will have the property of intersecting the curve in a second point.</p>
<p>To show you what you'd get if this were over the real numbers, here is a plot of the same curve equation for that case. Once with coordinates -128 through 128, once with -8 through 8.</p>
<p><img src="https://i.stack.imgur.com/sNHuH.png" alt="y² = x³ + 7 over R, -128 through 128" />
<img src="https://i.stack.imgur.com/nhutd.png" alt="y² = x³ + 7 over R, -8 through 8" /></p>
