# cryptography – What does the curve used in Bitcoin, secp256k1, look like?

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.

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