cryptography – What does the curve utilized in Bitcoin, secp256k1, appear to be?

Because the underlying subject for the elliptic curve for secp256k1 is F_p the place p=2²⁵⁶-2³²-977, and since p could be very near an influence of two,
it is smart to attempt to graph the elliptic curve y²=x³+7 over the sector of 2-adic numbers or associated rings. Since we solely have a finite quantity of room to submit right here, allow us to merely graph the elliptic curve over a hoop of the shape Z_2ⁿ.
The ring of 2-adic integers is the inverse restrict of the rings of the shape Z_2ⁿ. Due to this fact, the rings Z_2ⁿ may be considered finite approximations for the ring of 2-adic integers. Now, we wish to graph the elliptic curve in such a manner in order that two factors that are close to one another with respect to the 2-adic metric are additionally close to one another on the graphical illustration of the elliptic curve.

Let f:Z_2ⁿ->{0,…,2ⁿ-1} be the operate the place
f(a₀2⁰+…+a_n-12^n-1)=
a_n-12⁰+…+a₀2^n-1
at any time when a₀,…,a_n-1 are all components of the set {0,1}. In different phrases, f merely reverses the bits within the binary illustration of a component of
Z_2ⁿ. Then the white pixels within the following graph are exactly on the factors with coordinates (f(x),f(y)) the place y²=x³+7 mod 2ⁿ the place n=9. This image is an approximation of the picture of the elliptic curve over the ring of 2-adic integers.

Because the subject of advanced numbers is isomorphic to any ultraproduct of the algebraic closures of finite fields F_p by a non-principal ultrafilter on the set of all prime numbers, one ought to consider the finite fields as an approximation to the set of all advanced numbers. Moreover, the sector of p-adic numbers embeds into the sector of advanced numbers, so one might consider the finite fields as objects that approximate a subject that incorporates the sector of p-adic numbers as a sub-field. Due to this fact, it’s acceptable to make use of the graphs of the elliptic curve y²=x³+7 over the reals, advanced numbers, and even the p-adic numbers as a visualization for the fields utilized in elliptic curve cryptography. One simply wants to understand that one is working in a distinct subject for visualization functions.