Sunday, May 22

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

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50 Because the underlying subject for the elliptic curve for secp256k1 is Fp the place p=2256-232-977, and since p could be very near an influence of two,
it is sensible to attempt to graph the elliptic curve y2=x3+7 over the sphere of 2-adic numbers or associated rings. Since we solely have a finite quantity of room to put up right here, allow us to merely graph the elliptic curve over a hoop of the shape Z2n.
The ring of 2-adic integers is the inverse restrict of the rings of the shape Z2n. Due to this fact, the rings Z2n will be regarded as 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:Z2n->{0,…,2n-1} be the operate the place
f(a020+…+an-12n-1)=
an-120+…+a02n-1
at any time when a0,…,an-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
Z2n. 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 complicated 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 ought to consider the finite fields as an approximation to the set of all complicated numbers. Moreover, the sphere of p-adic numbers embeds into the sphere of complicated numbers, so one might consider the finite fields as objects that approximate a subject that comprises the sphere of p-adic numbers as a sub-field. Due to this fact, it’s applicable to make use of the graphs of the elliptic curve y2=x3+7 over the reals, complicated 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.