Understanding of elliptical curves and curve: guide
Elliptic curves are the basic concept of the theory of numbers, cryptography and coding theory. One of the most common types of elliptical curves is the SECP256K1 curve, which has gained a widespread reception in Bitcoin applications and other blockchain applications. In this article, we will delve into the world of elliptical curves, especially focusing on the rank of the SECP256K1 curve.
What is an elliptical curve?
The elliptical curve is a mathematical subject, which consists of a set of points in a two -dimensional space, called the affect plane. It is defined by a few points (x0, y0) and (x1, y1), where x0y1 = x1y0. The curve equation can be saved as:
y^2 - s (x) xy + t (x)^2 = 0
Where s (x) and t (x) are two polynomials in X.
ECP256K1 elliptical curve 
The SECP256K1 curve is a popular elliptical curve that has been chosen for Bitcoin cryptographic algorithms due to the high level of security. It is based on the problem of the logarithm of the discrete elliptical curve (ECDLP), which is considered one of the most difficult problems in the theory of numbers.
Rank of the curve
The rank of the elliptical curve refers to its maximum order, marked by k. In other words, represents the highest possible order of the point on the curve. The rank of the curve determines the difficulty of solving the ECDLP problem for points on the curve.
In the case of SECP256K1, the rank of the curve is K = 256. This means that the highest possible order of any point on the curve is 256.
rank of computing curve
Although the calculation of the rank of the curve is not trivial using internet tools, such as Sagemath or Pari/GP, we can derive the expression for him using algebraic techniques.
Let (X0, Y0) be a point on the SECP256K1 curve. We can prescribe the curve equation as:
y^2 - s (x) xxy + t (x)^2 = 0
Where s (x) and t (x) are polynomials in x.
Using the properties of elliptical curves, we can make an expression for the rank (k) of points on the curve:
k = lim (n → ∞) (1/n) \* ∑ [i = 0 to n-1] (-1)^i | x |^(2N-i-1)
Where x is a point on the curve, and the summary runs over all possible values and.
Calculation of the rank of the curve
To calculate the rank of the curve for SECP256K1, we must connect specific values. The most commonly used value is N = 255, which corresponds to the maximum order of points on the curve (i.e. K = 256).
After connecting these values and simplifying the expression, we get:
K off 225
Application
In this article, we studied the world of elliptical curves and especially focused on SECP256K1. Understanding how to calculate the rank of an elliptical curve, you will be better prepared to solve cryptographic problems, such as solving the ECDLP problem.
Although it may not be possible to calculate the exact value using online tools, we have made a simplified expression to calculate the rank of curve for SECP256K1. This gives a good sense of approach to the task and can help appreciate the complexity and beautiful elliptical curves in mathematics.