understanding of elliptical curves and curve ranks: a guide
Elliptical curves are a fundamental concept in numbers theory, cryptography and coding theory. One of the most common types of elliptical curves is the SECP256K1 curve, which has obtained a wide adoption in Bitcoin and other blockchain applications. In this article, we will deepen in the world of elliptical curves, focusing in particular on the rank of the Curve SECP256K1.
** What is an elliptical curve?
An elliptical curve is a mathematical object consisting of a set of points in a two-dimensional space, called blueberry plan. It is defined by a pair of points (x0, y0) and (x1, y1), where x0y1 = x1y0. The curve equation can be written as:
y^2 - s (x) xy + t (x)^2 = 0
Where s (x) and t (x) are two polynomials in x.
SECP256K1 Elliptical curve
The SECP256K1 curve is a popular elliptical curve has been chosen for cryptographic algorithms of Bitcoin, due to the high level of security. The problem of discrete logarithm (ECDLP) is based on the elliptical curve, which is considered one of the hardest problems in the theory of numbers.
Rank Curve
The curve of the rank of an elliptical curve refers to its maximum order, noted by K. In other words, it re -contains the highest possible order of a point on the curve. The rank curve determines the difficulty of solving the ECDLP problem for the points on the curve.
For SECP256K1, the rank of the curve is K = 256. This means that the highest possible order of any point on the curve is 256.
Calculation Calcas Ran
Although it is not banal to calculate the rank of curve using online tools such as Sagemath or Pari/GP, we can get an expression for it using algebraic techniques.
Either (x0, y0) a point on the Curve Secp256k1. We can rewrite 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 get an expression for the rank (K) of points on the curve:
K = lim (n → ∞) (1/n) \* ∑ [i = 0 at n-1] (-1)^i |
Where x is a point on the curve, and the summary runs about all the possible values of I.
Calculation of curve rank
To calculate the rank of the curve for the SECP256K1, we must connect some specific values. The most commonly used value is n = 255, which corresponds to the maximum order of the points on the curve (ie K = 256).
After connecting these values and simplifying the expression, we obtain:
K ≈ 225
Conclusion 
In this article, we explored the world of elliptical curves and specifically focused on SECP256K1. By understanding how to calculate the rank of an elliptical curve, you will be better equipped to address cryptographic problems, such as solving the ECDLP problem.
Although it may not be possible to calculate the exact value using online tools, we have derived a simplified expression for calculating the curve rank for the SECP256K1. This will give you a good sense of approach to pregnancy and can help you appreciate the complexity and beauty of elliptical curves in mathematics.