1. Overview of Secret Key Generation

After obtaining a 256-bit private key, the corresponding public key needs to be calculated within a finite field using elliptic curve encryption algorithms. The mathematical fundamentals involved in this process belong to the realm of number theory and include modular arithmetic, prime theory, Euler's function, Euler's theorem, Euclidean extended theorem, and so on. Essentially, it can be regarded as an advanced version of RSA cryptography.

2. Elliptic Curve Encryption Algorithm ( ECC ) Analysis

Elliptic curve cryptography is a type of one-way asymmetric encryption technology, with its core being the irreversibility of operations. Any operation with irreversibility characteristics can be applied to the field of asymmetric encryption. Currently, the mainstream irreversible operations include "modular arithmetic" and "point operations", which are also known as one-way functions or one-way operations.

1. The performance of elliptic curve functions in the real number domain

The most commonly used type of elliptic curve in cryptography is the Weierstrass standard form. There are different representations in various mathematical contexts, but the form typically used in cryptography is: y^2=x^3+ax+b (where x and y are real numbers).

The Bitcoin system uses a specific elliptic curve function that adheres to the SEC (Standards for Efficient Cryptography) specifications: y^2=x^3+7 (where a=0, b=7, and x and y are real numbers). This function exhibits a unique curved shape in the coordinate system.

Figure 1 shows the graph of the function y^2=x^3+7 (where x and y are real numbers). This type of curve is quite different from traditional elliptical shapes, but it has unique value in cryptographic applications.