Elliptic Curves: The Finite Field

Most people think of "Curves" as smooth lines on a graph. But in Bitcoin, we use Elliptic Curves over a Finite Field. This transforms the smooth curve into a scattering of discrete points that look like noise but follow strict mathematical rules.

1. The Equation

Bitcoin uses a specific type of elliptic curve called secp256k1. The base equation is: $$y^2 = x^3 + 7$$

2. The Finite Field (Modulo $p$)

On a normal graph, $x$ and $y$ could be any decimal number. In Bitcoin, we only use Integers within a specific range defined by a massive prime number $p$. $$y^2 \equiv x^3 + 7 \pmod p$$

• Result: The curve "Wraps around" whenever it hits the boundary of $p$.

• The Prime $p$: $2^{256} - 2^{32} - 977$. It is a number so large that it is almost $2^{256}$.

3. Symmetry

Even though the points are scattered, the curve remains Symmetric across the X-axis. For every point $(x, y)$, there is a corresponding point $(x, -y \pmod p)$. This symmetry is what allows for Public Key Compression.

4. Why use Elliptic Curves?

Elliptic Curve Cryptography (ECC) is significantly more efficient than older systems like RSA.

• A 256-bit Elliptic Curve key provides the same level of security as a 3072-bit RSA key.

• Impact: Smaller keys mean smaller transactions, lower fees, and faster verification for nodes.

Property	Elliptic Curve (ECC)	RSA
Key Size	256 bits	3072 bits
Security Level	High	High
Efficiency	Superior	Low

In the next section, we will analyze the geometric rules of Point Addition & Doubling.