Public Keys: The Elliptic Curve Point

While a Private Key is just a number, a Public Key is a specific Point on a mathematical map called the Elliptic Curve. In Bitcoin, this point represents your "Digital Identity" that anyone can see.

1. The Math of Derivation

The relationship between a private key ($k$) and a public key ($P$) is defined by the formula: $$P = k \times G$$

• $k$: Your 256-bit private key.

• $G$: The "Generator Point," a fixed starting coordinate on the secp256k1 curve.

• $P$: The resulting Public Key coordinate $(x, y)$.

2. One-Way Directionality

This multiplication is not like standard multiplication. It is "Elliptic Curve Multiplication," which is easy to do in one direction but impossible to reverse.

• Easy: If I give you $k$, you can find $P$ in microseconds.

• Impossible: If I give you $P$, you cannot find $k$ even with all the computers on Earth. This is why it is safe to share your public key with the world.

3. The $(x, y)$ Coordinate

The result of the math is a pair of extremely large numbers:

• $x$: The horizontal position on the curve.

• $y$: The vertical position on the curve.

4. SEC Standards

Bitcoin follows the SEC (Standards for Efficient Cryptography). These standards define how we should serialize (write down) these $(x, y)$ coordinates so that different wallets can talk to each other.

Attribute	Description
Point Type	Affine Coordinate $(x, y)$
Curve	secp256k1
Equation	$y^2 = x^3 + 7$
Output Size	64 bytes of raw data

In the next section, we will look at the original Uncompressed way of writing these coordinates.