Scalar Multiplication

The Generator Point

In ECC, everyone agrees on a starting point called $G$, the generator point.

This is just a specific point on the curve that’s been standardized. For Bitcoin’s secp256k1 curve, $G$ is a fixed point that everyone uses.

Multiplying a Point by a Number

What does $1000G$ mean?

It means: add $G$ to itself 1000 times.

This is called scalar multiplication. The scalar (1000) tells us how many times to add.

The Naive Way

To compute $1000G$:

• Start with $G$
• Add $G$: $2G$
• Add $G$: $3G$
• … repeat …
• Add $G$: $1000G$

This takes 999 additions. Too slow for cryptography.

The Smart Way: Double-and-Add

There’s a much faster method using doubling.

How Double-and-Add Works

Instead of adding $G$ one at a time, use doubling to jump ahead.

• $G \to 2G$ (one double)
• $2G \to 4G$ (one double)
• $4G \to 8G$ (one double)
• … keep doubling …

Each doubling gets you twice as far. By combining doubles and adds cleverly, you can reach any number in roughly $\log_2(k)$ steps.

For $1000G$: only ~14 operations instead of 999!

Why This Matters

For a 256-bit scalar (like in Bitcoin):

The double-and-add algorithm makes ECC practical.

The One-Way Property

Here’s what makes ECC secure:

Easy direction:

Given $G$ and $k$, computing $kG$ is fast (using double-and-add).

Hard direction:

Given $G$ and $Q = kG$, finding $k$ is practically impossible.

This is the Elliptic Curve Discrete Logarithm Problem (ECDLP).

You can multiply forward easily, but you can’t divide backward.