Elliptic curves

Here’s the trapdoor at last. Bitcoin’s keys live on an elliptic curve — a gently looping shape with one magical feature: you can “add” points on it, hopping around the curve in a way that’s easy to do forwards and effectively impossible to undo. That asymmetry is what links your private and public keys.

A curve with an addition rule

An elliptic curve is the set of points satisfying an equation like y² = x³ + ax + b. What makes it special is a rule for “adding” two points: draw a straight line through them, find the third place it crosses the curve, and reflect that point across the x-axis. The result is defined to be the sum. It sounds arbitrary, but it obeys tidy algebraic laws — and it lets us do repeatable, well-defined jumps around the curve.

Play with it

Start at a point P (the “generator”). Each press adds P again — plotting P, then 2P, 3P, 4P… Watch where the points land: they scatter unpredictably around the curve, even though each step is a simple geometric construction.

The trapdoor: scalar multiplication

Adding P to itself n times is called scalar multiplication: n × P. This is exactly how a public key is born:

• Your PRIVATE key is a giant secret number, n.
• Your PUBLIC key is the point you land on after adding the generator to itself n times: n × P.

Going forwards is easy — even for an astronomically large n, clever doubling lets a computer reach n × P in a flash. But going backwards — being shown the final point and asked “how many hops was that?” — is the “elliptic-curve discrete logarithm problem”, and nobody knows how to solve it. As the demo hints, the landing points give no clue about the number of steps.

You now have every ingredient: hard money’s missing digital form, peer-to-peer networks with no centre, hashes to fingerprint data, and keys and signatures to prove ownership. The next section assembles them into the thing itself — Bitcoin.