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Elliptic Curve Crypto

Elliptic Curve Cryptography (ECC) is the backbone of modern security, protecting everything from Bitcoin to your WhatsApp messages.

1. The Curve

We define a curve using the Weierstrass equation:

y² = x³ + ax + b

By adjusting a and b, we change the curve's shape. Use the sliders on the right to see this.

2. Geometric Mode

In this mode, we see the curve over real numbers. The magic is in Point Addition.

To add two points P and Q:

1. Draw a line through P and Q.
2. It intersects the curve at a third point, -R.
3. Flip -R vertically to get R = P + Q.

Try dragging the points P (pink) and Q (blue) to see the yellow sum point move!

3. Crypto Mode

Computers can't handle infinite precision real numbers. So we wrap the curve around a Finite Field of integers modulo a prime number p.

The equation becomes:

y² ≡ x³ + ax + b (mod p)

The smooth curve turns into a cloud of points. But the addition rule still applies mathematically!

4. The Trapdoor

Security comes from the Discrete Logarithm Problem.

It's easy to calculate Q = n * P (adding P to itself n times). This is like a billiard ball bouncing around the grid.

But if I give you Q and P, it is extremely hard to find n. The point jumps so chaotically that you can't retrace the steps.

Try it: Switch to Crypto Mode and click "Animate". Watch how just adding 1 to n makes the point jump wildly.