The Elliptic Curve

The Equation

An elliptic curve is defined by:

$y^2 = x^3 + ax + b$

Different values of $a$ and $b$ give different curves.

A Real Example

Bitcoin and Ethereum use a curve called secp256k1:

$y^2 = x^3 + 7$

Here $a = 0$ and $b = 7$.

What Does It Look Like?

The curve is symmetric across the x-axis.

If the point $(x, y)$ is on the curve, then $(x, -y)$ is also on the curve.

This symmetry is important for how we “add” points.

Points on the Curve

A point is just an $(x, y)$ coordinate that satisfies the equation.

Example: Is $(2, 3)$ on the curve $y^2 = x^3 + 7$?

$y^2 = 3^2 = 9$

$x^3 + 7 = 2^3 + 7 = 8 + 7 = 15$

$9 \neq 15$, so no, this point is not on the curve.

Why This Shape?

The curve must be non-singular, meaning it has no cusps or self-intersections.

The special shape of this curve makes point addition work consistently.