Point Addition & Doubling

To perform math on an elliptic curve, we can't use standard addition ($1+1=2$). Instead, we use a geometric set of rules called the Chord-and-Tangent method.

1. Point Addition ($P + Q = R$)

To add two different points $P$ and $Q$ on the curve:

1. Draw a straight line through $P$ and $Q$.

2. The line will intersect the curve at exactly one other point (let's call it $-R$).

3. Reflect $-R$ across the X-axis to find the result: $R$.

2. Point Doubling ($P + P = 2P$)

What if you want to add a point to itself? You can't draw a line through two points if they are the same.

1. Draw a Tangent Line to the curve at point $P$.

2. The tangent line will intersect the curve at exactly one other point ($-R$).

3. Reflect $-R$ across the X-axis to find the result: $2P$.

3. The Algebraic Reality

While we visualize this with lines and reflections, computers perform these steps using Modular Arithmetic.

• Slope ($\lambda$): Calculated using the derivative of the curve equation.

• New Coordinates: $x_3 = \lambda^2 - x_1 - x_2$ and $y_3 = \lambda(x_1 - x_3) - y_1$.

4. The Identity Point ($O$)

In normal math, $0$ is the identity ($5+0=5$). In Elliptic Curve math, we have the Point at Infinity ($O$).

• If you add a point $(x, y)$ to its reflection $(x, -y)$, the vertical line never hits the curve again. It goes to "Infinity."

• Any point $P + O = P$.

5. Associativity

Just like normal addition, $(P + Q) + R = P + (Q + R)$. This property is critical because it allows us to perform massive multiplications efficiently.

In the next section, we will explore Scalar Multiplication (The Trapdoor).