Associativity 1

Adding two points on an elliptic curve involves a specific process defined by the curve's geometry and algebraic properties. Here’s a brief description of the process for adding two distinct points P and Q on an elliptic curve:

1. Draw a line through P and Q: Begin by drawing a straight line that passes through both points P and Q. This line represents the "addition" of the two points in geometric terms.

2. Find the third intersection: This line will generally intersect the elliptic curve at a third point, say R'. The curve's symmetry about the x-axis means that this intersection point is not directly the sum of P and Q, but rather, in the context of the curve's group law, it is the point -(P + Q).

3. Reflect across the x-Axis: To find the actual sum P + Q, reflect R' across the x-axis. This reflection, R, is the result of the addition P + Q.

The process is guided by the elliptic curve equation, typically of the form y^2 = x^3 + ax + b, and involves algebraic calculations to determine the exact coordinates of R when P and Q are known. The key steps involve:

- Calculating the slope m of the line through P and Q.
- Using m to find the x-coordinate of R', x_R', through the formula x_R' = m^2 - x_P - x_Q.
- Finding the y-coordinate of R' and then determining the y-coordinate of R by using the curve's equation or directly reflecting R' across the x-axis, effectively calculating y_R = -y_R' + s where s is a constant determined by the specific form of the elliptic curve equation.

This geometric and algebraic process allows for the definition of a group structure on the set of points on an elliptic curve, which is fundamental to the curve's applications in cryptography, specifically in elliptic curve cryptography (ECC).