Point Addition And The Elliptic Curve Group

Core idea: Point addition turns the curve's valid points into a consistent arithmetic system.

Addition rules

• Different points: Draw the line through $P$ and $Q$, find its third curve intersection, then reflect that intersection across the x-axis to obtain $P+Q$.
• Doubling: When $P=Q$, use the tangent at $P$ instead of a line through two distinct points.
• Inverse: If $P=(x,y)$, then $-P=(x,-y)$ and $$P+(-P)=\mathcal{O}.$$
• Identity: The point at infinity acts like zero: $$P+\mathcal{O}=P.$$

Group view

$E(\mathbb{R})$ means all valid real-curve points together with $\mathcal{O}$. Under point addition they form an abelian group: the operation is closed, associative, has an identity and inverses, and is commutative.

Next: Reuse point addition efficiently to compute $kP$.