A Formula for Adding Points

We close this section with an explicit formula for adding two points in $E(K)$. If $E$ is an elliptic curve over a field $K$, given by an equation $y^2=x^3+ax+b$, then we can compute the group addition using the following algorithm.

Algorithm 10.1.2 (Elliptic Curve Group Law)   Given $P_1, P_2\in E(K)$, this algorithm computes the sum $R=P_1+P_2 \in E(K)$.

1. [One Point $\O$] If $P_1=\O$ set $R=P_2$ or if $P_2=\O$ set $R=P_1$ and terminate. Otherwise write $P_i=(x_i,y_i)$.
2. [Negatives] If $x_1 = x_2$ and $y_1 = -y_2$, set $R=\O$ and terminate.
3. [Compute $\lambda$]Set $$\lambda = \begin{cases} (3x_1^2+a)/(2y_1) & \text{if }P_1 = P_2,\\ (y_1-y_2)/(x_1-x_2) & \text{otherwise.} \end{cases}$$
   Note: If $y_1=0$ and $P_1=P_2$, output $\O$ and terminate.
4. [Compute Sum] Then $R = \displaystyle \left(\lambda^2 -x_1 - x_2, -\lambda x_3 - \nu\right)$, where $\nu = y_1 - \lambda x_1$ and $x_3$ is the $x$ coordinate of $R$.