Let $p \ne 2,3$, and let $E: y^2=x^3+ax+b$ be an elliptic curve over $\mathbb F_p$ with identity element $O$. For every affine point $P=(x_1,y_1) \in E(\mathbb F_p)$, define $-P=(x_1,-y_1)$, and define $-O=O$. If $P,Q \in E(\mathbb F_p)$, then $P+O=O+P=P$ and $P+(-P)=O$. For affine points $P=(x_1,y_1)$ and $Q=(x_2,y_2)$ with $Q \ne -P$, put

\begin{align*} \lambda = \begin{cases} \dfrac{y_2-y_1}{x_2-x_1}, & P \ne Q,\cr \dfrac{3x_1^2+a}{2y_1}, & P=Q, \end{cases} \end{align*}

\begin{align*} x_3 = \lambda^2-x_1-x_2 \end{align*}

\begin{align*} y_3 = \lambda(x_1-x_3)-y_1. \end{align*}

Then the chord-and-tangent addition law on $E(\mathbb F_p)$ is given by $P+Q=(x_3,y_3)$ in the nonvertical affine case and by the exceptional identity formulas above in the vertical and identity cases.