Elliptic Curve Math

For an elliptic curve of the form $y^{2}=x^{3}+ax+b$ .

Notes

An elliptic curve is any curve that can be described using an equation of the form $y^{2}=x^{3}+ax+b$ .
Addition over elliptic curves is performed using the following equations.
$\lambda=\begin{cases}\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\text{ if }x_{2}\ne x_{1} \\\frac{3x_{1}^{2}+a}{2y_{1}}\text{ if }x_{2}=x_{1}\end{cases}$
$x_{3}=\lambda^{2}-x_{1}-x_{2}$
$x_{3}=\lambda(x_{1}-x_{3})-y_{1}$