The explicit triplication formula for elliptic curves

Question

Does someone have the triplication formula, or its reference?

I know the explicit addition formula for elliptic curves, but it is too complex to get the triplication formula using it.

Answer 1

HINT.-Let $\Gamma$ be the elliptic curve $y^2=x^3+ax+b$ and $P=(x_p,y_p)$, $Q=(x_q,y_q)$ two points of $\Gamma$

►Case $P\ne Q$ (the case of a chord secant).

Making $\lambda=\dfrac{y_q-y_p}{x_q-x_p}$ you have $P+Q=(x_r,-y_r)$ where $$x_r=\lambda^2-x_p-x_q\\y_r=\lambda(x_p-x_r)-y_p\Rightarrow -y_r=\lambda^3-(2x_p+x_q)+y_p$$

►Case $P=Q=(x_1,y_1)$ (the case of a tangent).

Making now (by the limit for the tangent) $\lambda=\dfrac{3x_1^2+a}{2y_1}$ you have $2P=(x_2,-y_2)$ where $$x_2=\lambda^2-2x_1\\y_2=\lambda(x_1-(\lambda^2-2x_1))-y_1\Rightarrow-y_2=\lambda^3-3x_1\lambda+y_1$$

►Do you have $3P=P+2P$ and $P\ne2P$ ($P$ non torsion point) which is the first case above with $$P=(x_1,y_1)\\Q=2P=(\lambda^2-2x_1,\lambda^3-3x_1\lambda+y_1)$$

The resulting polynomial expressions are of the degree $9$ (in general the degree of $nP$, when $P$ non-torsion point, is of degree $n^2$).

Try to find $3P$ (belong to the fourth quadrant) in the attached figure.