i'm trying to find an algorithm that finds all points on an (projective) elliptic curve.
I find it hard to write an efficient algorithm that ignores points that are "Related".
For example, given the following (Projective) elliptic curve (a=b=1):
$${C = Y^2Z = X^3 + XZ^2 + Z^3}$$
under ${\mathbb{F}_3}$, the algorithm should return:
$${C(\mathbb{F}_3) = {(1:0:1),(0:1:2),(0:1:1),(0:1:0)}}$$ while ignoring:
$${(0:2:0),(0:2:1),(0:2:2),(2:0:2)}$$
One way to do so is to go over all options, but then when p is a large number (say >200), it takes way too long and impractical.
Does anyone have an idea?