exercise-66

[skaunov]

Hi, @erhant ! Please tell me if such issues is welcome, since it's just a way of discussions for me. Also I'm not a Pythonist, so I can't suggest a nice commit in case...

So. I feel like https://github.com/erhant/moonmath/tree/master/elliptic-curves#exercise-66 isn't done in the spirit of the exercise. To me it looks like the idea was take all projective plane points and filter them getting the set of those which are in the curve.

It's still a question for me what's the best way not to double check projective points... Is there an iterator over them in Sage? Or just find out a fine traversing algorithm, which seems not a huge headache in a small field.

[skaunov]

Though I managed to stitch a thing.

F = FiniteField(5)
P = ProjectiveSpace(2, F)

for p in P:
  if p[1]^2 * p[2] == p[0]^3 + p[0] * p[2]^2 + p[2]^3 :
    print(p)

[erhant]

Greetings @skaunov,

Please tell me if such issues is welcome, since it's just a way of discussions for me
Sure thing! This is precisely why I have gathered my answers in this repo ;)

Thank you for the suggestion, firstly I solved it by iterating over the affine points with $z=1$ because in the example it particularly says:

Moreover, a point at infinity [X : Y : 0] can only satisfy the equation in (5.19) for any a and
b, if X = 0, which implies that the only point at infinity relevant for Short Weierstrass elliptic
curves is [0 : 1 : 0], since [0 : k : 0] = [0 : 1 : 0] for all k ∈ F∗. Therefore, we can exclude all points at infinity except the point [0 : 1 : 0].

However, I liked your suggestion of using ProjectiveSpace; I think the snippet you wrote does the job! I also found the following ProjectiveSpace_field.curve online. Apparently we can construct a curve from the projective field, but I couldn't get it to work for this exercise.

Anyways, I wrote the following:

F5 = GF(5)
F5P2 = ProjectiveSpace(F5, 2)
points = [(x, y, z) for (x, y, z) in F5P2 if y^2 * z == x^3 + x * z^2 + z^3]

I will be adding it to the solution 👌🏻