How to determine whether a point is at infinity in homogenous coordinates?

Question

I'm implementing ECC in my spare time project. I'm referencing RFC-6090 for point arithmetic algorithm over homogeneous coordinates.

In Appendix F subsection 2, there are 5 case labels when determining which formula to use depending on how many if any point-at-infinity exists in operands. For me, implementing these case labels in constant-time isn't too big a problem, but I'm not too sure about determining whether a point is at infinity.

When the point-at-infinity occurs within point arithmetic, it has the homogeneous coordinates $(0,y,0)$ where $y\ne 0$. So Q: is it sufficient to check $Z = 0$ for point at infinity, or all 3 dimensions has to be checked?

Answer 1

In homogenous coordinates, we have the equivalence relation as $(X, Y, Z) \sim (\lambda X, \lambda Y, \lambda Z)$ where $\lambda \neq 0$.

• if $Z \neq 0$ then we can convert $(X, Y, Z)$ into the affine point as $(X/Z,Y/Z)$ ( notice that $\lambda$s cancels)

• if $Z = 0$ then with the equivalence class definition $(\lambda X, \lambda Y, 0)$ all represents the point at infinity. I.e. $$(x,y,0)\sim(2x,2y,0)\sim(-x,-y,0)$$

  Therefore, when one sees $Z=0$ in the projective coordinates it is the point at infinity.

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And, if the affine point $(0,0)$ doesn't satisfy the curve equation, i.e. $b \neq 0$ in the short Weierstrass $y^2 = x^3 + ax +b$, then it is a good place to store the point at infinity in the affine coordinates where normally it doesn't have a representation.

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Note that it is common to use $(X: Y: Z)$ notation for homogenous coordinates, that I did not use here.