I have a limited understanding of elliptic curve terminology, but if you can bear with me I'd like to ask this question anyway. I'll start with what I understand...
- You do EC multiplication with the private key to get the public key.
- EC Multiplication is just EC Addition (done over and over again).
- EC Multiplication can be achieved through a combination of EC Addition and EC Double (for speed).
From the perspective of the graph:
- EC Double = draw a tangent on your current point, take the intersect, invert it.
- EC Addition = draw a line through your current point and a second point, take the intersect, invert it.
EC Double I understand -- you have your current point and you draw a tangent -- no need for any extra points on the curve to get the intersect.
But in EC Addition, if I have my current point, where does the second point on the curve come from when attempting to draw a line across the graph to find the next intersect?