Ah, I tried to trim the page down as much as possible, but there's a million tangents like this I could have gone down.
Each point that you pick is going to have a different number of times it can be added to itself before it lands on a point that has the same x-value but different y-value, and then the "point addition" operation draws a vertical line and the point goes to infinity. The number of times you can add a point to itself before it happens and the cycle resets is called the point's "order".
Most of the points on the graph will repeat themselves after less than a dozen times. The one I picked repeats itself after 72 points, which is great because that's every point on the curve. I chose it by writing a little program that tried each point and returned the best one.
Compare that to a "real" curve like Curve25519: it has the base point at x=9 and can repeat itself over 2^252 times before repeating. The author of that curve used a different technique to find the point's order (obviously he didn't try adding the point to itself a trillion^6 times) but the idea's the same.
That's also where had to scroll up again; where do Alice and Bob know P from?
That's pre defined public knowledge, right? It belongs to the curve they use.
Many many thanks for this brilliantly depicted explanation!
Ot: I also looked up ulfheim after I realized your first name is Michael, not Ulf.
"ulfheim" is an old domain name that I've had for decades; there's a little explanation on my home page but the short version is that it's from an old BBS handle.
Unfortunately a few years ago a racist hate group also started using the name for their own purposes. Today I've started the process of moving all my hosts to a new domain name, xargs.org .
