My question is about elliptic curves over $GF(p)$: How is the order of a generating element $G$ (which is to my knowledge also the order of the cyclic subgroup $G^n$) calculated? Taking P-256 as an example, I know basically what the base point $G$ is and also what its order is.
- But how is the order of any point calculated? The order is really a very big number and I think that its value was not found by brute force.
- Under what circumstances is the order of the subgroup $G^n$ the same as the order of the whole group (co-factor = 1) ?
- Why must the order always be prime?
- is each group for elliptic curves over $GF(p)$ group cyclic? In this case there is a generating point G (a base point) and all sub-groups are cyclically, too.
- How are "good" points found and what is the cryprtographic meaning of "good" ?
From group theory I just know, that the order of a subgroup divides the order of the complete group, which gives rise to the definition of the co-factor h, but that doesnt help.
The questions are not of practical interest, but I want to understand what I'm doing. The information I read so far seem to appear from nowhere. If the mathematical background, needed to answer those questions is not totally out of scope for a non-mathematician, where can I read about that?