One thing that I wonder about for a long time and to which I did not find an answer after doing a search on the web and I hope to find an answer here.
When we construct the elliptic curve over a main field, why do we actually select a cyclic subgroup instead of taking the entire group of the elliptic curve?
On a side note what confuses me the most about this choice: we know that the first order cyclic subgroup
P is isomorphic to
Z / pZ and finding isomorphism would mean solving the discrete register.
Switching to a cyclic group actually seems more like making the problem easier compared to staying with the full elliptical curve.