Not sure if that made sense, but essentially you have 8 "different possibilities" for a point P.

P, P + S1 , P + S2 , P + S3 , ..., P + S7

S1 has an order of 8. So all of these points differ by an order of 8.

For example, 8P = 8 (P + S1) = 8P + 8*S1 = 8P + Identity = 8P

With Ritretto you use a 2-isogeny to map it into the Jacobi Quartic, which basically doubles the points.


2P, 2(P + S1) , 2(P + S2) , 2(P + S3) , ..., 2(P + S7)

Now all of these points differ by an order of 4. And it turns out that in the Jacobi Quartic under certain conditions, we can represent this set of points succintly. Should be on Ristretto.group.

It also turns out that if we enforce two checks, we can _force_ the representative to be the first point. Whenever you send someone a RistrettoPoint, you are sending the representative for that set and they also ensure that the checks I mentioned previously are satisfied.