[boris1]

Yes, just do not write (mod p), as it can be misleading to the reader. A mathematician doesn't care, but in RFCs they call the (mod p) groups a "prime group" to differentiate it from the Elliptic Curve group. (In fact, I think they call them "prime fields", not merely groups).

I think the keys can be smaller because every random coordinate is a valid value (valid key), but in the case of RSA, valid values are more sparse.