What is the hard problem for this algebraic encryption construct?
I want to get a good characterization of the construct so that I may describe it accurately.


I would characterize it as "insecure".

If someone has a ciphertext, and manages to guess the plaintext it corresponds to, then they can compute:

$$\text{gcd}(E_I(M) - M, N)$$

and they'll give then the factor $p$.