Can we do modulus switching for number theoretic encryption? Can we do modulus switching for number theoretic encryption such as Paillier or ElGamal?
Security goals like confidentiality is the reason to ask for encryption.
Modulus is chosen (for particular group types) on the grounds of hard problems associated with that groups; encryption strength is defined by level of hardness of that problems.

It would be nice to discover a hard problem related to some change of modulus, comparable to indistinguishability of Diffie-Hellman tuples of 4 group elements for ElGamal or $n$-th roots for Paillier with modulus $n^2$.