Regarding fully homomorphic encryption; only a small number of operations can be performed on FHE variables without the public key, and "bootstrapping" FHE from a somewhat homomorphic scheme requires the public key to be used in all operations as well as the secret key itself to be encrypted under the FHE scheme to allow bootstrapping, at least with the currently known schemes based on lattices and integer arithmetic by Gentry et al.

It seems unlikely that FHE could operate without knowledge of at least the public key. If it were possible to continue a simulation indefinitely without the public key then the implication is that one could evaluate O(2^N) simulations with O(N) work: Choose an N-bit scheme such that N >= the number of bits required for the state of the simulation and run the simulation on arbitrary FHE values. Decryption with any N-bit key would yield a different, valid simulation history assuming a mapping from decrypted states to simulated states.