This can be accomplished, completely generically, by running a Turing machine using fully-homomorphic encryption, and providing only the bootstrap/evaluation ke

This can be accomplished, completely generically, by running a Turing machine using fully-homomorphic encryption, and providing only the bootstrap/evaluation key, but not the encryption/decryption keys. (Note: you have to use a Turing machine rather than some application-specific circuit, or someone can use a circuit-bending attack to tamper with the data.) Unfortunately, this is infeasible even by FHE standards, and doesn't let the user get an output non-interactively.. might as well run it on your local machine. But the sorts of things SGX does may one day be doable with only high-powered maths, if we get good functional encryption or program obfuscation.