Consider the following exchange:
- Alice has some plaintext $x$. She sends the encrypted version $\mathcal E(x)$ to Bob.
- Bob returns $(\mathcal E(x), T)$, where $T$ is a turing machine encoding the function he wants to preform on the unencrypted data.
- To decrypt this result, Alice first decrypts $\mathcal E(x)$ to get the original plaintext $x$, and then runs the turing machine $T$ with $x$ as the input. The decrypted result is the output of the turing machine.
By sending back $T$, Bob essentially offloads the work of actual computation on the ciphertext to the decryption function. However, from what I can tell, this is technically still a fully homomorphic encryption system, as it allows Bob to execute an arbitrary function on the plaintext, without ever knowing the plaintext himself (since the computation of the Turing machine is done by Alice).
Is this cryptosystem actually considered homomorphic encryption? Is there a subtlety in the constraints for a FHE system that excludes this? Or is it just rejected for being pedantic and useless?