Indeed, and that directly contradicts the paper I originally cited. The paper seems to claim it's impossible. That's the mystery, and it's very interesting! There's a contradiction that no one seems to know the answer to, which is the best kind of mystery.
The program in this case is the FHE virtual machine and indeed we cannot obfuscate that per the result of the paper. The "plaintext" input is the encrypted program along with its data. It is implicit in the definition of a program is the fact that the input to it is "plain".
Additionally, you should keep in mind that when doing the weird virtual machine thing you are not dealing with a single program. There are in fact 4. The actual program, call this P, you want to run on your data, the program to encrypt your program P along with its data, call this E, the virtual machine to run this encrypted program on the server, call this V, and finally the program to decrypt the result from running V on the output of E, call this E'.
Clearly, if we compose all these programs and run it on some data D, we get E'(V(E(P, D))) = P(D). However, the server doesn't know E, D and P and certainly doesn't know E'. The only thing the server knows is the value of E(P, D) and what V is. I haven't looked at the paper to deeply but I suspect it's vitally important to their result that you have some type of oracle that given a D, tells you the output of P(D), but the server cannot have that or else it would be pointless wasting your time with this FHE thing.
There is no contradiction. You are conflating the concept of 'obfuscation' and the very specific (and rigidly formalized) cryptographic primitive called 'indistinguishability obfuscation'.
Comments
Indeed, and that directly contradicts the paper I originally cited. The paper seems to claim it's impossible. That's the mystery, and it's very interesting! There's a contradiction that no one seems to know the answer to, which is the best kind of mystery.
The program in this case is the FHE virtual machine and indeed we cannot obfuscate that per the result of the paper. The "plaintext" input is the encrypted program along with its data. It is implicit in the definition of a program is the fact that the input to it is "plain".
Additionally, you should keep in mind that when doing the weird virtual machine thing you are not dealing with a single program. There are in fact 4. The actual program, call this P, you want to run on your data, the program to encrypt your program P along with its data, call this E, the virtual machine to run this encrypted program on the server, call this V, and finally the program to decrypt the result from running V on the output of E, call this E'.
Clearly, if we compose all these programs and run it on some data D, we get E'(V(E(P, D))) = P(D). However, the server doesn't know E, D and P and certainly doesn't know E'. The only thing the server knows is the value of E(P, D) and what V is. I haven't looked at the paper to deeply but I suspect it's vitally important to their result that you have some type of oracle that given a D, tells you the output of P(D), but the server cannot have that or else it would be pointless wasting your time with this FHE thing.
There is no contradiction. You are conflating the concept of 'obfuscation' and the very specific (and rigidly formalized) cryptographic primitive called 'indistinguishability obfuscation'.