How can a system where the party performing a computation also possess the private key and still not know the answer of computation be designed ? Also the other party who does not have the private key know the correct result ?
Involving a trusted third part seems a possible solution. But are there other ways to design such a system ?
Consider the following scenario: A has an input $x_1$, the public key and private key. B comes in with another input $x_2$. A sends the public key to B. B sends the encrypted value $enc(x_2)$ for A to perform a computation $C(enc(x_1), enc(x_2))$. A performs the computation and sends the result to B. A should not know the correct value of result. But here if A is performing the computation and also has the private key, then A can decipher the answer.