I've often wondered about how to compute encrypted data, and appears that a "hack" to do so has been found: http://www.technologyreview.com/computing/37197/
Is anyone able to offer an better, yet still "simple" explanation of how Craig Gentry's approach is possible?
A simple explanation would be to imagine a bijective function $f_k: \mathbb{Z} \to \mathbb{Z}$ that depends upon a key $k$, such that $f_k(n)\pm f_k(m)=f_k(n\pm m)$ and similarly for multiplication and whatever other operations are embedded in the processor. Then I can send an insecure computer my program and encrypted data, it can do the computation and return the encrypted result, which I can decrypt. Each individual step along the way, it has the encrypted value of what I would get on my processor without encryption. I still need to convince myself that the computation was done correctly, but one can imagine embedding simple checks like $2+2=4$. Not that I have such a function-if so I would have Gentry's PhD.