Curious about a detail here. If this machine can evaluate AND and XOR, then how do you prevent a malicious attacker with the public key from performing encrypted each power of two, performing an AND and comparing with zero?

  {x&0x2} = {0x2}

Does this imply that hcrypt VM is not necessarily equal to a Turing machine? The program counter seems like it could definitely cover finite automata, but it doesn't seem expressive enough to simulate a stack. If PC is always encrypted then it ends up having to encode both current state and the information necessary to evaluate the branch implicitly as a state change, which again seems more like finite automata.

If you guys figured out how to securely implement control ( halting, loops, etc ), without functional encryption, that would be a huge breakthrough in FHE.