Would you say that this circuit was fully homomorphic if, for all inputs, the processor appeared to be doing the same work?
That is to say? If X = 1 or X = 2 the processor did not appear to “do nothing” but, instead, performed an operation on an encrypted string that was constant time?
I’m not really sure what you’re asking. By definition, FHE cannot be Turing complete. A Turing machine must be capable of performing operations that cannot be handled in constant time, and for which the timing reveals information about the input. That is a fundamental feature of Turing machines; it’s the whole basis for how they came about.
There’s no way to do a FHE style system with branching unless all branches are indistinguishable from one another. All branches must execute in the same amount of time, otherwise statistical analysis will leak information.
Forget Turing completeness for a second, that’s not actually what you want for a FHE style system. You want computational indistinguishability which can’t be achieved without side-channel resistance.
That is to say? If X = 1 or X = 2 the processor did not appear to “do nothing” but, instead, performed an operation on an encrypted string that was constant time?