Separate comment to address a subtlety that comes up a lot:
Often you'll hear about fully homomorphic encryption (FHE) being Turing-complete. But you can't actually have a Turing complete system with variable-run-time loops that's homomorphically encrypted, because that leaks information about the inputs.
When they say FHE is Turing-complete, what they mean is that you can take an arbitrary program requiring Turing completeness, then time-bound it, unroll it into a fixed-length circuit, and run that homomorphically. Since you can keep upping the time bound, you can compute any function. So the system that translates your programs into those circuits, with no limit on the bound you set, could then be called Turing-complete -- but you couldn't say that about any of those circuits individually.
Often you'll hear about fully homomorphic encryption (FHE) being Turing-complete. But you can't actually have a Turing complete system with variable-run-time loops that's homomorphically encrypted, because that leaks information about the inputs.
When they say FHE is Turing-complete, what they mean is that you can take an arbitrary program requiring Turing completeness, then time-bound it, unroll it into a fixed-length circuit, and run that homomorphically. Since you can keep upping the time bound, you can compute any function. So the system that translates your programs into those circuits, with no limit on the bound you set, could then be called Turing-complete -- but you couldn't say that about any of those circuits individually.
Earlier related comment: https://news.ycombinator.com/item?id=40078494