It lets you have someone add up your numbers, and give you the sum, without knowing the input numbers or their sum. Basically, any time you want someone else to

It lets you have someone add up your numbers, and give you the sum, without knowing the input numbers or their sum. Basically, any time you want someone else to host the data, while you can also do queries/computations on it. That now generalizes to any computation (not just addition), because there is a logically complete set of primitive operations for which fully homomorphic encryption (FHE) can be done. Caveats: 1) You can't actually do e.g. while loops with run time unknown. All such FHE computations are done by generating a fixed size boolean circuit for a given input size. It's "Turing-complete" in the sense that you can size up the circuit to any input size, but it wouldn't directly implement an unbounded while loop -- you have to generate a different "program" (circuit) for each input size. 2) All such computations must have a fixed output size -- else it leaks information about the data inside. So allowable computations would be like, "give me the first 30 bytes of the result of querying for names in the set that begin with A". If there are no results, the output still has to be 30 bytes. 3) For similar reasons, any FHE computation must look at all bits of the input (otherwise, it leaks info about what's in them). "See if this value is in the set" can't just jump to the relevant section. 4) The untrusted third party knows what computation you're doing (at least at a low level), just not the data it's being performed on or the result. 5) As you might have expected from 3), there's a massive blowup in resources to do it. There can be improvements, but some blowups are inherent to FHE.