If you send someone `key xor X`, they can reply with `key xor X xor Y`, and you can recover `X xor Y` – without them having a clue what X is.

Hopefully then it's not too hard to believe you can build an encryption scheme and operations where `dec(enc(X) ⊞ Y) = X + Y` and `dec(enc(X) ⊠ Y) = X × Y`. If these operations can also be composed, i.e. homomorphism, then they're Turing complete! You can turn any function f(X) into a ciphertext version `dec(F(enc(X))) = f(X)`. It can't leak any information about X, because you only ever provided enc(X). For example you transform 'branching' code to something which traverses every path, and paths which do nothing will still appear to randomize the ciphertext, because you can't invent operations that distinguish which path is taken, because it's a secure ciphertext.

Modern homomorphic encryption uses more complex encryption schemes and primitives, to achieve thousands/hundreds rather than millions of times slower than f. Nonetheless it's impossible for the provider to gain a single bit of information about X.