It does generalize, but not all the way. You can't perform while loops (or anything with unbounded runtime) or operate on anything with unbounded (secret) size,

It does generalize, but not all the way. You can't perform while loops (or anything with unbounded runtime) or operate on anything with unbounded (secret) size, so it isn't Turing complete. You can do anything expressible as a bounded-depth (non-cyclic) circut of eg NAND gates. You can also do something like [a][+ b][+ c][+ d], where each [] is a new homomorphic operation, which gets around the bounded-size problem somewhat, but the attacker can obviously see how many chunks you're working on, just not the content, which provides some traffic analysis vulnerabilities. All in all, it's a very useful tool but I don't really like the way people keep presenting it a silver bullet, like "yay, once we get this working we won't have to care that the cloud is full of phantom trolleys armed with hammers!".