Here’s a very simple one-time-pad style construction for homeomorphic (but not fully homeomorphic) encryption.

Suppose the plaintext, ciphertext, and key, are each a natural number modulo 5.

The key is selected uniformly at random. The ciphertext is obtained by adding the key to the plaintext (and as a result is also uniform random).

Then the ciphertext is sent to the server. The server only has the ciphertext, which is uniform random. The server then adds some integer mod 5 to the ciphertext, producing a new ciphertext, and sends it back to the user. The user then subtracts their key from the new ciphertext to obtain the new plaintext, which is their original plaintext plus the number the server added.

At no point in this process did the server learn anything about the user’s plaintext.

This is clearly secure.

It is also useless, because just adding a number isn’t a useful thing for a server to do (they may as well just send the user the number and let them add it themselves), but that’s because it is a toy example.

I suppose if mind uploading were possible, then under FHE it would in principle be possible to take the ciphertext and run the computation of “this mind upload of this person reads the data and provides some output”, and then presumably that uploaded person would have the experience of seeing the plaintext? But it wouldn’t be possible to get any information about the plaintext without the key (or enough brute force to find the key, but that’s why we use big enough keys to make this infeasible).

What does that have to do with incpa?

I send this server a question encrypted. It stays encrypted yet the server distinguished my ciphertext enough to produce a coherent reply. That means there was enough information in the ciphertext for the server to perform the inference, even if it was obscured to where it looks indistinguishable, it apparently wasn't.

This doesn't remove trust from the equation, it puts the trust directly on the algorithm. Probably one no one can explain to me like I'm 5. But because it's encrypted I am no longer breaking the law by processing on protected data. I imagine very large contracts await on the other side. Lots of money to be made.