(Disclaimer: I am not a cryptographer and this is a heavily simplified explanation). Homomorphic encryption is built on the foundation of 'hard problems' (e.g. the Learning with Errors Problem) - loosely, computational problems that are thought to be impossible to reverse without being in the possession of a secret key.

The crux of HE is that it provides a _homomorphism_: you map from the space of plaintext to the space of cipher texts, but the mapping preserves arithmetic properties such as addition and multiplication. To be clear - this means that the server can add and multiply the cipher texts, but the plaintext result of that operation is still irreversible without the private key. To the server, it looks like random noise.

I don't think it's helpful to think about this as connected to deep learning or embedding spaces. An excellent resource I'd recommend is Jeremy Kun's guide: https://www.jeremykun.com/2024/05/04/fhe-overview/