A very basic way of how it works: encryption is basically just a function e(m, k)=c. “m” is your plaintext and “c” is the encrypted data. We call it an encryption function if the output looks random to anyone that does not have the key
If we could find some kind of function “e” that preserves the underlying structure even when the data is encrypted you have the outline of a homomorphic system. E.g. if the following happens:
e(2,k)*e(m,k) = e(2m,k)
Here we multiplied our message with 2 even in its encrypted form. The important thing is that every computation must produce something that looks random, but once decrypted it should have preserved the actual computation that happened.
It’s been a while since I did crypto, so google might be your friend here; but there are situations when e.g RSA preserves multiplication, making it partially homomorphic.
I get how that works for arithmetic operations - what about stuff like sorting, finding an element in a set etc? This would require knowledge of the cleartext data, wouldn't it?
You can reduce anything happening on the computer to arithmetic operations. If you can do additions and multiplications, then it's turing complete. All others can be constructed from them.
While correct, that doesn't answer the question at all, though. If I have my address book submited into an FHE system and want to sort by name - how do you do that if the FHE system does not have access to cleartext names?
> If we could find some kind of function “e” that preserves the underlying structure even when the data is encrypted
But isn't such a function a weakened form of encryption? Properly encrypted data should be indistinguishable from noise. "Preserving underlying structure" seems to me to be in opposition to the goal of encryption.
Given that the operations you can execute on the ciphertext are Turing complete (it suffices to show that we can do addition and multiplication) then it follows that any conceivable computation can be performed on the ciphertext.
Oh this is outside the context of encryption. My curiosity was, is there such a compression function that permits operations on the compressed data without first decompressing it?
One that is kind of in this spirit is that you can describe sparse matrices by omitting all the zeros and only describe the indices that have data. In this compression you can still perform normal matrix operations without having to unpack them into the “normal form”. Now this is neither encryption nor a particularly interesting compression, but it does prove that it is possible in principle ;p
If we could find some kind of function “e” that preserves the underlying structure even when the data is encrypted you have the outline of a homomorphic system. E.g. if the following happens:
e(2,k)*e(m,k) = e(2m,k)
Here we multiplied our message with 2 even in its encrypted form. The important thing is that every computation must produce something that looks random, but once decrypted it should have preserved the actual computation that happened.
It’s been a while since I did crypto, so google might be your friend here; but there are situations when e.g RSA preserves multiplication, making it partially homomorphic.