[dachrillz]

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.

[littlecranky67]

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?

[barisozmen]

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.

[j2kun]

Comparisons can be implemented by approximating a < b with

    0.5 * (sign(a - b) + 1)

And the sign function can be approximated by a polynomial that uses only additions and multiplications and products with constants.

Other FHE schemes have support for small-bitwidth lookup tables that makes supporting comparison more direct.

[JohnFen]

> 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.

[xhrpost]

This made me wonder if there is such a thing as homomorphic compression. A cursory search says yes but seems like limited information.

[Tryk]

What do you mean by homomorphic compression?

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.

[paulrudy]

Thank you, this really clarified things for me!