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?

You can do that by encrypting the names. You send encrypted names to the FHE-server, and then the server does necessary sorting computations on it.

The point of FHE is it can operate on gibberish-looking ciphertext, and when this ciphertext decrypted afterwards, the result is correct.

Indeed, there are those working on faster FHE sorting: https://eprint.iacr.org/2021/551.pdf

Honestly it breaks my brain as well. I just have to take it on trust that it apparently works.

When comparing two ciphertexts A,B a FHE sorting function will output a sorted pair of two new ciphertexts:

E.g. FHE_SORT(A,B) -> (X,Y)

where Dec(X)<Dec(Y)

But without decoding, there's no way of knowing whether X (or Y) comes from A or B.

Source: II. D of https://eprint.iacr.org/2015/995.pdf

It’s not that simple. The client has to send the server the comparison function.

To do anything practical the server usually needs to provide the client with gigabytes of per-client-key encrypted seed data.

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.

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

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.

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

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.

Thank you, this really clarified things for me!

a simple example of partial homomorphic encryption (not full), would be if a system supports addition or multiplication. You know the public key, and the modulus, so you can respect the "wrap around" value, and do multiplication on an encrypted number.

other ones I imagine behave kinda like translating, stretching, or skewing a polynomial or a donut/torus, such that the point/intercepts are still solveable, still unknown to an observer, and actually represent the correct mathematical value of the operation.

just means you treat the []byte value with special rules

Thank you. So based on your examples it sounds like the "computation" term is quite literal. How would this apply at larger levels of complexity like interacting anonymously with a database or something like that?

There are FHE schemes which effectively allow putting together arbitrary logical circuits, so you can make larger algorithms FHE by turning them into FHE circuits -- Jeremy Kun's 2024 overview [1] has a good summary

[1] https://www.jeremykun.com/2024/05/04/fhe-overview/ - discussed previously: https://news.ycombinator.com/item?id=40262626