There's a proof-of-concept file sharing protocol (Monotone? I cannot remember the name. Edit: it's monolith.sourceforge.net) that distributes numbers as xor of legal numbers.

Alice publishes a truly random string of bits, Bob publishes a XOR of Alice's string with an "illegal" number. Information theory says you cannot tell which is which (and publishing a random string of bits is supposedly legal) but you can still retrieve the "illegal" number by xoring Alice's and Bob's strings.

You're probably thinking of the OFFSystem (https://en.wikipedia.org/wiki/OFFSystem)

This seems to use the same underlying idea as Monolith (which is the one I thought of - edited)

That's covered by this essay: https://ansuz.sooke.bc.ca/entry/23

Bob publishes encrypted data. If it’s legal, just use AES with known pass phrase, its output looks random enough.

The point of the protocol is that both Bob and Alice can claim they were publishing a fully random string of bits and there is no way to disprove it.

I don't see how the known pass phrase achieves that.

There's a way to prove it by XORing their streams. Just like there's a way to prove that given stream is AES-encrypted by trying to decrypt it with key published by Alice.

I think that a crucial point about XOR-based systems is that both Bob and Alice can produce the "other half" of their random-looking file which produces a legitimate non-random file (e.g. a video of their holiday), while claiming that the other party maliciously used the "first half" as the random-looking input against which they XORed their forbidden file.

Unless an omniscient observer knows the publication date of all the "halves" in the system, it is impossible to say which of Bob or Alice are telling the truth.

Of course, both parties are inherently slightly suspicious for using such a system in the first place, but if it existed as part of a free public hosting service, and lots of people used it for legitimate purposes, it might only be as incriminating as using BitTorrent.

That makes sense, thanks.

"Any integer can be expressed uniquely as a sum of unique powers of even primes."