If you compress a document that simply obscures the size of the original document.
That is not generally true - if you compress AAA and ABC then the compressed version also leaks information about the content exactly like in the case of an audio stream and it reveals more information then the lengths of the uncompressed documents which don't differ.
Feed that information into a prebuilt phoneme-in-context model trained on that codec and language, and said eavesdropper could probably reconstruct a pretty good estimated transcript - different phonemes compress differently.
There is no reason this would not work in bulk: I am under the impression GCHQ have done a fair amount of research in this area, for example.
These sorts of attacks don't matter against a compressed and then encrypted file of indeterminate data, correct? Similarly, an amalgamation of many types of data, such as a disk or archive?
For example, you know its type, and it's 150KB. Even if you know it's a text document, or a WAV file, or bitmap, I think you get far less information from the compressed version than from the non-compressed version. The non-compressed version of the text document may give you approximate word count, and the WAV file will give you some probable lengths of the recording at common sampling rates, and the bitmap will give you some probable dimensions. Compressed versions hide this information within the natural variance of the compression.
The problem seems less to do with compression, and more to do with splitting larger data (or streams of data) into smaller chunks to encrypt, which in itself loses some of the benefits of hiding information in the variance of the data presented. Compression seems to exacerbate this situation by amplifying variances in very small data sets in a way that yields additional information, but it seems to me that's just a natural progression of encrypting very small amounts of data being not nearly as effective as large amounts of data.
At least, that's what I can intuit about the situation. It may be partly (or totally) wrong given some more advanced security theory (I am not a security professional).
I might be wrong, but I think the parent had something like this in mind:
Assuming you are encrypting with AES-CFB your each plaintext block P[i] produces a ciphertext block C[i] (with key K and initialization vector IV) according to rule:
C[0] = AES_encrypt(K, IV) ^ P[0],
C[i] = AES_encrypt(K, C[i-1]) ^ P[i].
Given that IV is assumed to be known by the attacker, if the plaintext is a vanilla-compressed stream, it is more likely than not that most of bits in P[0] are going to be known as well, which might allow some sort of prunned brute-force attack on key K, given a small set of (C[0] ^ P'[0]), for all P'[0] that satisfy the known bits in P[0].This particular implementation would benefit then from adding a pseudorandom P[0] block (a "salt", if I understood correctly) that the receiver is to discard on arrival. I don't know enough cryptanalysis to tell if the above scenario is valid or not, but it sounds like a legitimate question at least.
I would guess that the phones and voip is a perfect target to do this, as those system are often designed to filter out noise. I guess it also would be almost impossible to use this to transcribe a encrypted movie or music, as the unpredictable sounds would break the prediction model.