The standard security property demanded of a blockcipher is that it be a pseudo-random permutation; i.e., given a uniformly random key, the blockcipher should be computationally indistinguishable from a random permutation under a chosen-plaintext or chosen-ciphertext attack.
However, what if the key is randomly selected from a non-uniform distribution? Has there been any research in analyzing the security of AES (or any other well-known blockcipher, for that matter) when the key is instead chosen from some other distribution? Clearly low-entropy distributions would permit brute-force attacks, but it seems there's a pretty big gulf between keys for which brute-force attacks are infeasible and uniformly distributed keys.
My question is motivated by the observation that encryption keys are frequently derived from passwords or other low-entropy sources by hashing them (perhaps with a salt, which itself may or may not fall into the hands of an attacker). The hashing may expand the key to the required length and deter brute-force attacks, but cannot introduce entropy into the result. (Edit: My question isn't about scenario in particular, it's just an example of when a non-uniform key distribution could happen in practice.)
I am aware that there has been work on related-key attacks, but to my knowledge the results aren't directly applicable to this question.