Various encryption schemes would be considered broken if an adversary could have a non-negligible edge in predicting the first (or any) bit of an encrypted message. I am looking for a slightly stronger guarantee.
In particular, does any cryptographic system or crypto assumption (that relies on keys, not a one time pad) guarantee that any linear combination of the encrypted bits cannot be predicted after an adversary sees only a polynomial number of messages (possibly of his choosing)?
Note -- I am not at all an expert in cryptography, and this may be considered trivial.