When using a "good", modern cipher (specifically one that provides ciphertext indistinguishability), is it a problem at all if there is some well-known structure in all plaintexts?
For example, consider a simple protocol that transmits many messages that all start with an identical, but secret string. All of the messages are encrypted with a different symmetric key, and the keys are in no way correlated (to either plaintext or previous keys).
Is there a way for an attacker to use his knowledge of the fact that the first n bytes of ciphertext always correspond to the same plaintext? Can those bytes even be decrypted? If so, is there a specific term for that kind of attack/cipher weakness? (Maybe something like "known identical plaintext"?)
My background for asking is section 5 of RFC 4345, which specifically advises against using RC4 in that kind of scenario. I know that RC4 has some known weaknesses regarding ciphertext indistinguishability, and it seems that my scenario is exactly one where those would allow the plaintext to be recovered.
So my question, in other words, is: Do other ciphers like AES share that weakness? If not, does resistance against that weakness, however it is called, follow from the fact that a cipher provides ciphertext indistinguishability?