Assume a communication session between Alice and Bob (being client and server, in my case). Alice uses a secure random generator to generate both a key and an IV for use with AES in CBC mode.
Alice transmits both the key (and IV) over a "secure channel" (RSA is being used), together with a message that has been encrypted using these parameters, to Bob.
Bob decrypts the message, and wants to send a message back, obviously using the same received session key. As I am working using a bandwidth constrained channel, every byte saved is a win.
Is it okay for Bob to use the last AES ciphertext block as initialization vector for his transmission, given that this is the way CBC works?
If not, why not? If so, can I keep on chaining using this method, assuming a perfect lossless channel?
Bonus question: if it works, is it possible to apply this to other block modes?