Okay, so the following theorem is going to show that in fact CBC mode encryption with a random IV is in fact semantically secure under a chosen plaintext attack, and so let's take that more precisely, basically if we start with a PRP, in other words, our block cipher E, that is defined over a space X, then we are gonna to end up with a encryption algorithm Ecbc that takes messages of length L and outputs ciphertexts of length L+1. 