We look at the different conditions under which the shift, mono-alphabetic substitution, and Vigen`ere ciphers are perfectly secret: 1. Prove that if only a single character is encrypted, then the shift cipher is perfectly secret. 2. What is the largest message space M for which the mono-alphabetic substitution cipher provides perfect secrecy? 3. Prove that the Vigen`ere cipher using (fixed) period t is perfectly secret when used to encrypt messages of length t. Reconcile this with the attacks shown in Problem 1(Problem 1:Show that the shift, substitution, and Vigen`ere ciphers are all trivial to break using a chosen-plaintext attack (CPA). How much chosen paintext is needed to recover the key for each of the ciphers?)