. Prove that of all the possible keys (K1, K2), the expected number of keys for which DESK1 (pi) = DES-1 K2 (ci) ?i, where 1 = i = l, is about 22n-lm. iv. Suppose l = 2n/m, what can you say to the attacker to help him in developing an attack against the composed cipher DES0 ? v. The attacker starts building up two lists: L1 and L2. Each entry in the list L1 and L2 has l tuples of elements of P followed by an element from K. The lists are filled with all possible keys. The lists are now sorted in a lexicographic manner on the l tuples. The attacker now does a linear search to find out the common l tuples in the lists. Explain how does the attacker maintain the list and how does this approach help him to find out the correct key? Show that the amount of memory required by the attacker is 2n+1(ml + n) bits and number of encryptions and/or decryptions required to identify the key is l2 n+1. (Hint: Use the distinguisher: for the correct key DESK1 (pi) = DES-1 K2 (ci) ?i) vi. Into what class does the above kind of attack fall?