Problem: Let the encrypt function of standard DES be C = E(P, K) where C is the ciphertext, P is the plaintext, and K is the key. Accordingly, let the decrypt function of DES be P = D(C, K). Suppose that Alice uses a customized cipher C ′ = D(E(P, K1), K2) where K1 and K2 are two keys, and Alice then sends C ′ to Bob. Suppose an attacker happens to know the plaintext P, but he does not know the two keys that was used in the cipher.
(a) What are the sizes of K1 and K2?
(b) Suppose an attacker knows the cipher text C ′. Show that the attacker can figure out the two keys. Please elaborate your attack with diagrams.
(c) How could you modify the customized cipher and improve its security without adding another key? Why it becomes more secure? Please explain your answer.