a) Let DES_k(P) denote the DES encryption of the plain text P by the key K.

i) If we set K to be the key consisting of all zeroes, explain why encrypting twice with this key for DES has the same effect as encrypting then decrypting. Equivalently, account for the following property when using the all-zero key

DES_k(DES_k(P) = P

ii) Find another key that also has this property, and explain the structure of the DES key schedule that permit such keys to exist.

b) Let b' denote the complement of a sequence of bits, so that for example. Then considering the round structure of DES, show that

In other words, complementing the key, plaintext and ciphertext produces the same ciphertext as when none of these values is complemented.Hint: Use the following property of the XOR function x + x = 0.