There is a perfectly secure cipher, known as the one-time pad, which is said to have been used for encrypting messages on the “hot line” between Moscow and Washington, DC, during the Cold War. In this cryptosystem, Alice and Bob each share a random bit string, K, as large as any message they might wish to communicate. The string K is the secret key. To compute a ciphertext, C, from a message, M, Alice computes 

C = M ⊕ K, 

where “⊕” denotes the bitwise exclusive-or operator. Show that Bob can decrypt the ciphertext, C, by computing C ⊕ K. Also, show that this scheme achieves perfect confidentiality, based on the facts that each bit of the output is independent, random, and every plaintext of length |M| is a possible plaintext for the ciphertext, C.