Problem 2.4 (The trouble with compression.) Consider a perfectly secure scheme that operates on messages in f0; 1gn (i.e. messages whose length is at most n bits). Suppose that the ciphertext output by the encryption algorithm is exactly the same as the input plaintext (e.g., one-time pad). To reduce ciphertext size, there is a strong desire to combine encryption with lossless compression. We can think of compression as a function from f0; 1gn to f0; 1gn where, for some messages, the output is shorter than the input. As always, the compression algorithm is publicly known to everyone. (a) Compress-then-encrypt: Suppose the encryptor compresses the plaintext message before passing it to the encryption algorithm. Some n-bit messages compress well, while other messages do not compress at all. Argue that the resulting system is not perfectly secure . (b) Encrypt-then-compress: Suppose that instead, the encryptor applies compression to the output of the compression algorithm. Explain why this proposal is of no use for reducing ciphertext size.