(a) Show that if p = 11, q = 5, x = 3, and k = 3, then (x k (mod p))(mod q) and (x k (mod q))(mod p) are different.

(b) Alice and Bob want to exchange encrypted signed messages. Alices public key is (N, e) and private decryption exponent d, whereas Bobs public key is (N0 , e0 ) and private decryption exponent d 0 . Alice wants to send a message x to Bob. She first signs a message encrypted by Bobs public key so sends y = (x e 0 (mod N0 ))d (mod N) to Bob. To read the message and verify the signature, Bob computes z = y e (mod N) = x e 0 (mod N0 ) and then computes z d 0 (mod N0 ) = x e 0 d 0 (mod N0 ) = x. Will this work? Explain why or why not.