$(10$ POINTS$)$ Consider the following authentication protocol, which uses a classical

cryptosystem. Alice generates a random message r$,$ enciphers it with the key k she shares with

Bob, and sends the encrypted message Ek$\{$r$\}$ to Bob. Bob deciphers it$,$ adds $1$ to r$,$ and sends

Ek$\{$r$+1\}$ back to Alice. Alice decrypts the message and compares it with r$.$ If the difference is

$1,$ she knows that her correspondent shares the same key k and is therefore Bob. If not, she

assumes that her correspondent does not share the key k and so is not Bob. Does this protocol

authenticate Bob to Alice? Discuss whether the authentication service of this protocol is

vulnerable to any attack. Please explain why or why not vulnerable.