Suppose that Alice and Bob share a 4-digit PIN number, X. To establish a shared symmetric key, Bob proposes the following protocol: Bob will generate a random key K that he will encrypt using the PIN number X, that is, E(K, X). Bob will send E(K, X) to Alice, who will decrypt it using the shared PIN number X to obtain K. Alice and Bob will then use the symmetric key K to protect their subsequent conversation. However, Trudy can easily determine K by a brute force attack on the PIN number X, so this protocol is insecure.

Modify the protocol to make it more secure. Note that Alice and Bob only share the 4-digit PIN number X and they do not have access to any other symmetric key or public keys. Hint: Use Diffie-Hellman.

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Consider the Diffie-Hellman key exchange protocol. Suppose that Alice sends her Diffie-Hellman value, g ^{a} mod p, to Bob. Further, suppose that Bob wants the resulting shared secret to be a specific value X. Can Bob choose his Diffie-Hellman value so that, following the protocol, Alice will compute the shared secret XI If so, provide precise details and if not, why not?

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