Question

Alice and Bob shared an n-bit secret key some time ago. Now they are no longer sure they still have the same key. Thus, they use the following method to communicate with each other over an insecure channel to verify that the key k_a held by Alice is the same as the key k_b held by Bob. Their goal is to prevent an attacker from learning the secret key. Alice generates a random n-bit value r. Alice computes x = k_a r, and sends x to Bob. Bob computes y = k_b x and sends y to Alice. Alice compares r and y. If r = y, she concludes that k_a = k_b, that is, she and Bob have indeed the same secret key. Show how an attacker eavesdropping on the channel can gain possession of the shared secret key. Also show how an attacker who can do more than eavesdropping can make Alice and Bob believe they do not share the same key.

Answer 1

The method described in the question is vulnerable to a man-in-the-middle attack, where an eavesdropper intercepts and manipulates the messages to gain access to the secret key. Using a secure key exchange protocol such as Diffie-Hellman can prevent this attack.

Step-by-step explanation:

The method described in the question is vulnerable to an attack known as a man-in-the-middle attack. In this attack, an eavesdropper intercepts the messages being sent between Alice and Bob and manipulates them to gain access to the secret key.

When Alice sends the value x = k_ar to Bob, the attacker can intercept this value and replace it with a different value x'. The attacker then computes y' = k_bx' and sends it to Alice. If Alice compares r and y', she will find that they are not equal and conclude that she and Bob do not share the same key.

To prevent this attack, Alice and Bob can use a secure key exchange protocol such as Diffie-Hellman key exchange, which ensures that an attacker cannot learn the secret key even if they eavesdrop on the communication.