Suppose that you know a MAC value X and the key K that was used to compute the MAC, but you do not know the original message. (It may be instructive to compare this problem to Problem 16 in Chapter 5.)

a. Show that you can construct a message M that also has its MAC equal to X. Note that we are assuming that you know the key K and the same key is used for both MAC computations.

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b. How much of the message M are you free to choose?

Suppose that we define triple 3DES with a 168-bit key as

C = E(E(E(P,K_{1}),K_{2}),K_{3}).

Suppose that we can compute and store a table of size 256, and a chosen plaintext attack is possible. Show that this triple 3DES is no more secure than the usual 3DES, which only uses a 112-bit key. Hint: Mimic the meet-in-the-middle attack on double DES.

Suppose that Alice and Bob have access to two secure block ciphers, say, Cipher A and Cipher B, where Cipher A uses a 64-bit key, while Cipher B uses a 128-bit key. Alice prefers Cipher A, while Bob wants the additional security provided by a 128-bit key, so he insists on Cipher B. As a compromise, Alice proposes that they use Cipher A, but they encrypt each message twice, using two independent 64-bit keys. Assume that no shortcut attack is available for either cipher.

Is Alice’s approach as secure as Bob’s?

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