4. (10 points) Suppose that we have a linear block cipher E that encrypts 128-bit blocks of plaintext into 128-bit blocks of ciphertext. Let Ek(m) denote the ciphertest of a 128-bit message m under a key k. Thus, for messages m1 and m2, we have

Ek(m1 m2) = Ek(m1) Ek(m2)

Describe how this cipher can be easily cracked using chosen-ciphertext attack, i.e., an adversary can decrypt any ciphertext without knowledge of the secret key k.

4. (10 points) Suppose that we have a linear block cipher E that encrypts 128-bit blocks of plaintext into 128-bit blocks of ciphertext. Let Ex(m) denote the ciphertest of a 128-bit message m under a key k. Thus, for messages m and m2, we have Ex(m o m2) = Ek(mi) e Ex(m2). Describe how this cipher can be easily cracked using chosen-ciphertext attack, i.e., an adversary can decrypt any ciphertext without knowledge of the secret key k