Alice and Bob agree to communicate privately via email using a scheme based on RC4, but they want to avoid using a new secret key for each transmission. Alice and Bob privately agree on a 128-bit key .To encrypt a message , consisting of a string of bits, the following procedure is used.

1. Choose a random 80-bit value ν

2. Generate the ciphertext c = RC4(v ||k) ⊕ m

3. Send the bit string (v || c)

a. Suppose Alice uses this procedure to send a message m to Bob. Describe how Bob can recover the message m from  (v || c) using k.

b. If an adversary observes several values (v₁ || c₁), (v₂ || c₂), . . . transmitted between Alice and Bob, how can he/she determine when the same key stream has been used to encrypt two messages?

c. Approximately how many messages can Alice expect to send before the same key stream will be used twice?

d. What does this imply about the lifetime of the key (i.e., the number of messages that can be encrypted using k)?