$2.$ For RC$4$ stream cipher, what is the value of the key that will leave the initial state $S$ unchanged during initialization? That is$,$ after the initial permutation of $S,$ the entries of $S$ will be equal to the values from $0$ through $255$ in ascending order?

$3.$ Assume Alice and Bob agree to communicate privately via e$-$mail using a scheme based on RC$4,$ but they want to avoid using a new secret key for each transmission. Alice and Bob privately agree on a $128-$bit key $k.$ To encrypt a message $m$ consisting of a string of bits, the following procedure is used: Choose a random initial vector of $80-$bit value $v$ Generate the ciphertext $c=RC4(v||k)o+m,$ where $"||"$ indicates concatenation. 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 $k.$ If an adversary observes several values $(v1||c1),(v2||c2),$dots,$(v_i||c_i),$dots$(v_j||c_j)$ transmitted between Alice and Bob, how can he$/$she determine when the same key stream $RC4(v||k)$ has been used to encrypt two messages $c_i,$ and $c_j?$

b$)$ If $v$ is a random vector of $80$ bits, the random generator will likely generate the same value of $v$ every ~~$2^{40},$ so what should Alice do to avoid breaking the communication?