$1.$ Provide a formal definition of the Gen, Enc, and Dec algorithms for the mono$-$alphabetic substitution cipher.

$2.$ Provide a formal definition of the Gen, Enc, and Dec algorithms for the Vigen$$re cipher. $($Note: there are several plausible choices for Gen; choose one.$)$

$3.$ Say you are given a ciphertext that corresponds to English$-$language text that was encrypted using either the shift cipher or the Vigen$$re cipher with period greater than $1.$ How could you tell which was the case?

$4.$ The shift and Vigen$$re ciphers can also be defined on the $256-$character alphabet consisting of all possible bytes $(8-$bit strings$),$ and using XOR instead of modular addition.

a$)$ Provide a formal definition of both schemes in this case.

b$)$ Discuss how the attacks we have shown in this chapter can be modified to break these schemes.