$4.$ We now consider the relation between passwords and key size. For this purpose we consider a cryptosystem where the user enters a key in the form of a password.

a$.$ Assume a password consisting of $8$ letters, where each letter is encoded by the ASCII scheme $(7$ bits per character, i$.$e$.,128$ possible characters$).$ What is the size of the key space which can be constructed by such passwords?

b$.$ What is the corresponding key length in bits?

c$.$ Assume that most users use only the $26$ lowercase letters from the alphabet instead of the full $7$ bits of the ASCII$-$encoding. What is the corresponding key length in bits in this case?

d$.$ At least how many characters are required for a password in order to generate a key length of $128$ bits in case of letters consisting of

i$.7-$bit characters?

ii$.26$ lowercase letters from the alphabet?