For n$-$bit block size, there are $2$n possible plaintext blocks and $2$n possible ciphertext blocks. If the plaintext or

ciphertext is represented by an integer, then its value ranges from $0$ to $2$n $1.$

a$.$ How many different n$-$bit block to n$-$bit block mappings are there?

b$.$ How many different reversible n$-$bit block to n$-$bit block mappings are there?

c$.$ If a reversible n$-$bit block mapping is used as a key for the substitution, then what would its minimum bit$-$

length be$?$

d$.$ In class we considered the representation of a reversible $4-$bit mapping: this was a $16\backslash$times $4=64-$bit string $(=$

the key length$=$size of key$).$ How much shorter is the key length computed in c$.$ when n$=4?$

Please explain the solution process in detail and let me know the answer at the end.