There is nothing special about using XOR for the one$-$time pad. We can get the same security

properties using addition modn and multiplication modn. Specifically, consider a variant of the one$-$

time pad, in which keys, plaintexts, and ciphertexts are all elements of $Z_n$ instead of $\{0,1{\}}^n($where

$Z_n$ consists of all integers between $0$ and $n-1,$ and addition and multiplication are performed modulo

$n.$ In what follows, define Gen$(n)$ to output $klarr{Z}_n.$

$($a$)$ For $k,min{Z}_n,$ define Enc$(k,m)=(k+m)modn.$ What is the decryption algorithm that

corresponds to this encryption algorithm? The decryption algorithm in your answer should ensure

correctness for the modified scheme.

$($b$)$ Prove that when using the encryption algorithm from part $($a$),$ the ciphertexts in the resulting

scheme are uniformly distributed in $Z_n($where the probability is over the choice of the key$).$

$($c$)$ Suppose we instead defined Enc$(k,m)=(k*m)modn.$ Show that the ciphertexts in the resulting

scheme are not necessarily uniformly distributed.