$1\mathrm{.}12.$ There is nothing exclusively special about strings and XOR in OTP. We can get the same

properties using integers modn and addition modn.

This problem considers a variant of one$-$time pad, in which the keys, plaintexts, and ci$-$

phertexts are all elements of $Z_n$ instead of $\{0,1{\}}^{}.$

$($a$)$ What is the decryption algorithm that corresponds to the following encryption algo$-$

rithm?

$($b$)$ Show that the output of the following subroutine is uniformly distributed in $Z_n$ :

$\left[\begin{array}{c}\frac{EAVESDRO{P}^{\text{'}}(min{Z}_n)\text{:}}{k}larr{Z}_n\\ c\text{:}\\ \text{r}\text{e}\text{t}\text{u}\text{r}\text{n}c\end{array}\right]=(k+m)\%n.$

$($c$)$ It's not just the distribution of keys that is important. The way that the key is combined

with the plaintext is also important. Show that the output of the following subroutine

is not necessarily uniformly distributed in $Z_n$ :

$)\text{'}$