$($a$)$ The complementation property of DES : Given $Y=DE{S}_K(x),$ prove that $\frac{?}{\text{b}\text{a}\text{r}}(Y)=DE{S}_{\frac{?}{\text{b}\text{a}\text{r}}(K)}(\stackrel{}{x}),$

where $\stackrel{}{x}$ denotes the bitwise complement of $x($i$.$e$.,\stackrel{}{x}$ is obtained from $x$ by changing all

the $1s$ to $0s$ and all the $0s$ to $1s).$

$($Hint: This has nothing to do with the structure of the S$-$boxes. You also need to leverage

the fact that operations such as permutations and shifts are not affected by bitwise com$-$

plementation. To solve the problem, just work through all the other steps of the encryption

algorithm and provide a detailed justification for your answer.

You might also need to use the fact the for any two bits a and $b,$ we have $ao+b=o\frac{+}{b}ar(b),$ and

$\frac{?}{\text{b}\text{a}\text{r}}(ao+b)=o+b$