Let $DE{S}_k(x)$ represent the encryption of plaintext $x$ with key $k$ using the DES cryptosystem.

Suppose $y=DE{S}_k(x)$ and $y^{\text{'}}=DE{S}_{\frac{?}{\text{b}\text{a}\text{r}}(k)}(\stackrel{}{x}),$ where ${?}^{-}$denotes the bitwise complement of its argument.

Prove that $y^{\text{'}}\frac{?}{b}=ar(y)($i$.$e$.,$ if we complement the plaintext and the key, then the ciphertext is also

complemented$).$ Note that this can be proved using only the "high$-$level" description of DES

the actual structure of $S-$boxes and other components are irrelevant.