$($b$)$ Use the complementation property in point $($a$)$ above to show how can an adversary reduce

in half the expected number of keys to be tried in a brute$-$force chosen$-$plaintext attack $($from

$2^{55}$ to $2^{54}$ keys$).$

$($Hint: You can assume that the attacker can execute a chosen$-$plaintext attack, so the adver$-$

sary has access to the following pairs of plaintext$/$ciphertext: $(M_1,C_1)$ and $(\frac{?}{\text{b}\text{a}\text{r}}(M_1),C_2),$ where

$C_1=E_K(M_1)$ and $\{$:$C_2=E_K(\frac{?}{\text{b}\text{a}\text{r}}(M_1)).)$