Suppose that the S$-$Box of Example $4\mathrm{.}1$ is replaced by the S$-$Box defined by following

substitution ${}_{S^{\text{'}\text{'}}}$ :

$($a$)$ Compute the table of values $N_D($as defined in Definition $4\mathrm{.}3)$ for this S$-$Box.

$($b$)$ Find a differential trail using four active S$-$Boxes, namely, $S_1,S_4^1,S_4^2,$ and $S^3,$

that has propagation ratio ${?}^{27}$ and will find eight subkey bits in the last round.

$($c$)$ Describe a differential attack, analogous to Algorithm $4\mathrm{.}3,$ that will find eight

subkey bits in the last round. Your algorithm should have similar syntax as

shown in Algorithm $4\mathrm{.}3.$