Suppose that the S$-$Box of Example $4\mathrm{.}1$ is replaced by the S$-$Box defined by following substitution $\backslash$pi $\_($S$^())$ : $($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 $4\mathrm{.}3.[50$ Points$]$ Suppose that the S$-$Box of Example $4\mathrm{.}1$ is replaced by the S$-$Box defined by

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

$($a$)[21$ Points$]$ Compute the table of values $N_D($as defined in Definition $4\mathrm{.}3)$ for this

S$-$Box.

$($b$)[21$ Points$]$ Find a differential trail using four active S$-$Boxes, namely, $S_1^1,S_4^1,S_4^2,$ and

$S_4^3,$ that has propogation ratio $\frac{27}{2048}$ and will find eight subkey bits in the last round.

$($c$)[8$ Points$]$ 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 $4\mathrm{.}3.$