$1.$

a$.$ Determine gcd$(24140,16762)$

b$.$ Determine gcd$(4655,12075)$

$2.$ Using the extended Euclidean algorithm, find the multiplicative inverse of$?$

a$.24140$ mod $40902$

b$.1234$ mod $4321$

c$.550$ mod $1769$

$3.$ Determine the multiplicative inverse of x$3+$ x $+1$ in GF$(24$

$)$ with m$($x$)=$ x

$4+$ x $+1$

$4.$

a$.$ Addition in GF$(24$

$)$: Compute A$($x$)+$B$($x$)$ mod P$($x$)$ in GF$(24$

$)$ using the irreducible polynomial

P$($x$)=$ x$4+$ x $+1$

$1.$ A$($x$)=$x$2+1,$ B$($x$)=$x$3+$x$2+1$

$2.$ A$($x$)=$ x$2+1,$ B$($x$)=$ x $+1$

b$.$ Multiplication in GF$(24$

$)$: Compute A$($x$)$B$($x$)$ mod P$($x$)$ in GF$(24$

$)$ using the irreducible

polynomial P$($x$)=$ x$4+$ x $+1$

$1.$ A$($x$)=$x$2+1,$ B$($x$)=$x$3+$x$2+1$

$2.$ A$($x$)=$ x$2+1,$ B$($x$)=$ x $+1$

$5.$

Given the plaintext $\{000102030405060708090$A$0$B$0$C$0$D$0$E$0$F$\}$ and the key

$\{01010101010101010101010101010101\}$:

a$.$ Show the original contents of State, displayed as a $4\backslash$times $4$ matrix.

b$.$ Show the value of State after initial AddRoundKey.

c$.$ Show the value of State after SubBytes.

d$.$ Show the value of State after ShiftRows.

e$.$ Show the value of State after MixColumns.