Given the following S$-$Box and the answers for the State displayed by a $44$ matrix, the

initial RoundKey, Sub Bytes, and ShiftRows, find the MixColumns Transformation. To

find MixColumns, each element in the product matrix is the sum of products of elements

of one row and one column.

The S$-$bix used in AES is shown as follows:

Given the plaintext $\{0$F$0$E$0$D$0$C$0$B$0$A$09080706050403020100\}$ and the key

$\{02020202020202020202020202020202\}$:

Original state displayed as a $44$ matrix:

$[$OF OB $0703]$

$[$OE OA $0602]$

$[$OD $090501]$

$[0$C $080400]$

State after initial AddRoundKey:

$[0$D $090501]$

$[0$C $080400]$

$[$OF OB $0703]$

$[$OE OA $0602]$

State after SubBytes:

$[$D$7016$B $7$C$]$

$[$FE $30$ F$263]$

$[762$B C$57$B$]$

$[$AB $676$F $77]$

State after ShiftRows:

$[$D$7016$B $7$C$]$

$[30$ F$263$ FE$]$

$[$C$57$B $762$B$]$

$[77$ AB $676F]$