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I am trying to read and understand the textbook's exercise questions. The sample solution at the end the textbook has a final answer and it does not clarify how it got the result. I am trying to workout the question but I am not successful. Any help would be appreciated.

Questions: Show the value of State after MixColumns:

$\begin{bmatrix}7C & 6B & 01 & D7\\F2 & 30 & FE & 63\\2B & 76 & 7B & C5\\AB & 77 & 6F &67\end{bmatrix}$

Answer: Final answer of textbook:

$\begin{bmatrix}75 & 87 & 0F & B2\\55 & E6 & 04 & 22\\3E & 2E & B8 & 8C\\10 & 15 & 58 & 0A\end{bmatrix}$

My answer:

I used the path that has been mentioned here to find $2\times 0x7C$ and $3\times 0xF2$.

$\begin{bmatrix} 2&3&1&1\\1&2&3&1\\1&1&2&3\\3&1&1&2\end{bmatrix} \times \begin{bmatrix}7C \\ F2 \\ 2B \\ AB\end{bmatrix} = \begin{bmatrix} ?\\?\\?\\?\end{bmatrix}$

    7C = 0111 1100
2 * 7C = 1111 1000

    F2 = 1111 0010
2 * F2 = 1110 0100
3 * F2 = 0001 0110  

    2B = 0010 1011
    AB = 1010 1011
-----------------------

Now find value for Cell (1,1)

    2 * 7C = 1111 1000  
    3 * F2 = 0001 0110
        2B = 0010 1011
XOR     AB = 1010 1011
        --------------
             0110 1110 = 0x6E <-- is not equal to 0x75
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2 * F2 = 1110 0100

That's your problem; 2 * F2 is not F2 shifted left one. Instead, because the msbit is set, it is F2 shifted left one, xor'ed with the feedback polynomial (1B).

So, 2 * F2 = 1110 0100 $\oplus$ 0001 1011 = 1111 1111

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you can done the multiplication operations for the mixcolumn matrices by two methods: either by using the x-time of shifting steps as it you mentioned in your questions or by straightforward multiplication module the (11b) that represent the first irreducible polynomial in the list of polynomial with degree eight for more details see these two manuscripts may be find what you need: (The Euphrates Cipher) and (The New Block cipher Design (Tigris Cipher))

Best regards

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