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In the encryption of MixColumn you use the constants 01, 02 and 03. I don't understand why the inverse of 01 becomes both 0D and 09. I also don't understand how you get 01, 02 and 03 in the first place which is probably why the decryption doesn't make sense to me. I would appreciate any help with understanding how to calculate these values.

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  • $\begingroup$ Note that the elements of the inverse matrix are not the inverses of the individual elements; hence 0D and 09 are not "inverses" of 01 $\endgroup$ – poncho Dec 10 '18 at 19:35
  • $\begingroup$ and, you calculate the inverse of the matrix in $GF(2^8)$ see How to invert matrix in finite field $\endgroup$ – kelalaka Dec 10 '18 at 19:36
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The MixColumns $MC$ and InvMixColumns $IMC$ implement matrix multiplications. These two matrices defined over Galois Field $GF(2^8)$ and they are inverse of each other.

$$MC \times IMC = I_4$$

To find the inverse, one can use the Gaussian Elimination method. But finding the inverse a bit tricky, since you will need the inverse of the pivot elements. The inverse in $GF(2^8)$ can be found by using the Extended Euclidean Algorithm.

In general, instead of calculating the inverse, the invMixColumn matrix copied from FIPS 197 or AES submission of Rijndael.

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