<s>[INST] MixColumn AES manually with GF(2^8) In the step of MixColumn in AES, calculate d0,0 if given c0,0 = c2,0 =
1100 0101, and c1,0 = c3,0 = 0100 1100.
Converting to polynomials:
c0,0 = c2,0 = x^7 + x^6 + x^2 + 1
c1,0 = c3,0 = x^6 + x^3 + x^2
now using the Galois field to get d0,0 we will get 4 results and then XOR all of them
c0,0 * 10 = (x^7 + x^6 + x^2 + 1)*(x) mod (x^8 + x^4 + x^3 + x + 1) mod 2 = 10010001
c1,0 * 11 = (x^6 + x^3 + x^2) * (x+1) mod (x^8 + x^4 + x^3 + x + 1) mod 2 = 11100100
c2,0 * 01 = (x^7 + x^6 + x^2 + 1)*(1) mod (x^8 + x^4 + x^3 + x + 1) mod 2 = 11000101
c3,0 * 01 = (x^6 + x^3 + x^2) * (1) mod (x^8 + x^4 + x^3 + x + 1) mod 2 = 01001100
Now XOR all of those 8 bit binary values I get 11111100 but the answer should be 11001100 where am I making my mistake?
Any help is appreciated
 [/INST] The second product is incorrect. There is no overflow to degree $8$ in that product, and it comes out as
$$
\begin{aligned}
c_{1,0}\cdot 11&=\overline{(x^6+x^3+x^2)(x+1)}\\
&=\overline{x^7+x^6+x^4+2x^3+x^2}\\
&=\overline{x^7+x^6+x^4+x^2}=11010100.
\end{aligned}
$$
I use overline to denote the coset of a polynomial because iterated mods give me pimples.
Anyway, this differs exactly in those two bit positions your end result differs from the given answer, so I'm optimistic about it being the only error.
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