Editing Advanced Encryption Standard (section)

=== The {{mono|MixColumns}} step ===
{{main|Rijndael MixColumns}}
[[Image:AES-MixColumns.svg|right|320px|thumbnail|In the {{mono | MixColumns}} step, each column of the state is multiplied with a fixed polynomial <math>c(x)</math>.]]
In the {{mono | MixColumns}} step, the four bytes of each column of the state are combined using an invertible [[linear transformation]]. The {{mono | MixColumns}} function takes four bytes as input and outputs four bytes, where each input byte affects all four output bytes. Together with {{mono | ShiftRows}}, {{mono | MixColumns}} provides [[diffusion (cryptography)|diffusion]] in the cipher.

During this operation, each column is transformed using a fixed matrix (matrix left-multiplied by column gives new value of column in the state):

::<math>
\begin{bmatrix}
b_{0,j} \\ b_{1,j} \\ b_{2,j} \\ b_{3,j}
\end{bmatrix} = \begin{bmatrix}
2 & 3 & 1 & 1 \\
1 & 2 & 3 & 1 \\
1 & 1 & 2 & 3 \\
3 & 1 & 1 & 2
\end{bmatrix} \begin{bmatrix}
a_{0,j} \\ a_{1,j} \\ a_{2,j} \\ a_{3,j}
\end{bmatrix}
\qquad 0 \le j \le 3
</math>

Matrix multiplication is composed of multiplication and addition of the entries. Entries are bytes treated as coefficients of polynomial of order <math>x^7</math>. Addition is simply XOR. Multiplication is modulo irreducible polynomial <math>x^8+x^4+x^3+x+1</math>. If processed bit by bit, then, after shifting, a conditional [[Exclusive or|XOR]] with 1B<sub>16</sub> should be performed if the shifted value is larger than FF<sub>16</sub> (overflow must be corrected by subtraction of generating polynomial). These are special cases of the usual multiplication in <math>\operatorname{GF}(2^8)</math>.

In more general sense, each column is treated as a polynomial over <math>\operatorname{GF}(2^8)</math> and is then multiplied modulo <math>{01}_{16} \cdot z^4+{01}_{16}</math> with a fixed polynomial <math>c(z) = {03}_{16} \cdot z^3 + {01}_{16} \cdot z^2 +{01}_{16} \cdot z + {02}_{16}</math>. The coefficients are displayed in their [[hexadecimal]] equivalent of the binary representation of bit polynomials from <math>\operatorname{GF}(2)[x]</math>. The {{mono | MixColumns}} step can also be viewed as a multiplication by the shown particular [[MDS matrix]] in the [[finite field]] <math>\operatorname{GF}(2^8)</math>. This process is described further in the article [[Rijndael MixColumns]].