MixColumns - garden

Mixing Within Columns

ShiftRows moved bytes across columns. Now MixColumns blends the bytes within each column.

Each column has 4 bytes. After MixColumns, every output byte is influenced by all 4 input bytes.

For each column:

Input	Output
4 bytes	4 new bytes

Every output byte is calculated from all 4 input bytes using a fixed formula.

Change any one input → all 4 outputs change.

The mixing is done by multiplying each column by this matrix:

\begin{bmatrix} 2 & 3 & 1 & 1 \\ 1 & 2 & 3 & 1 \\ 1 & 1 & 2 & 3 \\ 3 & 1 & 1 & 2 \end{bmatrix}

The multiplication uses finite field arithmetic, not regular math. But the key point: this specific matrix guarantees maximum mixing.

It’s an MDS matrix (Maximum Distance Separable).

This means: change any input byte, and all output bytes must change. Not some. All.

This gives AES the strongest possible diffusion within each column.

Next, we’ll look at AddRoundKey, where the key gets XORed into the state.