1. **Problem Statement:** We want to understand the inverse MixColumns operation used in AES encryption, its utility, and demonstrate it with an example matrix by applying MixColumns followed by inverse MixColumns. 2. **Background:** - MixColumns is a transformation in AES that mixes the bytes of each column of the state matrix using matrix multiplication over a finite field $GF(2^8)$. - The constant matrix used in MixColumns is fixed and ensures diffusion. - The inverse MixColumns uses a different constant matrix to reverse the effect of MixColumns. 3. **Utility of the Constant Matrix:** - The constant matrix in MixColumns defines how bytes are combined. - It ensures that each output byte is a function of all input bytes in the column, providing diffusion. - The inverse constant matrix is designed so that when multiplied by the MixColumns matrix, it yields the identity matrix, thus reversing the operation. 4. **Example:** Let the state column be: $$\begin{bmatrix}0xdb \\ 0x13 \\ 0x53 \\ 0x45\end{bmatrix}$$ The MixColumns constant matrix is: $$M = \begin{bmatrix}2 & 3 & 1 & 1 \\ 1 & 2 & 3 & 1 \\ 1 & 1 & 2 & 3 \\ 3 & 1 & 1 & 2\end{bmatrix}$$ The inverse MixColumns constant matrix is: $$M^{-1} = \begin{bmatrix}14 & 11 & 13 & 9 \\ 9 & 14 & 11 & 13 \\ 13 & 9 & 14 & 11 \\ 11 & 13 & 9 & 14\end{bmatrix}$$ 5. **Apply MixColumns:** Multiply $M$ by the state column in $GF(2^8)$ (multiplication and addition modulo the AES polynomial). The result is: $$\begin{bmatrix}0x8e \\ 0x4d \\ 0xa1 \\ 0xbc\end{bmatrix}$$ 6. **Apply Inverse MixColumns:** Multiply $M^{-1}$ by the result above: $$M^{-1} \times \begin{bmatrix}0x8e \\ 0x4d \\ 0xa1 \\ 0xbc\end{bmatrix} = \begin{bmatrix}0xdb \\ 0x13 \\ 0x53 \\ 0x45\end{bmatrix}$$ This recovers the original state column, confirming the inverse operation. **Final answer:** The inverse MixColumns reverses the diffusion caused by MixColumns using the inverse constant matrix, restoring the original state matrix column.