# How was the AES MixColumns matrix constructed?

How was the constant matrix in AES MixColumns constructed?

\begin{bmatrix} 02 & 03 & 01 & 01 \\ 01 & 02 & 03 & 01 \\ 01 & 01 & 02 & 03 \\ 03 & 01 & 01 & 02 \\ \end{bmatrix}

How can I construct in $GF(2^4)$ an invertible matrix for the MixColumns step?

• how was it constructed? with a circulant matrix containing the smallest possible values to get the job done – Richie Frame Jun 16 '16 at 20:33
• There's an entire book (230 pages) about the design of AES, it's called: "The Design of Rijndael: AES – The Advanced Encryption Standard.", chances are the answer is in there somewhere... – SEJPM Jun 16 '16 at 21:51
• Read at least the Rinjdael submission, section 7.3 – fgrieu Jun 16 '16 at 22:02

Any full rank $n \times n$ circulant matrix $A$ with all entries nonzero generates an MDS code. The entries here were picked for efficient fast multiplication, as far as I can recall.