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Can zero be one of the elements used in an MDS matrix (in the context of AES)?

Based on what I have read all entries of an MDS Matrix need to be non-zero. Also, I would appreciate any help in understanding the reason behind this requirement if zero cannot be an element of MDS.

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  • $\begingroup$ In any MDS matrix, any square submatrix (subset of rows/columns) must be invertible. This includes 1x1 matrices. $\endgroup$
    – Fractalice
    Nov 2 '20 at 8:45
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No, for MDS codes used in the way it is used in AES there's no other choice, i.e., an MDS code with these dimension must have all entries nonzero in the $M$ matrix.

The MDS matrix $M$ has to be square, mapping 4 bytes to 4 bytes and must have 4 nonzero entries in each row by MDS property.

It also means changing each input byte affects each output byte, helping in diffusion.

Background: by coding theory the code is generated by the $k\times n$ matrix $$G=[I|M]$$ The Matrix converts a length $k$ message $m$ to the codeword $c$ via $$ c=mG $$ where $m,c$ are row vectors. This code has minimum weight $n-k+1$, the maximum possible by singleton bound. Since each row is a codeword the $M$ submatrix must have rows of minimum weight $n-k$. For AES, $n=2k=8$ so $n-k=4.$ So the rows of $M$ must have 4 nonzero entries.

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  • $\begingroup$ Thank you @kodlu, really appreciate your help. $\endgroup$ Nov 2 '20 at 3:45
  • $\begingroup$ ok, you can upvote the answer :-) $\endgroup$
    – kodlu
    Nov 2 '20 at 3:49
  • $\begingroup$ I tried doing that but the platform does not allow me unfortunately. Here is the message I keep getting: "Thanks for the feedback! Votes cast by those with less than 15 reputation are recorded, but do not change the publicly displayed post score." $\endgroup$ Nov 2 '20 at 3:58
  • $\begingroup$ @kodlu what do you mean by "no other choice"? $\endgroup$
    – Fractalice
    Nov 2 '20 at 8:45
  • $\begingroup$ @kodlu could you take the advantage of the title to write this answer for beginners? $\endgroup$
    – kelalaka
    Nov 2 '20 at 10:42
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After digging deeper here is what I found:

Let M be an MDS matrix, for any square sub-matrix Mi,j of M. From the property of MDS matrices, every sub-matrix should be non-singular.

No Mi,j elements belonging to M can be equal to zero, since this would lead to a singular submatrix of type 1×1 in M and thus would contradict the non-singularity requirement.

Simply put the cofactor matrix cannot be zero for a 1x1 sub-matrix of M if M is MDS since every sub-matrix of M should be non-singular (invertible).

https://www.aimsciences.org/article/doi/10.3934/amc.2019045

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