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.
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.
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).