The construction of circulant matrices related to MDS matrices

The objective of this paper is to suggest a method of the construction of circulant matrices, which are appropriate for being MDS (Maximum Distance Separable) matrices utilising in cryptography. Thus, we focus on designing so-called bi-regular circulant matrices, and furthermore, impose additional restraints on matrices in order that they have the maximal number of some element occurrences and the minimal number of distinct elements. The reason to construct bi-regular matrices is that any MDS matrix is necessarily the bi-regular one, and two additional restraints on matrix elements grant that matrix-vector multiplication for the samples constructed may be performed efficiently. The results obtained include an upper bound on the number of some element occurrences for which the circulant matrix is bi-regular. Furthermore, necessary and sufficient conditions for the circulant matrix bi-regularity are derived. On the basis of these conditions, we developed an efficient bi-regularity verification procedure. Additionally, several bi-regular circulant matrix layouts of order up to 31 with the maximal number of some element occurrences are listed. In particular, it appeared that there are no layouts of order 32 with more than 5 occurrences of any element which yield a bi-regular matrix (and hence an MDS matrix).