On construction of bi-regular circulant matrices, relating to MDS matrices

The objective of this work is to design bi-regular circulant matrices with the maximum number of occurrences of an arbitrary element. The reason to examine bi-regular matrices is that any MDS matrix is necessarily the bi-regular one, and MDS matrices are substantial to cryptography. Simultaneously, the reason to maximize the number of occurrences of an arbitrary element for matrices is that such matrices allow to perform matrix-vector multiplication more efficiently. The results obtained include the upper bound of the number of arbitrary element occurrences for which the bi-regularity of the circulant matrix preserves. Furthermore, necessary and sufficient conditions for the bi-regularity of the circulant matrix is derived. Those conditions provide with the efficient procedure of bi-regularity property verification, which is described within the paper. Additionally, paper lists several bi-regular circulant matrices templates of order up to 31 with the maximum number of arbitrary element occurrences. It was revealed that there are no square templates of order 32 of the structure mentioned.