A reduced set of submatrices for a faster evaluation of the MDS property of a circulant matrix with entries that are powers of two

In this paper a reduced set of submatrices for a faster evaluation of the MDS property of a circulant matrix, with entries that are powers of two, is proposed. A proposition is made that under the condition that all entries of a t × t circulant matrix are powers of 2, it is sufficient to check only its 2x2 submatrices in order to evaluate the MDS property in a prime field. Although there is no theoretical proof to support this proposition at this point, the experimental results conducted on a sample of 100 thousand randomly generated matrices indicate that this proposition is true. There are benefits of the proposed MDS test on the efficiency of search methods for the generation of circulant MDS matrices, regardless of the correctness of this proposition. However, if this proposition is correct, its impact on the speed of search methods for circulant MDS matrices will be huge, which will enable generation of MDS matrices of large sizes. Also, a modified version of the make_binary_powers function is presented. Based on this modified function and the proposed MDS test, some examples of efficient 16 x 16 MDS matrices are presented. Also, an examples of efficient 24 x 24 matrices are generated, whose MDS property should be further validated

On Efficient Constructions of Lightweight MDS Matrices

The paper investigates the maximum distance separable (MDS) matrix over the matrix polynomial residue ring. Firstly, by analyzing the minimal polynomials of binary matrices with 1 XOR count and element-matrices with few XOR counts, we present an efficient method for constructing MDS matrices with as few XOR counts as possible. Comparing with previous constructions, our corresponding constructions only cost 1 minute 27 seconds to 7 minutes, while previous constructions cost 3 days to 4 weeks. Secondly, we discuss the existence of several types of involutory MDS matrices and propose an efficient necessary-and-sufficient condition for identifying a Hadamard matrix being involutory. According to the condition, each involutory Hadamard matrix over a polynomial residue ring can be accurately and efficiently searched. Furthermore, we devise an efficient algorithm for constructing involutory Hadamard MDS matrices with as few XOR counts as possible. We obtain many new involutory Hadamard MDS matrices with much fewer XOR counts than optimal results reported before