Construction of Lightweight MDS Matrices over Matrix Polynomial Residue Ring

In this article, we investigate the construction of lightweight MDS matrices over the matrix polynomial residue ring. According to distributions of the minimum polynomial, distributions of XOR count and equivalence classes of MDS matrices, we propose an algorithm, which not only can construct lightest MDS matrices, but also is evidently more efficient than previous methods. Moreover, we investigate existences of involutory MDS matrices over the matrix polynomial residue ring. According to quadratic congruence, over the matrix polynomial residue ring, we propose a simplified necessary-and-sufficient condition for deciding whether a Hadamard matrix is invorlutory. With this method, we propose another efficient and special algorithm to construct lightweight Hadamard involutory MDS matrices. Over the 8× 8 matrix polynomial residue ring, we construct vast 4×4 Hadamard involutory MDS matrices with 20 XORs, which are much lighter than previous results. In addition, we obtain a series of propositions about the parity of XOR count.