A new matrix form to generate all 3 × 3 involutory MDS matrices over [formula omitted

Abstract

In this paper, we propose a new matrix form to generate all 3×3 involutory and MDS matrices over F2m and prove that the number of all 3×3 involutory and MDS matrices over F2m is (2m−1)2⋅(2m−2)⋅(2m−4), where m>2. Moreover, we give 3×3 involutory and MDS matrices over F23, F24 and F28 defined by the irreducible polynomials x3+x+1, x4+x+1 and x8+x7+x6+x+1, respectively, by considering the minimum XOR count, which is a metric used in the estimation of hardware implementation cost. Finally, we provide the maximum number of 1s in 3×3 involutory MDS matrices.