A New Matrix Form to Generate All 3 × 3 Involutory MDS Matrices Over F2m

Abstract

In this paper, we propose a new matrix form to generate all 3×3 involutory and MDS matrices over F<inf>2m </inf> and prove that the number of all 3×3 involutory and MDS matrices over F<inf>2m </inf> is (2m−1)2⋅(2m−2)⋅(2m−4), where m>2. Moreover, we give 3×3 involutory and MDS matrices over F<inf>23 </inf>, F<inf>24 </inf> and F<inf>28 </inf> 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. © 2019 Elsevier B.V.