On circulant involutory MDS matrices

Abstract

We give a new algebraic proof of the non-existence of circulant involutory MDS matrices with coefficients in fields of characteristic 2. In odd characteristics we give parameters for the potential existence. If we relax circulancy to $$\theta $$ -circulancy, then there is no restriction to the existence of $$\theta $$ -circulant involutory MDS matrices even for fields of characteristic 2. Finally, we relax further the involutory definition and propose a new direct construction of almost involutory $$\theta $$ -circulant MDS matrices. We show that they can be interesting in hardware implementations.