Direct Constructions of (Involutory) MDS Matrices from Block Vandermonde and Cauchy-Like Matrices

Abstract

MDS matrices are important components in the design of linear diffusion layers of many block ciphers and hash functions. Recently, there have been a lot of work on searching and construction of lightweight MDS matrices, most of which are based on matrices of special types over finite fields. Among all those work, Cauchy matrices and Vandermonde matrices play an important role since they can provide direct constructions of MDS matrices. In this paper, we consider constructing MDS matrices based on block Vandermonde matrices. We find that previous constructions based on Vandermonde matrices over finite fields can be directly generalized if the building blocks are pairwise commutative. Different from previous proof method, the MDS property of a matrix constructed by two block Vandermonde matrices is confirmed adopting a Lagrange interpolation technique, which also sheds light on a relationship between it and an MDS block Cauchy matrix. Those constructions generalize previous ones over finite fields as well, but our proofs are much simpler. Furthermore, we present a new type of block matrices called block Cauchy-like matrices, from which MDS matrices can also be constructed. More interestingly, those matrices turn out to have relations with MDS matrices constructed from block Vandermonde matrices and the so-called reversed block Vandermonde matrices. For all these constructions, we can also obtain involutory MDS matrices under certain conditions. Computational experiments show that lightweight involutory MDS matrices can be obtained from our constructions.