Circulant, circulant-like and orthogonal MDS generalized cauchy matrices

Abstract

A matrix M over the filed Fq is called maximum distance separable (MDS) if all of its square submatrices are invertible. MDS generalized Cauchy (MDS-GC) matrices are important in both cryptography and coding theory. As an application, these matrices are used to provide diffusion in block ciphers. In this paper MDS-GC matrices over F 2 k are constructed which are involutory, Hadamard, circulant or orthogonal. The construction is based on using linearly related Vandermonde matrices. First, the construction of n×n circulant MDS-GC matrices over F 2 k is given for n a divisor of 2 k − 1; and an n × n circulant MDS-GC matrix C and its inverse C −1 are used to obtain the n × 2n matrix H = (C ∥ C −1 ) with square nonsingular submatrices. A class of 2 m × 2 m orthogonal MDS-GC matrices over F 2 k is given subject to the divisibility of 2 k − 1 by 2 m − 1. These orthogonal MDS-GC matrices are used to construct a new class of multiple MDS matrices over F 2 k which have applications in the design of Feistel ciphers. The constructed n × n circulant matrices are also used to obtain (n + 1) × (n + 1) circulant-like MDS matrices over F 2 k , and a closed-form expression for the inverse of these matrices is given. This is the first construction of circulant-like MDS matrices and their inverses which does not employ exhaustive search.