A new construction of lightweight MDS matrices

Since maximum distance separable (MDS) matrices can be used as the best building blocks of diffusion layer of block cipher or hash functions, extensive studies have been investigated. In this article, we provide a new method to search lightweight MDS matrices. By providing a new matrix representation of elements of finite fields and observing the special form of the matrices, we exhibit new search algorithms that greatly reduce the search space and make lightweight MDS matrices with low XOR counts. We provide the non-involutory MDS matrices with the least possible XOR gates for dimension 4×4 over finite fields GF(2⁴) and GF(2⁸) respectively. Compared to the best known matrices, our new candidates have advantages for hardware implementation.