The lightest 4 × 4 MDS matrices over GL(4, 𝔽_2 )

Maximal distance separable (MDS) matrices are important components for block ciphers. In this paper, we present an algorithm for searching 4 × 4 MDS matrices over GL(4, F2). By this algorithm, we find all the lightest MDS matrices have only 10 XOR counts. Besides, all these lightest MDS matrices are classified to 3 types, and some necessary and sufficient conditions are presented for them as well. Some theoretical results can be generalized to the case GL(m,F2) easily, and 4× 4 MDS matrices with 10 XOR counts can be constructed directly.