Applications of design theory for the constructions of MDS matrices for lightweight cryptography

Abstract

Abstract In this paper, we observe simple yet subtle interconnections among design theory, coding theory and cryptography. Maximum distance separable (MDS) matrices have applications not only in coding theory but are also of great importance in the design of block ciphers and hash functions. It is nontrivial to find MDS matrices which could be used in lightweight cryptography. In the SAC 2004 paper [12], Junod and Vaudenay considered bi-regular matrices which are useful objects to build MDS matrices. Bi-regular matrices are those matrices all of whose entries are nonzero and all of whose 2 × 2 {2\times 2} submatrices are nonsingular. Therefore MDS matrices are bi-regular matrices, but the converse is not true. They proposed the constructions of efficient MDS matrices by studying the two major aspects of a d × d {d\times d} bi-regular matrix M, namely v 1 ⁢ ( M ) {v_{1}(M)} , i.e. the number of occurrences of 1 in M, and c 1 ⁢ ( M ) {c_{1}(M)} , i.e. the number of distinct elements in M other than 1. They calculated the maximum number of ones that can occur in a d × d {d\times d} bi-regular matrices, i.e. v 1 d , d {v_{1}^{d,d}} for d up to 8, but with their approach, finding v 1 d , d {v_{1}^{d,d}} for d ≥ 9 {d\geq 9} seems difficult. In this paper, we explore the connection between the maximum number of ones in bi-regular matrices and the incidence matrices of Balanced Incomplete Block Design (BIBD). In this paper, tools are developed to compute v 1 d , d {v_{1}^{d,d}} for arbitrary d. Using these results, we construct a restrictive version of d × d {d\times d} bi-regular matrices, introducing by calling almost-bi-regular matrices, having v 1 d , d {v_{1}^{d,d}} ones for d ≤ 21 {d\leq 21} . Since, the number of ones in any d × d {d\times d} MDS matrix cannot exceed the maximum number of ones in a d × d {d\times d} bi-regular matrix, our results provide an upper bound on the number of ones in any d × d {d\times d} MDS matrix. We observe an interesting connection between Latin squares and bi-regular matrices and study the conditions under which a Latin square becomes a bi-regular matrix and finally construct MDS matrices from Latin squares. Also a lower bound of c 1 ⁢ ( M ) {c_{1}(M)} is computed for d × d {d\times d} bi-regular matrices M such that v 1 ⁢ ( M ) = v 1 d , d {v_{1}(M)=v_{1}^{d,d}} , where d = q 2 + q + 1 {d=q^{2}+q+1} and q is any prime power. Finally, d × d {d\times d} efficient MDS matrices are constructed for d up to 8 from bi-regular matrices having maximum number of ones and minimum number of other distinct elements for lightweight applications.