Constructing Recursive MDS Matrices Effective for Implementation from Reed-Solomon Codes and Preserving the Recursive Property of MDS Matrix of Scalar Multiplication

Abstract

MDS matrices from Maximum Distance Separable codes (MDS codes) and MDS matrix transformations have important applications in cryptography. However, MDS matrices always have a large description and cannot be sparse, causing costly hardware/software implementations. Recursive MDS matrices allow to solve this problem as they can be a power of a simple serial matrix, so there is a compact description suitable even for constrained processing environments. In this paper, the method for constructing recursive MDS matrices effective for implementation from Reed-Solomon codes is presented. In addition, preserving the recursive property of MDS matrix of scalar multiplication transformation is given. The recursive MDS matrices effective for implementation are meaningful in hardware implementation, and the ability to preserve recursive property of MDS matrix of scalar multiplication transformation also has important applications for efficiently building dynamic block ciphers to improve the security of block ciphers.