Paper 2018/1072

Construction of Lightweight MDS Matrices from Generalized Feistel Structures

Mahdi Sajadieh and Mohsen Mousavi

Abstract

This paper investigates the construction of lightweight MDS matrices with generalized Feistel structures (GFS). The approach developed by this paper consists in deriving MDS matrices from the product of several sparser ones. This can be seen as a generalization to several matrices of the recursive construction which derives MDS matrices as the powers of a single companion matrix. The first part of this paper gives some theoretical results on the iteration of GFS. In second part, using GFS and primitive matrices, we propose some types of sparse matrices that are called extended primitive GFS (EGFS) matrices. Then, by applying binary linear functions to several round of EGFS matrices, lightweight $4\times 4$, $6\times 6$ and $8\times 8$ MDS matrices are proposed which are implemented with 67, 156 and 260 XOR for 8-bit input, respectively. The results match the best known lightweight $4\times 4$ MDS matrix and improve the best known $6\times 6$ and $8\times 8$ MDS matrices. Also, the proposed $6\times 6$ MDS matrix is implemented with 114 XOR for 6-bit input. Moreover, we propose an $6\times 6$ MDS matrix such that the implementation cost of the proposed matrix is 90 XOR for 4-bit input. Furthermore, we determine the implementation cost of the inverses of the proposed matrices, since none of them are involutions. Generally, the construction presented in this paper is relatively general and can be applied for other matrix dimensions and finite fields as well.

Note: This is the first revised version of the paper.