Lightweight Recursive MDS Matrices with Generalized Feistel Network

Abstract

Maximum distance separable (MDS) matrices are often used to construct optimal linear diffusion layers in many block ciphers. With the development of lightweight cryptography, the recursive MDS matrices play as good candidates. The recursive MDS matrices can be computed as powers of sparse matrices. In this paper, we consider searching recursive MDS matrices from Generalized Feistel Structure (GFN) matrices. The advantage of constructing MDS matrices based on GFN matrices mainly displays two aspects. First, the recursive GFN MDS matrix can be implemented by parallel computation that would reduce the running time. Second, the inverse matrix of recursive GFN MDS matrix is also a recursive GFN MDS matrix and they have the same XOR count. We provide some computational experiments to show we do find some lightweight $4\times 4$ and $8\times 8$ recursive GFN MDS matrices over $\mathbb {F}_{2^{n}}$. Especially, the $8\times 8$ recursive GFN MDS matrices over $\mathbb {F}_{2^{8}}$ have lower XOR count than the previous recursive MDS matrices.