**Construction of MDS Matrices from Generalized Feistel Structures**

*Mahdi Sajadieh and Mohsen Mousavi*

**Abstract: **This paper investigates the construction of 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.
Moreover, we propose $8\times 8$ Near-MDS matrices such that
the implementation cost of the proposed matrices are $108$ and $204$ XOR
for 4 and $8$-bit input, respectively.
Although none of the presented matrices are involutions,
the implementation cost of the inverses of the proposed matrices is
equal to the implementation cost of the given matrices.
Furthermore, the construction presented in this paper is relatively general
and can be applied for other matrix dimensions and finite fields as well.

**Category / Keywords: **implementation / MDS matrix, Generalized Feistel Structures, Diffusion layer.

**Date: **received 4 Nov 2018, last revised 29 May 2019

**Contact author: **m mousavi at mut-es ac ir

**Available format(s): **PDF | BibTeX Citation

**Note: **This is the last revised version of the paper.

**Version: **20190529:231209 (All versions of this report)

**Short URL: **ia.cr/2018/1072

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