MDS matrix

About: MDS matrix is a research topic. Over the lifetime, 102 publications have been published within this topic receiving 2000 citations.

Lightweight MDS Involution Matrices

Abstract: In this article, we provide new methods to look for lightweight MDS matrices, and in particular involutory ones. By proving many new properties and equivalence classes for various MDS matrices constructions such as circulant, Hadamard, Cauchy and Hadamard-Cauchy, we exhibit new search algorithms that greatly reduce the search space and make lightweight MDS matrices of rather high dimension possible to find. We also explain why the choice of the irreducible polynomial might have a significant impact on the lightweightness, and in contrary to the classical belief, we show that the Hamming weight has no direct impact. Even though we focused our studies on involutory MDS matrices, we also obtained results for non-involutory MDS matrices. Overall, using Hadamard or Hadamard-Cauchy constructions, we provide the (involutory or non-involutory) MDS matrices with the least possible XOR gates for the classical dimensions \(4 \times 4\), \(8 \times 8\), \(16 \times 16\) and \(32 \times 32\) in \(\mathrm {GF}(2^4)\) and \(\mathrm {GF}(2^8)\). Compared to the best known matrices, some of our new candidates save up to 50 % on the amount of XOR gates required for an hardware implementation. Finally, our work indicates that involutory MDS matrices are really interesting building blocks for designers as they can be implemented with almost the same number of XOR gates as non-involutory MDS matrices, the latter being usually non-lightweight when the inverse matrix is required.

Recursive Diffusion Layers for (Lightweight) Block Ciphers and Hash Functions

Abstract: Diffusion layers with maximum branch numbers are widely used in block ciphers and hash functions. In this paper, we construct recursive diffusion layers using Linear Feedback Shift Registers (LFSRs). Unlike the MDS matrix used in AES, whose elements are limited in a finite field, a diffusion layer in this paper is a square matrix composed of linear transformations over a vector space. Perfect diffusion layers with branch numbers from 5 to 9 are constructed. On the one hand, we revisit the design strategy of PHOTON lightweight hash family and the work of FSE 2012, in which perfect diffusion layers are constructed by one bundle-based LFSR. We get better results and they can be used to replace those of PHOTON to gain smaller hardware implementations. On the other hand, we investigate new strategies to construct perfect diffusion layers using more than one bundle-based LFSRs. Finally, we construct perfect diffusion layers by increasing the number of iterations and using bit-level LFSRs. Since most of our proposals have lightweight examples corresponding to 4-bit and 8-bit Sboxes, we expect that they will be useful in designing (lightweight) block ciphers and (lightweight) hash functions.

On the Construction of Lightweight Circulant Involutory MDS Matrices

Abstract: In the present paper, we investigate the problem of constructing MDS matrices with as few bit XOR operations as possible. The key contribution of the present paper is constructing MDS matrices with entries in the set of $$m\times m$$ non-singular matrices over $$\mathbb {F}_2$$ directly, and the linear transformations we used to construct MDS matrices are not assumed pairwise commutative. With this method, it is shown that circulant involutory MDS matrices, which have been proved do not exist over the finite field $$\mathbb {F}_{2^m}$$, can be constructed by using non-commutative entries. Some constructions of $$4\times 4$$ and $$5\times 5$$ circulant involutory MDS matrices are given when $$m=4,8$$. To the best of our knowledge, it is the first time that circulant involutory MDS matrices have been constructed. Furthermore, some lower bounds on XORs that required to evaluate one row of circulant and Hadamard MDS matrices of order 4 are given when $$m=4,8$$. Some constructions achieving the bound are also given, which have fewer XORs than previous constructions.

A New Involutory MDS Matrix for the AES

Abstract: This paper proposes a new, large diffusion layer for the AES block cipher. This new layer replaces the ShiftRows and MixColumns operations by a new involutory matrix in every round. The objective is to provide complete diffusion in a single round, thus sharply improving the overall cipher security. Moreover, the new matrix elements have low Hamming-weight in order to provide equally good performance for both the encryption and decryption operations. We use the Cauchy matrix construction instead of circulant matrices such as in the AES. The reason is that circulant matrices cannot be simultaneously MDS and involutory.

Lightweight Multiplication in $$GF2^n$$ with Applications to MDS Matrices

Abstract: In this paper we consider the fundamental question of optimizing finite field multiplications with one fixed element. Surprisingly, this question did not receive much attention previously. We investigate which field representation, that is which choice of basis, allows for an optimal implementation. Here, the efficiency of the multiplication is measured in terms of the number of XOR operations needed to implement the multiplication. While our results are potentially of larger interest, we focus on a particular application in the second part of our paper. Here we construct new MDS matrices which outperform or are on par with all previous results when focusing on a round-based hardware implementation.