On the automorphisms and isomorphisms of MDS matrices and their efficient implementations

Sakallı, Muharrem Tolga; Akleylek, Sedat; Akkanat, Kemal; Rijmen, Vincent

doi:10.3906/elk-1906-151

Cited by 9 publications

(3 citation statements)

References 23 publications

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“…Then, 70,344 ( ) representative involutory MDS matrices cannot be generated by Hadamard matrix form. That means, by using the method given in Sakallı et al (2020) , one can map these representative involutory MDS matrices in order to obtain directly isomorphic counterparts over . Note that there are 4 isomorphisms from the finite field (defined by any irreducible polynomial) to the finite field (defined by any irreducible polynomial).…”

Section: Resultsmentioning

confidence: 99%

“…To handle this problem, the Generalized Hadamard (shortly GHadamard) matrix form, a hybrid construction method, was proposed in Pehlivanoğlu et al (2018) . Overall, in Sakallı et al (2020) , the authors proposed a complementary method for the current construction methods in the literature, which generates isomorphic MDS matrices (new MDS matrices from the implementation point of view) from any existing MDS matrix (due to its ground field structure). All these methods can be evaluated within the local optimization category that focuses on the coefficients of a given matrix.…”

Section: Introductionmentioning

confidence: 99%

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A new hybrid method combining search and direct based construction ideas to generate all 4 × 4 involutory maximum distance separable (MDS) matrices over binary field extensions

Tuncay,

Büyüksaraçoğlu Sakallı,

Kurt Pehlivanoğlu

et al. 2023

PeerJ Computer Science

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This article presents a new hybrid method (combining search based methods and direct construction methods) to generate all $4 \times 4$ involutory maximum distance separable (MDS) matrices over $\mathbf{F}_{2^m}$. The proposed method reduces the search space complexity at the level of $$\sqrt n $$, where n represents the number of all $4 \times 4$ invertible matrices over $\mathbf{F}_{2^m}$ to be searched for. Hence, this enables us to generate all $4 \times 4$ involutory MDS matrices over $\mathbf{F}_{2^3}$ and $\mathbf{F}_{2^4}$. After applying global optimization technique that supports higher Exclusive-OR (XOR) gates (e.g., XOR3, XOR4) to the generated matrices, to the best of our knowledge, we generate the lightest involutory/non-involutory MDS matrices known over $\mathbf{F}_{2^3}$, $\mathbf{F}_{2^4}$ and $\mathbf{F}_{2^8}$ in terms of XOR count. In this context, we present new $4 \times 4$ involutory MDS matrices over $\mathbf{F}_{2^3}$, $\mathbf{F}_{2^4}$ and $\mathbf{F}_{2^8}$, which can be implemented by 13 XOR operations with depth 5, 25 XOR operations with depth 5 and 42 XOR operations with depth 4, respectively. Finally, we denote a new property of Hadamard matrix, i.e., (involutory and MDS) Hadamard matrix form is, in fact, a representative matrix form that can be used to generate a small subset of all $2^k\times 2^k$ involutory MDS matrices, where k > 1. For k = 1, Hadamard matrix form can be used to generate all involutory MDS matrices.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A new hybrid method combining search and direct based construction ideas to generate all 4 × 4 involutory maximum distance separable (MDS) matrices over binary field extensions

Tuncay,

Büyüksaraçoğlu Sakallı,

Kurt Pehlivanoğlu

et al. 2023

PeerJ Computer Science

Self Cite

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“…Furthermore, MDS matrices are used in stream ciphers ( Watanabe et al, 2002 ) and hash functions ( Barreto, Rijmen & Nv, 2000 ; Choy et al, 2012 ; Gazzoni Filho, Barreto & Rijmen, 2006 ; Gauravaram et al, 2009 ; Guo, Peyrin & Poschmann, 2011 ), as indicated in various studies. In the literature, there are different types of methods to construct efficient MDS matrices, such as direct construction ( Cui, Jin & Kong, 2015 ; Sajadieh et al, 2012 ; Gupta & Ray, 2013 ; Güzel et al, 2019 ), search based construction ( Wu, Wang & Wu, 2013 ; Chand Gupta & Ghosh Ray, 2014 ; Li & Wang, 2016 ; Sarkar & Syed, 2016 , 2017 ; Sakalli et al, 2020 ), and hybrid construction ( Pehlivanoğlu et al, 2018 ).…”

Section: Introductionmentioning

confidence: 99%

Optimizing implementations of linear layers using two and higher input XOR gates

Kurt Pehlivanoğlu,

Demir

2024

PeerJ Computer Science

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Maximum distance separable (MDS) matrices are often used in the linear layer of a block cipher due to their good diffusion property. A well-designed lightweight MDS matrix, especially an involutory one, can provide both security and performance benefits to the cipher. Finding the corresponding effective linear straight-line program (SLP) of the circuit of a linear layer is still a challenging problem. In this article, first, we propose a new heuristic algorithm called Superior Boyar-Peralta (SBP) in the computation of the minimum number of two-input Exclusive-OR (XOR) gates with the minimum circuit depth for the SLPs. Contrary to the existing global optimization methods supporting only two-input XOR gates, SBP heuristic algorithm provides the best global optimization solutions, especially for extracting low-latency circuits. Moreover, we give a new 4 × 4 involutory MDS matrix over F24, which requires only 41 XOR gates and depth 3 after applying SBP heuristic, whereas the previously best-known cost is 45 XOR gates with the same depth. In the second part of the article, for further optimization of the circuit area of linear layers with multiple-input XOR gates, we enhance the recently proposed BDKCI heuristic algorithm by incorporating circuit depth awareness, which limits the depth of the circuits created. By using the proposed circuit depth-bounded version of BDKCI, we present better circuit implementations of linear layers of block ciphers than those given in the literature. For instance, the given circuit for the AES MixColumn matrix only requires 44 XOR gates/depth 3/240.95 GE in the STM 130 nm (simply called ASIC4) library, while the previous best-known result is 55 XOR gates/depth 5/243.00 GE. Much better, our new 4 × 4 involutory MDS matrix requires only 19 XOR gates/depth3/79.75 GE in the STM 90 nm (simply called ASIC1) library, which is the lightest and superior to the state-of-the-art results.

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On the Construction of $$4 \times 4$$ Lightweight Involutory MDS Matrices Over $$\mathbb {F}_{2^{8}}$$

Pehlıvanoğlu

Sakallı

Akleylek

et al. 2022

Proceedings of the Seventh International Conference on Mathematics and Computing

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On the automorphisms and isomorphisms of MDS matrices and their efficient implementations

Cited by 9 publications

References 23 publications

A new hybrid method combining search and direct based construction ideas to generate all 4 × 4 involutory maximum distance separable (MDS) matrices over binary field extensions

A new hybrid method combining search and direct based construction ideas to generate all 4 × 4 involutory maximum distance separable (MDS) matrices over binary field extensions

Optimizing implementations of linear layers using two and higher input XOR gates

On the Construction of $$4 \times 4$$ Lightweight Involutory MDS Matrices Over $$\mathbb {F}_{2^{8}}$$

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