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Journals / Turkish Journal of Electrical Engineering and Computer Science / 2020 / Cilt: 28 - Sayı: 1

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

Muharrem Tolga SAKALLI
Sedat AKLEYLEK
Kemal AKKANAT
Vincent RIJMEN
Pages
275–289
Publication Date
2020-01-01
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The proposed method, due to its ground field structure, is intended to be a complementary method for the current construction methods in the literature.

📌 3 Key Takeaways

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Objective & Hypothesis:In this paper, we explicitly define the automorphisms of MDS matrices over the same binary extension field.
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Key Findings & Data:By extending this idea, we present the isomorphisms between MDS matrices over F2m and MDS matrices over F2mt , where t ≥ 1 and m > 1, which preserves the software implementation properties in view of XOR operations and table lookups of any given MDS matrix over F2m .
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Conclusion & Impact:The proposed method, due to its ground field structure, is intended to be a complementary method for the current construction methods in the literature.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:MDS matrix · branch number · block cipher

Abstract

In this paper, we explicitly define the automorphisms of MDS matrices over the same binary extension field. By extending this idea, we present the isomorphisms between MDS matrices over F2m and MDS matrices over F2mt , where t ≥ 1 and m > 1, which preserves the software implementation properties in view of XOR operations and table lookups of any given MDS matrix over F2m . Then we propose a novel method to obtain distinct functions related to these automorphisms and isomorphisms to be used in generating isomorphic MDS matrices new MDS matrices in view of implementation properties using the existing ones. The comparison with the MDS matrices used in AES, ANUBIS, and subfield-Hadamard construction shows that we generate an involutory 4 × 4 MDS matrix over F28 from an involutory 4 × 4 MDS matrix over F24 whose required number of XOR operations is the same as that of ANUBIS and the subfield-Hadamard construction, and better than that of AES. The proposed method, due to its ground field structure, is intended to be a complementary method for the current construction methods in the literature.

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Cite This Article

SAKALLI, M. T., AKLEYLEK, S., AKKANAT, K., RIJMEN, V. (2020). On the automorphisms and isomorphisms of MDS matrices and their efficient implementations. Turkish Journal of Electrical Engineering and Computer Science, Cilt: 28 - Sayı: 1, 275–289. https://ajindex.com/articles/186716

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References (1)

  1. Shannon CE. Communication theory of secrecy systems. Bell System Technical Journal 1949; 28: 656-715. doi: 10.1002/j.1538-7305.1949.tb00928.x [2] Daemen J, Rijmen V. The Design of Rijndael, AES - The Advanced Encryption Standard. Berlin, Germany: Springer-Verlag, 2002. [3] Barreto PSLM, Rijmen V. Whirlpool. In: van Tilborg HCA, Jajodia S. (editors). Encyclopedia of Cryptography and Security. 2nd ed. Boston, MA, USA: Springer, 2011, pp. 1384-1385. [4] Guo J, Peyrin T, Poschmann A. The PHOTON family of lightweight hash functions. In: Proceedings of CRYPTO; Santa Barbara, CA, USA; 2011. pp. 222-239. [5] Barreto PSLM, Nikov V, Nikova S, Rijmen V, Tischhauser E. Whirlwind: A new cryptographic hash function. Design, Codes and Cryptography 2010; 56: 141–162. doi: 10.1007/s10623-010-9391-y [6] Biham E, Shamir A. Differential cryptanalysis of DES-like cryptosystems. In: Proceedings of CRYPTO; Santa Barbara, CA, USA; 1990. pp. 2-21. [7] Matsui M. Linear cryptanalysis method for DES cipher. In: Proceedings of EUROCRYPT; Lofthus, Norway; 1993. pp. 386-397. [8] Youssef AM, Mister S, Tavares SE. On the design of linear transformation for substitution permutation encryption networks. In: Proceedings of SAC; Ottowa, Canada; 1997. pp. 40-48. [9] Gupta KC, Ray IG. On constructions of circulant MDS matrices for lightweight cryptography. In: Proceedings of ISPEC; Fuzhou, China; 2014. pp. 564-576. [10] Lacan J, Fimes J. Systematic MDS erasure codes based on Vandermonde matrices. IEEE Transactions on Communications Letters 2004; 8 (9): 570-572. doi: 10.1109/LCOMM.2004.833807 [11] Sajadieh M, Dakhilalian M, Mala H, Omoomi B. On construction of involutory MDS matrices from Vandermonde matrices in GF(2q ). Design, Codes and Cryptography 2012; 64 (3): 287-308. doi: 10.1007/s10623-011-9578-x [12] Augot D, Finiasz M. Direct construction of recursive MDS diffusion layers using shortened BCH codes. In: Proceedings of FSE; London, UK; 2014. pp. 3-17. [13] Berger TP. Construction of recursive MDS diffusion layers from Gabidulin codes. In: Proceedings of INDOCRYPT; Mumbai, India; 2013. pp. 274-285. [14] Cauchois V, Loidreau P, Merkiche N. Direct construction of quasi-involutory recursive-like MDS matrices from 2- cyclic codes. IACR Transactions on Symmetric Cryptology 2016; 2016 (2): 80-98. doi: 10.13154/tosc.v2016.i2.80-98 [15] Sajadieh M, Dakhilalian M, Mala H, Sepehrdad P. Recursive diffusion layers for block ciphers and hash functions. In: Proceedings of FSE; Washington, DC, USA;