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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.08250 |
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| _version_ | 1866910490025787392 |
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| author | Kumar, Yogesh Mishra, P. R. Samanta, Susanta Gaur, Atul |
| author_facet | Kumar, Yogesh Mishra, P. R. Samanta, Susanta Gaur, Atul |
| contents | Maximum distance separable (MDS) matrices play a crucial role not only in coding theory but also in the design of block ciphers and hash functions. Of particular interest are involutory MDS matrices, which facilitate the use of a single circuit for both encryption and decryption in hardware implementations. In this article, we present several characterizations of involutory MDS matrices of even order. Additionally, we introduce a new matrix form for obtaining all involutory MDS matrices of even order and compare it with other matrix forms available in the literature. We then propose a technique to systematically construct all $4 \times 4$ involutory MDS matrices over a finite field $\mathbb{F}_{2^m}$. This method significantly reduces the search space by focusing on involutory MDS class representative matrices, leading to the generation of all such matrices within a substantially smaller set compared to considering all $4 \times 4$ involutory matrices. Specifically, our approach involves searching for these representative matrices within a set of cardinality $(2^m-1)^5$. Through this method, we provide an explicit enumeration of the total number of $4 \times 4$ involutory MDS matrices over $\mathbb{F}_{2^m}$ for $m=3,4,\ldots,8$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Systematic Construction Approach for All $4\times 4$ Involutory MDS Matrices Kumar, Yogesh Mishra, P. R. Samanta, Susanta Gaur, Atul Cryptography and Security Maximum distance separable (MDS) matrices play a crucial role not only in coding theory but also in the design of block ciphers and hash functions. Of particular interest are involutory MDS matrices, which facilitate the use of a single circuit for both encryption and decryption in hardware implementations. In this article, we present several characterizations of involutory MDS matrices of even order. Additionally, we introduce a new matrix form for obtaining all involutory MDS matrices of even order and compare it with other matrix forms available in the literature. We then propose a technique to systematically construct all $4 \times 4$ involutory MDS matrices over a finite field $\mathbb{F}_{2^m}$. This method significantly reduces the search space by focusing on involutory MDS class representative matrices, leading to the generation of all such matrices within a substantially smaller set compared to considering all $4 \times 4$ involutory matrices. Specifically, our approach involves searching for these representative matrices within a set of cardinality $(2^m-1)^5$. Through this method, we provide an explicit enumeration of the total number of $4 \times 4$ involutory MDS matrices over $\mathbb{F}_{2^m}$ for $m=3,4,\ldots,8$. |
| title | A Systematic Construction Approach for All $4\times 4$ Involutory MDS Matrices |
| topic | Cryptography and Security |
| url | https://arxiv.org/abs/2404.08250 |