Analysis of Toeplitz MDS Matrices

Sarkar, Sumanta; Syed, Habeeb

doi:10.1007/978-3-319-59870-3_1

Cited by 15 publications

(13 citation statements)

References 14 publications

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“…Toeplitz matrices have a deep interconnection with circulant matrices and interested reader may consult [18,39] for more information about the connection. Recently Toeplitz matrices are used to construct MDS matrices using search technique [55,56]. It has similarity to the constructions of circulant MDS matrices discussed in [24].…”

Section: Constructing Mds Matrices From Toeplitz and Hankel Matricesmentioning

confidence: 99%

“…Toeplitz matrices and Hankel matrices are deeply interconnected with circulant matrices. Recently, Toeplitz matrices have been used to construct MDS matrices [55,56]. Similar to circulant, circulant-like and generalized circulant MDS matrices, search methods are used to construct Toeplitz MDS matrices.…”

Section: Introductionmentioning

confidence: 99%

“…However, the corresponding Hankel matrix is symmetric. MDS matrices have been constructed from Toeplitz matrices in [55,56]. In Section 7, we use the above interconnection to easily extend these constructions for MDS matrices from Hankel matrices.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cryptographically significant mds matrices over finite fields: A brief survey and some generalized results

Gupta¹,

Pandey²,

Ray³

et al. 2019

Advances in Mathematics of Communications

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A matrix is MDS or super-regular if and only if every square submatrices of it are nonsingular. MDS matrices provide perfect diffusion in block ciphers and hash functions. In this paper we provide a brief survey on cryptographically significant MDS matrices-a first to the best of our knowledge. In addition to providing a summary of existing results, we make several contributions. We exhibit some deep and nontrivial interconnections between different constructions of MDS matrices. For example, we prove that all known Vandermonde constructions are basically equivalent to Cauchy constructions. We prove some folklore results which are used in MDS matrix literature. Wherever possible, we provide some simpler alternative proofs. We do not discuss efficiency issues or hardware implementations; however, the theory accumulated and discussed here should provide an easy guide towards efficient implementations.

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Section: Constructing Mds Matrices From Toeplitz and Hankel Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cryptographically significant mds matrices over finite fields: A brief survey and some generalized results

Gupta¹,

Pandey²,

Ray³

et al. 2019

Advances in Mathematics of Communications

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“…At IACR Transactions on Symmetric Cryptology 2016, Sarkar et al [27] showed that Toeplitz matrix cannot be both involutory and MDS and investigated the lower bounds of XOR counts of 4 × 4 MDS matrices over GF (2 4 ) and GF (2 8 ). At S&P 2017, Sarkar et al [28] further presented characterizations of Toeplitz matrices in light of MDS property and improved the lower bounds of XOR counts of 8 × 8 MDS matrices by obtaining Toeplitz MDS matrices with lower XOR counts over GF (2 4 ) and GF (2 8 ).…”

Section: Matrix Polynomialmentioning

confidence: 99%

“…Additionally, they showed that there are MDS matrices with higher Hamming weight than the AES diffusion matrix, but fewer XORs. After that, many works [23,11,25,21,22,27,28,31] measured the lightweight of MDS matrices with XOR counts.…”

Section: Introductionmentioning

confidence: 99%

On Efficient Constructions of Lightweight MDS Matrices

Zhou

Wang

Sun

2018

ToSC

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The paper investigates the maximum distance separable (MDS) matrix over the matrix polynomial residue ring. Firstly, by analyzing the minimal polynomials of binary matrices with 1 XOR count and element-matrices with few XOR counts, we present an efficient method for constructing MDS matrices with as few XOR counts as possible. Comparing with previous constructions, our corresponding constructions only cost 1 minute 27 seconds to 7 minutes, while previous constructions cost 3 days to 4 weeks. Secondly, we discuss the existence of several types of involutory MDS matrices and propose an efficient necessary-and-sufficient condition for identifying a Hadamard matrix being involutory. According to the condition, each involutory Hadamard matrix over a polynomial residue ring can be accurately and efficiently searched. Furthermore, we devise an efficient algorithm for constructing involutory Hadamard MDS matrices with as few XOR counts as possible. We obtain many new involutory Hadamard MDS matrices with much fewer XOR counts than optimal results reported before.

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