Construction of Recursive MDS Matrices Using DLS Matrices

Gupta, Kishan Chand; Pandey, Sumit Kumar; Samanta, Susanta Kumar

doi:10.1007/978-3-031-17433-9_1

Cited by 3 publications

(7 citation statements)

References 27 publications

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“…The authors proposed a specific form of recursive MDS matrix to perform nearly identical encryption and decryption and analyze some experimental results. In [ 29 ], the authors presented a very sparse and diagonal type of matrix. From there, they gave the minimum fixed number of XORs of the MDS matrices generated from the above matrix form.…”

Section: Related Workmentioning

confidence: 99%

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Efficient implementation of the linear layer of block ciphers with large MDS matrices based on a new lookup table technique

Luong,

Van Long,

2024

PLoS ONE

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Block cipher is a cryptographic field that is now widely applied in various domains. Besides its security, deployment issues, implementation costs, and flexibility across different platforms are also crucial in practice. From an efficiency perspective, the linear layer is often the slowest transformation and requires significant implementation costs in block ciphers. Many current works employ lookup table techniques for linear layers, but they are quite costly and do not save memory storage space for the lookup tables. In this paper, we propose a novel lookup table technique to reduce memory storage when executing software. This technique is applied to the linear layer of block ciphers with recursive Maximum Distance Separable (MDS) matrices, Hadamard MDS matrices, and circulant MDS matrices of considerable sizes (e.g. sizes of 16, 32, 64, and so on). The proposed lookup table technique leverages the recursive property of linear matrices and the similarity in elements of Hadamard or circulant MDS matrices, allowing the construction of a lookup table for a submatrix instead of the entire linear matrix. The proposed lookup table technique enables the execution of the diffusion layer with unchanged computational complexity (number of XOR operations and memory accesses) compared to conventional lookup table implementations but allows a substantial reduction in memory storage for the pre-computed tables, potentially reducing the storage needed by 4 or 8 times or more. The memory storage will be reduced even more as the size of the MDS matrix increases. For instance, analysis shows that when the matrix size is 64, the memory storage ratio with the proposed lookup table technique decreases by 87.5% compared to the conventional lookup table technique. This method also allows for more flexible software implementations of large-sized linear layers across different environments.

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Section: Related Workmentioning

confidence: 99%

“…In recent years, and mainly to gain advantages in hardware implementation, there have been many studies focusing on building recursive linear layers [26][27][28][29]. Accordingly, their linear matrix is a power of a companion matrix [30,31] that has a simple form and is easy for hardware implementation.…”

Section: Introductionmentioning

confidence: 99%

Efficient implementation of the linear layer of block ciphers with large MDS matrices based on a new lookup table technique

Luong,

Van Long,

2024

PLoS ONE

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“…Let 𝐵 = (𝛽 1 , 𝛽 2 , … , 𝛽 𝑘 ) be a basis for 𝐺, then it holds that 𝐺 𝛽 1 ,𝛽 2 ,…,𝛽 𝑘 = Efficient multiplication of a vector by a matrix MDS {𝐼 𝑛 }, where 𝐼 𝑛 is the identity matrix of size 𝑛 × 𝑛. This determines a descending chain of stabilizers: 𝐺 = 𝐺 (1) ⊇ 𝐺 (2) ⊇ ⋯ ⊇ 𝐺 (𝑘+1) = {𝐼 𝑛 },…”

Section: Preliminariesmentioning

confidence: 99%

“…Writing L β 2 in the form shown, the right transversal U 2 of G (3) in G (2) is determined and given 𝑢 2 ∈ U 2 , ((a 0 , a 1 , … . , a n−1 ) u 2 ) is as shown below:…”

Section: Proofmentioning

confidence: 99%

“…However, MDS matrices are not sparse and have a large description, which induces costly implementation in software and hardware. Research focuses on circulant [1] and recursive matrices [2] to reduce implementation costs, but the solutions found to date are only for small dimensions. When the dimensions considered grow, both generating the MDS matrices and reducing their implementation costs, become extremely complex problems that have not yet been resolved.…”

Section: Introductionmentioning

confidence: 99%

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Efficient multiplication of a vector by a matrix MDS

Arrozarena¹,

Fiallo²

2023

ISJ

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Abstract— An algorithm is proposed for the efficient multiplication of a vector by an n x n MDS matrix defined on Fq or by its inverse. The algorithm is based on the multiplication of two polynomials modulo a polynomial of degree n Reed-Solomon code generator and has complexity O(nlog2log2(nlog2q)). The algorithm only needs to store n values of the Fq field for the multiplication of a vector by an n x n MDS matrix and 2n values for the multiplication of the vector by the inverse matrix.

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A Dynamic Algorithm for the Linear Layer of SPN Block Ciphers Based on Self-Reciprocal Recursive MDS Matrices

Luong

2023

2023 15th International Conference on Knowledge and Systems Engineering (KSE)

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Construction of Recursive MDS Matrices Using DLS Matrices

Cited by 3 publications

References 27 publications

Efficient implementation of the linear layer of block ciphers with large MDS matrices based on a new lookup table technique

Efficient implementation of the linear layer of block ciphers with large MDS matrices based on a new lookup table technique

Efficient multiplication of a vector by a matrix MDS

A Dynamic Algorithm for the Linear Layer of SPN Block Ciphers Based on Self-Reciprocal Recursive MDS Matrices

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