On the Construction of New Lightweight Involutory MDS Matrices in Generalized Subfield Form

The XOR operator is a simple yet crucial computation in computer science, especially in cryptography. In symmetric cryptographic schemes, particularly in block ciphers, the AddRoundKey transformation is commonly used to XOR an internal state with a round key. One method to enhance the security of block ciphers is to diversify this transformation. In this paper, we propose some straightforward yet highly effective techniques for generating t-bit random XOR tables. One approach is based on the Hadamard matrix, while another draws inspiration from the popular intellectual game Sudoku. Additionally, we introduce algorithms to animate the XOR transformation for generalized block ciphers. Specifically, we apply our findings to the AES encryption standard to present the key-dependent AES algorithm. Furthermore, we conduct a security analysis and assess the randomness of the proposed key-dependent AES algorithm using NIST SP 800-22, Shannon entropy based on the ENT tool, and min-entropy based on NIST SP 800-90B. Thanks to the key-dependent random XOR tables, the key-dependent AES algorithm have become much more secure than AES, and they also achieve better results in some statistical standards than AES.

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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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On the Construction of New Lightweight Involutory MDS Matrices in Generalized Subfield Form

Cited by 3 publications

References 51 publications

Introducing a new connection between the entries of MDS matrices constructed by generalized Cauchy matrices in $$GF(2^q)$$

Introducing a new connection between the entries of MDS matrices constructed by generalized Cauchy matrices in $$GF(2^q)$$

Enhancing block cipher security with key-dependent random XOR tables generated via hadamard matrices and Sudoku game

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

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