On the Counting of Involutory MDS Matrices

• URL: http://arxiv.org/abs/2310.00090v3
• Date: Tue, 01 Oct 2024 06:08:45 GMT
• Title: On the Counting of Involutory MDS Matrices
• Authors: Susanta Samanta,
• Abstract summary: This paper enumerates Hadamard MDS and involutory Hadamard MDS matrices of order $4$ within the field $mathbbF_2r$. It also derives the count of Hadamard-MDS (NMDS) and involutory Hadamard NMDS matrices, each with exactly one zero in each row, of order $4$ over $mathbbF_2r$.
• Score: 0.0
• License:
• Abstract: The optimal branch number of MDS matrices has established their importance in designing diffusion layers for various block ciphers and hash functions. As a result, numerous matrix structures, including Hadamard and circulant matrices, have been proposed for constructing MDS matrices. Also, in the literature, significant attention is typically given to identifying MDS candidates with optimal implementations or proposing new constructions across different orders. However, this paper takes a different approach by not emphasizing efficiency issues or introducing new constructions. Instead, its primary objective is to enumerate Hadamard MDS and involutory Hadamard MDS matrices of order $4$ within the field $\mathbb{F}_{2^r}$. Specifically, it provides an explicit formula for the count of both Hadamard MDS and involutory Hadamard MDS matrices of order $4$ over $\mathbb{F}_{2^r}$. Additionally, it derives the count of Hadamard Near-MDS (NMDS) and involutory Hadamard NMDS matrices, each with exactly one zero in each row, of order $4$ over $\mathbb{F}_{2^r}$. Furthermore, the paper discusses some circulant-like matrices for constructing NMDS matrices and proves that when $n$ is even, any $2n \times 2n$ Type-II circulant-like matrix can never be an NMDS matrix. While it is known that NMDS matrices may be singular, this paper establishes that singular Hadamard matrices can never be NMDS matrices. Moreover, it proves that there exist exactly two orthogonal Type-I circulant-like matrices of order $4$ over $\mathbb{F}_{2^r}$.

Related papers

• A note on MDS Property of Circulant Matrices [3.069335774032178]
  In $2014$, Gupta and Ray proved that the circulant involutory matrices over the finite field $mathbbF_2m$ can not be maximum distance separable (MDS) This article delves into circulant matrices possessing these characteristics over the finite field $mathbbF_2m$.
  arXiv  Detail & Related papers  (2024-06-22T16:00:00Z)
• On MDS Property of g-Circulant Matrices [3.069335774032178]
  We first discuss $g$-circulant matrices with involutory and MDS properties. We then delve into $g$-circulant semi-involutory and semi-orthogonal matrices with entries from finite fields.
  arXiv  Detail & Related papers  (2024-06-22T15:18:31Z)
• A Systematic Construction Approach for All $4\times 4$ Involutory MDS Matrices [1.3332839594069594]
  We present several characterizations of involutory MDS matrices of even order. We propose a technique to systematically construct all $4 times 4$ involutory MDS matrices over a finite field.
  arXiv  Detail & Related papers  (2024-04-12T05:37:42Z)
• Construction of all MDS and involutory MDS matrices [7.171901763517741]
  We propose two algorithms for a hybrid construction of all $ntimes n$ MDS and involutory MDS matrices over a finite field $mathbbF_pm$. The proposed algorithms effectively narrow down the search space to identify $(n-1) times (n-1)$ MDS matrices.
  arXiv  Detail & Related papers  (2024-03-15T15:03:02Z)
• Large-scale gradient-based training of Mixtures of Factor Analyzers [67.21722742907981]
  This article contributes both a theoretical analysis as well as a new method for efficient high-dimensional training by gradient descent. We prove that MFA training and inference/sampling can be performed based on precision matrices, which does not require matrix inversions after training is completed. Besides the theoretical analysis and matrices, we apply MFA to typical image datasets such as SVHN and MNIST, and demonstrate the ability to perform sample generation and outlier detection.
  arXiv  Detail & Related papers  (2023-08-26T06:12:33Z)
• On the Direct Construction of MDS and Near-MDS Matrices [6.356706243374352]
  Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer. This paper introduces some direct constructions of NMDS matrices in both nonrecursive and recursion settings. We propose a method for constructing involutory MDS and NMDS matrices using generalized Vandermonde matrices.
  arXiv  Detail & Related papers  (2023-06-22T12:43:53Z)
• One-sided Matrix Completion from Two Observations Per Row [95.87811229292056]
  We propose a natural algorithm that involves imputing the missing values of the matrix $XTX$. We evaluate our algorithm on one-sided recovery of synthetic data and low-coverage genome sequencing.
  arXiv  Detail & Related papers  (2023-06-06T22:35:16Z)
• Non-PSD Matrix Sketching with Applications to Regression and Optimization [56.730993511802865]
  We present dimensionality reduction methods for non-PSD and square-roots" matrices. We show how these techniques can be used for multiple downstream tasks.
  arXiv  Detail & Related papers  (2021-06-16T04:07:48Z)
• Can Single-Shuffle SGD be Better than Reshuffling SGD and GD? [77.82009268160053]
  We conjecture that the means of matrix products corresponding to with- and without-replacement variants of SGD satisfy a series of spectral norm inequalities. We present theorems that support our conjecture by proving several special cases.
  arXiv  Detail & Related papers  (2021-03-12T04:34:45Z)
• What if Neural Networks had SVDs? [66.91160214071088]
  Various Neural Networks employ time-consuming matrix operations like matrix inversion. We present an algorithm that is fast enough to speed up several matrix operations.
  arXiv  Detail & Related papers  (2020-09-29T12:58:52Z)
• Sparsifying Parity-Check Matrices [60.28601275219819]
  We consider the problem of minimizing the number of one-entries in parity-check matrices. In the maximum-likelihood (ML) decoding method, the number of ones in PCMs is directly related to the time required to decode messages. We propose a simple matrix row manipulation which alters the PCM, but not the code itself.
  arXiv  Detail & Related papers  (2020-05-08T05:51:40Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.