CONSTRUCTION OF ALL MDS AND INVOLUTORY MDS MATRICES

Authors

• Yogesh Kumar, DRDO Scientific Analysis Group
• Prasanna Raghaw Mishra, DRDO Scientific Analysis Group
• Susanta Samanta, Indian Statistical Institute, Kolkata
• Kishan Chand Gupta, Indian Statistical Institute, Kolkata
• Atul Gaur, University of Delhi

Article Type

Research Article

Publication Title

Advances in Mathematics of Communications

Abstract

In this paper, we propose two algorithms for a hybrid construction of all n × n MDS and involutory MDS matrices over a finite field Fpm, respectively. The proposed algorithms effectively narrow down the search space to identify (n − 1) × (n − 1) MDS matrices, facilitating the generation of all n × n MDS and involutory MDS matrices over Fpm. To the best of our knowledge, existing literature lacks methods for generating all n × n MDS and involutory MDS matrices over Fpm. In our approach, we introduce a representative matrix form for generating all n × n MDS and involutory MDS matrices over Fpm. The determination of these representative MDS matrices involves searching through all (n − 1) × (n − 1) MDS matrices over Fpm. Our contributions extend to proving that the count of all 3 × 3 MDS matrices over F2m is precisely (2^m −1)⁵ (2^m −2)(2^m −3)(2^2m −9 ·2^m +21). Furthermore, we explicitly provide the count of all 4 × 4 MDS and involutory MDS matrices over F2m for m = 2, 3, 4.

First Page

922

Last Page

941

DOI

10.3934/amc.2024033

Publication Date

6-1-2025

Recommended Citation

Kumar, Yogesh; Mishra, Prasanna Raghaw; Samanta, Susanta; Gupta, Kishan Chand; and Gaur, Atul, "CONSTRUCTION OF ALL MDS AND INVOLUTORY MDS MATRICES" (2025). Journal Articles. 5288.
https://digitalcommons.isical.ac.in/journal-articles/5288