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A note on MDS Property of Circulant Matrices

Tapas Chatterjee, Ayantika Laha

Abstract

In $2014$, Gupta and Ray proved that the circulant involutory matrices over the finite field $\mathbb{F}_{2^m}$ can not be maximum distance separable (MDS). This non-existence also extends to circulant orthogonal matrices of order $2^d \times 2^d$ over finite fields of characteristic $2$. These findings inspired many authors to generalize the circulant property for constructing lightweight MDS matrices with practical applications in mind. Recently, in $2022,$ Chatterjee and Laha initiated a study of circulant matrices by considering semi-involutory and semi-orthogonal properties. Expanding on their work, this article delves into circulant matrices possessing these characteristics over the finite field $\mathbb{F}_{2^m}.$ Notably, we establish a correlation between the trace of associated diagonal matrices and the MDS property of the matrix. We prove that this correlation holds true for even order semi-orthogonal matrices and semi-involutory matrices of all orders. Additionally, we provide examples that for circulant, semi-orthogonal matrices of odd orders over a finite field with characteristic $2$, the trace of associated diagonal matrices may possess non-zero values.

A note on MDS Property of Circulant Matrices

Abstract

In , Gupta and Ray proved that the circulant involutory matrices over the finite field can not be maximum distance separable (MDS). This non-existence also extends to circulant orthogonal matrices of order over finite fields of characteristic . These findings inspired many authors to generalize the circulant property for constructing lightweight MDS matrices with practical applications in mind. Recently, in Chatterjee and Laha initiated a study of circulant matrices by considering semi-involutory and semi-orthogonal properties. Expanding on their work, this article delves into circulant matrices possessing these characteristics over the finite field Notably, we establish a correlation between the trace of associated diagonal matrices and the MDS property of the matrix. We prove that this correlation holds true for even order semi-orthogonal matrices and semi-involutory matrices of all orders. Additionally, we provide examples that for circulant, semi-orthogonal matrices of odd orders over a finite field with characteristic , the trace of associated diagonal matrices may possess non-zero values.

Paper Structure

This paper contains 6 sections, 7 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Contribution
  3. Preliminaries
  4. Circulant matrices with MDS and semi-orthogonal properties
  5. Circulant matrices with MDS and semi-involutory properties
  6. Conclusion

Key Result

Theorem 3.7

Let $A$ be an $n \times n$ circulant matrix over a finite field. Then $A$ is semi-involutory if and only if there exist non-singular diagonal matrices $D_1,D_2$ such that $D_1^n=k_1I$ and $D_2^n=k_2I$ for non-zero scalars $k_1,k_2$ in the finite field, and $A^{-1}=D_1AD_2$.

Theorems & Definitions (22)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Proposition 4.1
  • proof
  • ...and 12 more