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<p>In 1998,&#xa0; Daemen <i> et al.</i> introduced a circulant Maximum Distance Separable (MDS) matrix in the diffusion layer of the Rijndael block cipher, drawing significant attention to circulant MDS matrices. This block cipher is now universally acclaimed as the AES block cipher. In 2016,&#xa0; Liu and Sim introduced cyclic matrices by modifying the permutation of circulant matrices and established the existence of MDS property for orthogonal left-circulant matrices, a notable subclass within cyclic matrices. While circulant matrices have been well-studied in the literature, the properties of cyclic matrices are not. Back in 1961, Friedman introduced <i>g</i>-circulant matrices which form a subclass of cyclic matrices. In this article, we first establish a permutation equivalence between a cyclic matrix and a circulant matrix. We explore properties of cyclic matrices similar to <i>g</i>-circulant matrices. Additionally, we determine the determinant of <i>g</i>-circulant matrices of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_806_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(2^d \times 2^d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mi>d</mi> </msup> <mo>×</mo> <msup> <mn>2</mn> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and prove that they cannot be simultaneously orthogonal and MDS over a finite field of characteristic 2. Furthermore, we prove that this result holds for any cyclic matrix.</p>

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A note on cyclic MDS and non-MDS matrices

  • 期刊论文

  • 发表日期：

    2025年5月27日

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  • Tapas Chatterjee,
  • Ayantika Laha

摘要

In 1998,  Daemen et al. introduced a circulant Maximum Distance Separable (MDS) matrix in the diffusion layer of the Rijndael block cipher, drawing significant attention to circulant MDS matrices. This block cipher is now universally acclaimed as the AES block cipher. In 2016,  Liu and Sim introduced cyclic matrices by modifying the permutation of circulant matrices and established the existence of MDS property for orthogonal left-circulant matrices, a notable subclass within cyclic matrices. While circulant matrices have been well-studied in the literature, the properties of cyclic matrices are not. Back in 1961, Friedman introduced g-circulant matrices which form a subclass of cyclic matrices. In this article, we first establish a permutation equivalence between a cyclic matrix and a circulant matrix. We explore properties of cyclic matrices similar to g-circulant matrices. Additionally, we determine the determinant of g-circulant matrices of order \(2^d \times 2^d\) 2 d × 2 d and prove that they cannot be simultaneously orthogonal and MDS over a finite field of characteristic 2. Furthermore, we prove that this result holds for any cyclic matrix.