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On MDS Property of g-Circulant Matrices

Tapas Chatterjee, Ayantika Laha

TL;DR

The paper investigates $g$-circulant matrices over finite fields as potential MDS diffusion-layer components in cryptographic primitives. It derives the structure of $A^2$ for a $g$-circulant matrix and shows that involutory MDS $g$-circulant matrices can exist only when $g^2 \equiv 1 \pmod k$, with the feasible nontrivial case being $g \equiv -1 \pmod k$, while ruling out others. It extends circulant results to semi-involutory and semi-orthogonal $g$-circulant matrices, proving that the associated diagonal matrices raised to the $k$-th power are scalar, and provides explicit small-field constructions illustrating the theory. Overall, the work generalizes known circulant results to the $g$-circulant setting, offering insights for efficient diffusion-layer implementations and inverse computations in cryptographic schemes.

Abstract

Circulant Maximum Distance Separable (MDS) matrices have gained significant importance due to their applications in the diffusion layer of the AES block cipher. In $2013$, Gupta and Ray established that circulant involutory matrices of order greater than $3$ cannot be MDS. This finding prompted a generalization of circulant matrices and the involutory property of matrices by various authors. In $2016$, Liu and Sim introduced cyclic matrices by changing the permutation of circulant matrices. In $1961,$ Friedman introduced $g$-circulant matrices which form a subclass of cyclic matrices. In this article, we first discuss $g$-circulant matrices with involutory and MDS properties. We prove that $g$-circulant involutory matrices of order $k \times k$ cannot be MDS unless $g \equiv -1 \pmod k.$ Next, we delve into $g$-circulant semi-involutory and semi-orthogonal matrices with entries from finite fields. We establish that the $k$-th power of the associated diagonal matrices of a $g$-circulant semi-orthogonal (semi-involutory) matrix of order $k \times k$ results in a scalar matrix. These findings can be viewed as an extension of the results concerning circulant matrices established by Chatterjee {\it{et al.}} in $2022.$

On MDS Property of g-Circulant Matrices

TL;DR

The paper investigates -circulant matrices over finite fields as potential MDS diffusion-layer components in cryptographic primitives. It derives the structure of for a -circulant matrix and shows that involutory MDS -circulant matrices can exist only when , with the feasible nontrivial case being , while ruling out others. It extends circulant results to semi-involutory and semi-orthogonal -circulant matrices, proving that the associated diagonal matrices raised to the -th power are scalar, and provides explicit small-field constructions illustrating the theory. Overall, the work generalizes known circulant results to the -circulant setting, offering insights for efficient diffusion-layer implementations and inverse computations in cryptographic schemes.

Abstract

Circulant Maximum Distance Separable (MDS) matrices have gained significant importance due to their applications in the diffusion layer of the AES block cipher. In , Gupta and Ray established that circulant involutory matrices of order greater than cannot be MDS. This finding prompted a generalization of circulant matrices and the involutory property of matrices by various authors. In , Liu and Sim introduced cyclic matrices by changing the permutation of circulant matrices. In Friedman introduced -circulant matrices which form a subclass of cyclic matrices. In this article, we first discuss -circulant matrices with involutory and MDS properties. We prove that -circulant involutory matrices of order cannot be MDS unless Next, we delve into -circulant semi-involutory and semi-orthogonal matrices with entries from finite fields. We establish that the -th power of the associated diagonal matrices of a -circulant semi-orthogonal (semi-involutory) matrix of order results in a scalar matrix. These findings can be viewed as an extension of the results concerning circulant matrices established by Chatterjee {\it{et al.}} in
Paper Structure (6 sections, 21 theorems, 30 equations)

This paper contains 6 sections, 21 theorems, 30 equations.

Table of Contents

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

Key Result

Lemma 3.5

If $A$ is semi-orthogonal and $P$ is a permutation matrix, then both $PA$ and $AP$ are semi-orthogonal.

Theorems & Definitions (47)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • Definition 3.6
  • Lemma 3.7
  • Definition 3.8
  • Theorem 3.9
  • Theorem 3.10
  • ...and 37 more