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Towards a conjecture on a special class of matrices over commutative rings of characteristic 2

Baofeng Wu

TL;DR

The paper addresses proving the Keller–Rosemarin conjecture on the nilpotent behavior of a matrix polynomial arising from a block matrix with Hadamard-type blocks over commutative rings of characteristic $2$, motivated by invariant-subspace attacks on Starkad/HADES. It presents a simple, algebraic proof that $q(M)^2=0$ where $q$ is the characteristic polynomial of $M''$, by analyzing the Hadamard algebra $\mathcal{H}_k(R)$ and its augmentation via a group-algebra isomorphism $ \mathcal{H}_k(R)\cong R[G]$. The authors also relate block-Hadamard matrices to Hadamard-block matrices, deriving corollaries such as an improved invariant-subspace lower bound $\dim U \ge t-2s$, and discuss structural insights and potential generalizations to other Abelian groups. These results deepen the theory of Hadamard matrices over rings and have potential implications for diffusion-layer designs in arithmetization-oriented symmetric ciphers.

Abstract

In this paper, we prove the conjecture posed by Keller and Rosemarin at Eurocrypt 2021 on the nullity of a matrix polynomial of a block matrix with Hadamard type blocks over commutative rings of characteristic 2. Therefore, it confirms the conjectural optimal bound on the dimension of invariant subspace of the Starkad cipher using the HADES design strategy. Moreover, we reveal the algebraic structure formed by Hadamard matrices over commutative rings from the perspectives of group algebra and polynomial algebra. An interesting relation between block-Hadamard matrices and Hadamard-block matrices is obtained as well.

Towards a conjecture on a special class of matrices over commutative rings of characteristic 2

TL;DR

The paper addresses proving the Keller–Rosemarin conjecture on the nilpotent behavior of a matrix polynomial arising from a block matrix with Hadamard-type blocks over commutative rings of characteristic , motivated by invariant-subspace attacks on Starkad/HADES. It presents a simple, algebraic proof that where is the characteristic polynomial of , by analyzing the Hadamard algebra and its augmentation via a group-algebra isomorphism . The authors also relate block-Hadamard matrices to Hadamard-block matrices, deriving corollaries such as an improved invariant-subspace lower bound , and discuss structural insights and potential generalizations to other Abelian groups. These results deepen the theory of Hadamard matrices over rings and have potential implications for diffusion-layer designs in arithmetization-oriented symmetric ciphers.

Abstract

In this paper, we prove the conjecture posed by Keller and Rosemarin at Eurocrypt 2021 on the nullity of a matrix polynomial of a block matrix with Hadamard type blocks over commutative rings of characteristic 2. Therefore, it confirms the conjectural optimal bound on the dimension of invariant subspace of the Starkad cipher using the HADES design strategy. Moreover, we reveal the algebraic structure formed by Hadamard matrices over commutative rings from the perspectives of group algebra and polynomial algebra. An interesting relation between block-Hadamard matrices and Hadamard-block matrices is obtained as well.
Paper Structure (10 sections, 6 theorems, 26 equations)

This paper contains 10 sections, 6 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Backgrounds on Conjecture \ref{['conj1']}
  3. Our work
  4. Outline
  5. Known properties of Hadamard matrices
  6. Proof of Conjecture \ref{['conj1']}
  7. Further discussions on Conjecture \ref{['conj1']}
  8. Structure of the algebra $\mathcal{H}_k(R)$
  9. Block-Hadamard matrices and Hadamard-block matrices
  10. Conclusion and further work

Key Result

Proposition 2.1

Let $R$ be a commutative ring of characteristic 2 and $H, H_1, H_2 \in\mathcal{H}_k(R)$ where $k\in\mathbb{N}$. Let $\hbox{\rm{det}}(\cdot)$ and $\lambda(\cdot)$ denote the determinant and an eigenvalue of any matrix over a commutative ring. Then we have

Theorems & Definitions (7)

  • Conjecture 1: See KR21
  • Proposition 2.1
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Theorem 6.1
  • Corollary 6.2