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

TL;DR

This paper proves a conjecture about the nullity of certain matrices over rings of characteristic 2, confirming an optimal bound relevant to cryptographic cipher analysis and exploring algebraic structures of Hadamard matrices.

Contribution

It establishes the conjecture on matrix nullity over rings of characteristic 2 and analyzes the algebraic structure of Hadamard matrices in this context.

Findings

Confirmed the conjectural bound on invariant subspace dimension.

Revealed algebraic structures of Hadamard matrices over rings.

Established relations between block-Hadamard and Hadamard-block matrices.

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.