## Cryptology ePrint Archive: Report 2021/1689

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

Baofeng Wu

Abstract: In this note, 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. We also give characterizations of the algebraic structure formed by Hadamard matrices over commutative rings.

Category / Keywords: secret-key cryptography / Hadamard matrix, block matrix, Characteristic polynomial, Cayley–Hamilton theorem