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

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.