  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.
