import hashlib


def bin_key(key):
    """
    Converts a key to its binary representation by hashing it against the SHA-256 hashing algorithm.
    :param key: The key to convert.
    :return: The binary representation of the keys corresponding SHA-256 hash.
    """
    digest = hashlib.sha256()
    digest.update(bytes(key))
    hk = digest.hexdigest()
    return bin(int(hk, 16))[2:]