  In identity-based encryption (IBE) systems, an efficient key delegation
method to manage a large number of users and an efficient key revocation method
to handle the dynamic credentials of users are needed. Revocable hierarchical
IBE (RHIBE) can provide these two methods by organizing the identities of users
as a hierarchy and broadcasting an update key for non-revoked users per each
time period. To provide the key revocation functionality, previous RHIBE
schemes use a tree-based revocation scheme. However, this approach has an
inherent limitation such that the number of update key elements depends on the
number of revoked users.
  In this paper, we propose two new RHIBE schemes in multilinear maps that use
the public-key broadcast encryption scheme instead of using the tree-based
revocation scheme to overcome the mentioned limitation. In our first RHIBE
scheme, the number of private key elements and update key elements is reduced
to $O(\ell)$ and $O(\ell)$ respectively where $\ell$ is the depth of a
hierarchical identity. In our second RHIBE scheme, we can further reduce the
number of private key elements from $O(\ell)$ to $O(1)$.
