  We introduce a family of QKD protocols for distributing shared random keys
within a network of $n$ users. The advantage of these protocols is that any
possible key structure needed within the network, including broadcast keys
shared among subsets of users, can be implemented by using a particular
multi-partite high-dimensional quantum state. This approach is more efficient
in the number of quantum channel uses than conventional quantum key
distribution using bipartite links. Additionally, multi-partite
high-dimensional quantum states are becoming readily available in quantum
photonic labs, making the proposed protocols implementable using current
technology.
