  In multi-source multi-terminal key-dissemination, here called ``key-cast,''
introduced by the authors in [ITW2022], network nodes hold independent random
bits, and one seeks a communication scheme that allows all terminal nodes to
share a secret key K. The work at hand addresses positive (albeit, arbitrarily
small) rate key-cast under the security requirement that no single non-terminal
network node can gain information about the shared key K; this scenario is
useful in cryptographic settings. Specifically, key-dissemination protocols
based on secure multicast network coding are designed. The analysis presented
yields two combinatorial characterizations. In each, we assume a network in
which an eavesdropper may access any individual network node. The first
characterization captures all networks that support positive-rate secure
multicast; computing the secure-multicast capacity in the setting studied is a
known open problem. The second characterizes all networks that support
positive-rate secure key-cast.
