  In the secure groupcast problem, a transmitter wants to securely groupcast a
message with the maximum rate to the first $N$ of $K$ receivers by broadcasting
with the minimum bandwidth, where the $K$ receivers are each equipped with a
key variable from a known joint distribution. Examples are provided to prove
that different instances of secure groupcast that have the same entropic
structure, i.e., the same entropy for all subsets of the key variables, can
have different maximum groupcast rates and different minimum broadcast
bandwidth. Thus, extra-entropic structure matters for secure groupcast. Next,
the maximum groupcast rate is explored when the key variables are generic
linear combinations of a basis set of independent key symbols, i.e., the keys
lie in generic subspaces. The maximum groupcast rate is characterized when the
dimension of each key subspace is either small or large, i.e., the extreme
regimes. For the intermediate regime, various interference alignment schemes
originated from wireless interference networks, such as eigenvector based and
asymptotic schemes, are shown to be useful.
