Compound Secure Groupcast: Key Assignment for Selected Broadcasting

The compound secure groupcast problem is considered, where the key variables at
$K$
receivers are designed so that a transmitter can securely groupcast a message to any
$N$
out of the
$K$
receivers through a noiseless broadcast channel. The metric is the information theoretic tradeoff between key storage
$\alpha$
, i.e., the number of bits of the key variable per message bit, and broadcast bandwidth
$\beta$
, i.e., the number of bits of the broadcast information per message bit. We present two results. First, when broadcast bandwidth is minimized, i.e., when
$\beta=1$
, we show that the minimum key storage is
$\alpha=N$
. Second, when key storage is minimized, i.e., when
$\alpha=1$
, we show that broadcast bandwidth
$\beta=\min(N, K-N+1)$
is achievable.