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 stored at each receiver per message bit, and broadcast bandwidth
$\beta $
, i.e., the number of bits of the broadcast information sent by the transmitter per message bit. We have three main 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 and is optimal (minimum) if
$N=2$
or
$K-1$
. Third, when
$N=2$
, the optimal key storage and broadcast bandwidth tradeoff is characterized as
$\alpha +\beta \geq 3, \alpha \geq 1, \beta \geq 1$
.