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 $\alpha +\beta \geq 3, \alpha \geq 1, \beta \geq 1$ .