We consider a transmitter and $K$ receivers, each of which shares a key variable with the transmitter. Through a noiseless broadcast channel, the transmitter wishes to send a common message $W$ securely to $N$ out of the $K$ receivers while the remaining $K-N$ receivers learn no information about $W$. We are interested in the maximum message rate, i.e., the maximum number of bits of $W$ that can be securely groupcast to the legitimate receivers per key block and the minimum broadcast bandwidth, i.e., the minimum number of bits of the broadcast information required to securely groupcast the message bits. We focus on the setting of combinatorial keys, where every subset of the $K$ receivers share an independent key of arbitrary size. Under this combinatorial key setting, the maximum message rate is characterized for the following scenarios - 1) $N=1$ or $N=K-1$, i.e., secure unicast to 1 receiver with $K-1$ eavesdroppers or secure groupcast to $K-1$ receivers with $1$ eavesdropper, 2) $N=2, K=4$, i.e., secure groupcast to $2$ out of 4 receivers, and 3) the symmetric setting where the key size for any subset of the same cardinality is equal for any $N,K$. Further, for the latter two cases, the minimum broadcast bandwidth for the maximum message rate is characterized.

中文翻译：

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使用共享密钥进行安全组播

我们考虑一个发送器和 $K$ 接收器，每个接收器与发送器共享一个关键变量。通过无噪声广播信道，发射器希望安全地向 $K$ 接收器中的 $N$ 发送一条公共消息 $W$，而其余的 $KN$ 接收器不了解有关 $W$ 的信息。我们感兴趣的是最大消息速率，即每个关键块可以安全地群播给合法接收者的最大比特数 W$ 和最小广播带宽，即所需的广播信息的最小比特数安全地群播消息位。我们专注于组合密钥的设置，其中 $K$ 接收器的每个子集共享一个任意大小的独立密钥。在这种组合键设置下，最大消息速率的特征在于以下场景 - 1) $N=1$ 或 $N=K-1$，即安全单播到具有 $K-1$ 窃听者的 1 个接收者或安全组播到 $K-1$具有 $1$ 窃听者的接收器，2) $N=2，K=4$，即安全组播到 4 个接收器中的 $2$，以及 3) 对称设置，其中相同基数的任何子集的密钥大小等于任何 $N,K$。此外，对于后两种情况，表征了最大消息速率的最小广播带宽。