The multicast key-dissemination problem over noiseless networks, introduced by Langberg and Effros [ITW 2022], here called the ``key-cast'' problem, captures the task of disseminating a shared secret random key to a set of terminals over a given network. Unlike traditional communication, where messages must be delivered from source to destination(s) unchanged, key-cast is more flexible since key-cast need not require source reconstruction at destination nodes. For example, the distributed keys can be mixtures of sources from which the sources themselves may be unrecoverable. The work at hand considers key dissemination in the single-source, multiple-multicast network coding setting, i.e., the ``multiple key-cast'' problem. Here, distinct keys are to be simultaneously transmitted from a single source node to multiple terminal sets, one shared random key per multicast set. Scenarios include the secure setting, in which only the source and intended destinations gain information about a given key; and the non-secure setting in which the only requirement is that the knowledge of one key does not reveal information about another. In both settings, we present combinatorial conditions for key dissemination and design corresponding multiple key-cast schemes. In addition, we compare the multiple key-cast rate with and without the restriction of source reconstruction, the former corresponding to traditional forms of communication; key-cast achieves a strict advantage in rate when source reconstruction is relaxed.

翻译：朗伯格（Langberg）与埃夫罗斯（Effros）于2022年《ITW》提出的无噪声网络多播密钥分发问题（本文称为“密钥广播问题”），旨在研究通过给定网络向一组终端分发共享随机密钥的任务。与传统通信中需将消息原封不动地从源节点传输至目的节点不同，密钥广播具有更高灵活性，因其无需在目的节点重建源信息。例如，分布式密钥可以是无法恢复的源信号的混合。本研究聚焦于单源-多播网络编码场景中的密钥分发问题，即“多次密钥广播”问题。在此场景中，需同时从单一源节点向多个终端集合传输不同密钥，每个多播集对应一个共享随机密钥。研究涵盖两种场景：安全性场景中，仅源节点与指定目的节点可获取特定密钥信息；非安全性场景中，仅需确保一个密钥的知识不泄露另一密钥信息。针对两种场景，我们提出了密钥分发的组合条件并设计了相应的多次密钥广播方案。此外，我们比较了在是否施加源重建约束条件下的多次密钥广播速率——前者对应传统通信形式；当放宽源重建约束时，密钥广播在速率方面呈现显著优势。