${ \kappa }${\ell$}-cast: On the Foundations of Byzantine Reliable Broadcast in the Presence of Message Adversaries

This article is on reliable broadcast in asynchronous signature-free $n$-process message-passing systems in which up to \tb processes are Byzantine and where, at the network level, for each message broadcast by a correct process, an adversary can prevent up to \tm processes from receiving it (the integer \tm defines the power of the message adversary). So, differently from previous works, this article considers that not only computing entities can be faulty (Byzantine processes), but also that the network can lose messages. To this end the article first introduces a new basic communication abstraction denoted \kl-cast and studies its properties in the previous bi-dimensional adversary context. Then, the article deconstructs existing Byzantine-tolerant asynchronous broadcast algorithms and, with the help of the \kl-cast communication abstraction, reconstructs versions of them that tolerate both Byzantine processes and message adversaries. Interestingly, some of these reconstructed algorithms are also more efficient than the Byzantine-tolerant-only algorithms from which they originate. The article also shows that the condition $n>3\tb+2\tm$ is necessary and sufficient (with signatures) to design such reliable broadcast algorithms.