Unconditionally Reliable Message Transmission in Directed Hypergraphs

We study the problem of unconditionally reliable message transmission (URMT), where two non-faulty players, the sender S and the receiver R are part of a synchronous network modeled as a directedhypergraph, a part of which may be under the influence of an adversary having unbounded computingpower. S intends to transmit a message m to R, such that R should correctly obtain S’s message withprobability at least (1 − δ) for arbitrarily small δ > 0. However, unlike most of the literature on thisproblem, we assume the adversary modeling the faults is threshold mixed, and can corrupt differentset of nodes in Byzantine, passive and fail-stop fashion simultaneously. The main contribution of thiswork is the complete characterization of URMT in directed hypergraph tolerating such an adversary.Working out a direct characterization of URMT over directed hypergraphs tolerating threshold mixedadversary is highly un-intuitive. So we first propose a novel technique, which takes as input a di-rected hypergraph and a threshold mixed adversary on that hypergraph and outputs a correspondingdigraph, along with a non-threshold mixed adversary, such that URMT over the hypergraph tolerat-ing the threshold mixed adversary is possible iff a special type of URMT is possible over the obtaineddigraph, tolerating the corresponding non-threshold mixed adversary. We then characterize URMT inarbitrary digraphs tolerating non-threshold mixed adversary and modify it to obtain the characteriza-tion for special type of URMT over digraphs tolerating non-threshold mixed adversary. This completesthe characterization of URMT over the original hypergraph. Surprisingly, our results indicate thateven passive corruption, in collusion with active faults, substantially affects the reliability of URMTprotocols! This is interesting because it is a general belief that passive corruption (eavesdropping)does not affect reliable communication.