This paper presents a novel leaderless protocol (FPC-BI: Fast Probabilistic Consensus within Byzantine Infrastructures) with a low communicational complexity and which allows a set of nodes to come to a consensus on a value of a single bit. The paper makes the assumption that part of the nodes are Byzantine, and are thus controlled by an adversary who intends to either delay the consensus, or break it (this defines that at least a couple of honest nodes come to different conclusions). We prove that, nevertheless, the protocol works with high probability when its parameters are suitably chosen. Along this the paper also provides explicit estimates on the probability that the protocol finalizes in the consensus state in a given time. This protocol could be applied to reaching consensus in decentralized cryptocurrency systems. A special feature of it is that it makes use of a sequence of random numbers which are either provided by a trusted source or generated by the nodes themselves using some decentralized random number generating protocol. This increases the overall trustworthiness of the infrastructure. A core contribution of the paper is that it uses a very weak consensus to obtain a strong consensus on the value of a bit, and which can relate to the validity of a transaction.
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