  Consensus is arguably one of the most important notions in distributed
computing. Among asynchronous, randomized, and signature-free implementations,
the protocols of Most\'efaoui et al. (PODC 2014 and JACM 2015) represent a
landmark result, which has been extended later and taken up in practical
systems. The protocols achieve optimal resilience and takes, in expectation,
only a constant expected number of rounds of quadratic message complexity.
Randomization is provided through a common-coin primitive. In traditional
consensus protocols, all involved processes adhere to a global, symmetric
failure model, typically only defined by bounds on the number of faulty
processes. Motivated by applications to blockchains, however, more flexible
trust assumptions have recently been considered. In particular, with asymmetric
trust, a process is free to choose which other processes it trusts and which
ones might collude against it. This paper revisits the optimal asynchronous
protocol of Most\'efaoui et al. and shows how to realize it with asymmetric
trust. The paper starts by pointing out in detail why some versions of this
protocol may violate liveness. Then it proposes a fix for the protocol that
does not affect its properties, but lets it regain the simplicity of its
original version (PODC 2014). At the same time, the paper shows how to realize
randomized signature-free asynchronous Byzantine consensus with asymmetric
quorums. This results in an optimal consensus protocol with subjective,
asymmetric trust and constant expected running time. It is suitable for
applications to blockchains, for instance.
