  At PODC 2014, A. Most\'efaoui, H. Moumen, and M. Raynal presented a new and
simple randomized signature-free binary consensus algorithm (denoted here MMR)
that copes with the net effect of asynchrony Byzantine behaviors. Assuming
message scheduling is fair and independent from random numbers MMR is optimal
in several respects: it deals with up to t Byzantine processes where t < n/3
and n is the number of processes, O(n\^2) messages and O(1) expected time. The
present article presents a non-trivial extension of MMR to an even more
fault-prone context, namely, in addition to Byzantine processes, it considers
also that the system can experience transient failures. To this end it
considers self-stabilization techniques to cope with communication failures and
arbitrary transient faults (such faults represent any violation of the
assumptions according to which the system was designed to operate).
  The proposed algorithm is the first loosely-self-stabilizing Byzantine
fault-tolerant binary consensus algorithm suited to asynchronous
message-passing systems. This is achieved via an instructive transformation of
MMR to a self-stabilizing solution that can violate safety requirements with
probability Pr= O(1/(2\^M)), where M is a predefined constant that can be set
to any positive integer at the cost of 3 M n + log M bits of local memory. In
addition to making MMR resilient to transient faults, the obtained
self-stabilizing algorithm preserves its properties of optimal resilience and
termination, (i.e., t < n/3, and O(1) expected time). Furthermore, it only
requires a bounded amount of memory.
