  This work describes two randomized, asynchronous, round based, Binary
Byzantine faulty tolerant consensus algorithms based on the algorithms of [25]
and [26]. Like the algorithms of [25] and [26] they do not use signatures, use
$O(n^2)$ messages per round (where each message is composed of a round number
and a constant number of bits), tolerate up to one third failures, and have
expected termination in constant number of rounds.
  The first, like [26], uses a weak common coin (i.e. one that can return
different values at different processes with a constant probability) to ensure
termination. The algorithm consists of $5$ to $7$ message broadcasts per round.
An optimization is described that reduces this to $4$ to $5$ broadcasts per
round for rounds following the first round. Comparatively, [26] consists of $8$
to $12$ message broadcasts per round.
  The second algorithm, like [25], uses a perfect common coin (i.e. one that
returns the same value at all non-faulty processes) for both termination and
correctness. Unlike [25], it does not require a fair scheduler to ensure
termination. Furthermore, the algorithm consists of $2$ to $3$ message
broadcasts for the first round and $1$ to $2$ broadcasts for the following
rounds, while [29] consists of $2$ to $3$ broadcasts per round.
