  In extending fast digital clock synchronization to the bounded-delay model,
the expected constant time Byzantine pulse resynchronization problem is
investigated. In this problem, the synchronized state of the system should not
only be deterministically maintained but be reached from arbitrary states with
expected constant time in the presence of Byzantine faults. An intuitive
geometric representation of the problem is introduced, with which the classical
approximate agreement, randomized Byzantine agreement, and random walk are
integrated with some geometric operations. Efficient realizations are also
provided for practical uses. Compared with the state-of-the-art solutions, the
assumed common pulses need not be regularly generated, the message complexity
can be lowered as approximate agreement, and the expected stabilization time is
optimal. With this, the provided solution can efficiently convert irregularly
generated common pulses to self-stabilizing Byzantine pulse synchronization.
