  In this paper we prove lower and matching upper bounds for the number of
servers required to implement a regular shared register that tolerates
unsynchronized Mobile Byzantine failures. We consider the strongest model of
Mobile Byzantine failures to date: agents are moved arbitrarily by an
omniscient adversary from a server to another in order to deviate their
computation in an unforeseen manner. When a server is infected by an Byzantine
agent, it behaves arbitrarily until the adversary decides to move the agent to
another server. Previous approaches considered asynchronous servers with
synchronous mobile Byzantine agents (yielding impossibility results), and
synchronous servers with synchronous mobile Byzantine agents (yielding optimal
solutions for regular register implementation, even in the case where servers
and agents periods are decoupled). We consider the remaining open case of
synchronous servers with unsynchronized agents, that can move at their own
pace, and change their pace during the execution of the protocol. Most of our
findings relate to lower bounds, and characterizing the model parameters that
make the problem solvable. It turns out that unsynchronized mobile Byzantine
agent movements requires completely new proof arguments, that can be of
independent interest when studying other problems in this model. Additionally,
we propose a generic server-based algorithm that emulates a regular register in
this model, that is tight with respect to the number of mobile Byzantine agents
that can be tolerated. Our emulation spans two awareness models: servers with
and without self-diagnose mechanisms. In the first case servers are aware that
the mobile Byzantine agent has left and hence they can stop running the
protocol until they recover a correct state while in the second case, servers
are not aware of their faulty state and continue to run the protocol using an
incorrect local state.
