  The synchronization process of two mutually delayed coupled deterministic
chaotic maps is demonstrated both analytically and numerically. The
synchronization is preserved when the mutually transmitted signal is concealed
by two commutative private filters that are placed on each end of the
communication channel. We demonstrate that when the transmitted signal is a
convolution of the truncated time delayed output signals or some powers of the
delayed output signals synchronization is still maintained. The task of a
passive attacker is mapped onto Hilbert's tenth problem, solving a set of
nonlinear Diophantine equations, which was proven to be in the class of
NP-Complete problems. This bridge between two different disciplines,
synchronization in nonlinear dynamical processes and the realm of the NPC
problems, opens a horizon for a new type of secure public-channel protocols.
