  In this chapter, we consider a class of discrete dynamical systems defined on
the homogeneous space associated with a regular tiling of $\R^N$, whose most
familiar example is provided by the $N-$dimensional torus $\T ^N$. It is proved
that any dynamical system in this class is chaotic in the sense of Devaney, and
that it admits at least one positive Lyapunov exponent. Next, a
chaos-synchronization mechanism is introduced and used for masking information
in a communication setup.
