  Applying chaos theory for secure digital communications is promising and it
is well acknowledged that in such applications the underlying chaotic systems
should be carefully chosen. However, the requirements imposed on the chaotic
systems are usually heuristic, without theoretic guarantee for the resultant
communication scheme. Among all the primitives for secure communications, it is
well-accepted that (pseudo) random numbers are most essential. Taking the
well-studied two-dimensional coupled map lattice (2D CML) as an example, this
paper performs a theoretical study towards pseudo-random number generation with
the 2D CML. In so doing, an analytical expression of the Lyapunov exponent (LE)
spectrum of the 2D CML is first derived. Using the LEs, one can configure
system parameters to ensure the 2D CML only exhibits complex dynamic behavior,
and then collect pseudo-random numbers from the system orbits. Moreover, based
on the observation that least significant bit distributes more evenly in the
(pseudo) random distribution, an extraction algorithm E is developed with the
property that, when applied to the orbits of the 2D CML, it can squeeze uniform
bits. In implementation, if fixed-point arithmetic is used in binary format
with a precision of $z$ bits after the radix point, E can ensure that the
deviation of the squeezed bits is bounded by $2^{-z}$ . Further simulation
results demonstrate that the new method not only guide the 2D CML model to
exhibit complex dynamic behavior, but also generate uniformly distributed
independent bits. In particular, the squeezed pseudo random bits can pass both
NIST 800-22 and TestU01 test suites in various settings. This study thereby
provides a theoretical basis for effectively applying the 2D CML to secure
communications.
