  We propose a framework for joint entropy coding and encryption using Chaotic
maps. We begin by observing that the message symbols can be treated as the
symbolic sequence of a discrete dynamical system. For an appropriate choice of
the dynamical system, we could back-iterate and encode the message as the
initial condition of the dynamical system. We show that such an encoding
achieves Shannon's entropy and hence optimal for compression. It turns out that
the appropriate discrete dynamical system to achieve optimality is the
piecewise-linear Generalized Luroth Series (GLS) and further that such an
entropy coding technique is exactly equivalent to the popular Arithmetic Coding
algorithm. GLS is a generalization of Arithmetic Coding with different modes of
operation.
  GLS preserves the Lebesgue measure and is ergodic. We show that these
properties of GLS enable a framework for joint compression and encryption and
thus give a justification of the recent work of Grangetto et al. and Wen et al.
Both these methods have the obvious disadvantage of the key length being equal
to the message length for strong security. We derive measure preserving
piece-wise non-linear GLS (nGLS) and their skewed cousins, which exhibit Robust
Chaos. We propose a joint entropy coding and encryption framework using
skewed-nGLS and demonstrate Shannon's desired sensitivity to the key parameter.
Potentially, our method could improve the security and key efficiency over
Grangetto's method while still maintaining the total compression ratio. This is
a new area of research with promising applications in communications.
