In chapter 4 of his exciting book
Anticipatory Systems, R. Rosen elaborates in considerable detail on the quality of time.
It admits a multitude of different kinds of encoding, which differ vastly from one
another. We can consider reversible Hamiltonian time, irreversible dynamical time,
thermodynamic time, probabilistic time and sequential or logic time. Each of these capture
some particular aspects of our time sense, at least as these aspects are manifested in
particular kinds of situations. The conclusion drawn by R.~Rosen is that, while certain
formal relations could be established between these various kinds of time, none of them
could be reduced to any of the others.
This contribution takes a somewhat different standpoint. Using the example of Kolmogorov
systems which are some of the most unstable systems currently known in chaos theory,
fulfilling special conditions like ergodicity and the mixing property, these reversible
conservative systems exhibit additional important properties which allow for the
construction of equivalent irreversible dissipative systems by means of a necessarily
non-unitary but invertible transformation. This transformation is derived as an operator
based on the notion of internal time of the system under consideration which relates
entropy to age and essentially implements a convolution operation in an adequately chosen
set of basis functions.
As an application we mention the field of
cryptography. It will be shown how Kolmogorov systems can be utilized to implement highly
efficient computationally secure ciphers for bulk encryption applications. The highly
unstable dynamics associated with Kolmogorov systems is thereby taken as a chaotic
nonlinear permutation operator, while substitution is implemented using an adaption of a
standard shift register based pseudo random number generator. In this way we obtain secure
product ciphers which are firmly grounded on systems theoretic concepts, offering many
features that should make them superior to contemporary bulk encryption systems,
especially in applications where an appropriate combination of efficiency and security
plays a major role.