Dynamical characteristics of discretized chaotic permutations

Chaos theory has been applied to various fields where appropriate random sequences are required. The randomness of chaotic sequences is characteristic of continuous-state systems. Accordingly, the discrepancy between the characteristics of spatially discretized chaotic dynamics and those of original analog dynamics must be bridged to justify applications of digital orbits generated, for example, from digital computers simulating continuous-state chaos. The present paper deals with the chaotic permutations appearing in a chaotic cryptosystem. By analysis of cycle statistics, the convergence of the invariant measure and periodic orbit skeletonization, we show that the orbits in chaotic permutations are ergodic and chaotic enough for applications. In the consequence, the systematic differences in the invariant measures and in the Lyapunov exponents of two infinitesimally L^∞-close maps are also investigated.