Cycle Lengths and Correlation Properties of Finite Precision Chaotic Maps

This paper investigates cycle and transient lengths of spatially discretized chaotic maps with respect to different initial condition values. By investigating cycle lengths and their correlation properties using fixed point arithmetic, it is observed that for different initial condition values only a limited number of cycles occur for each resolution and rounding type. Additionally, it is seen that cycles with the same length are actually identical due to intersections between pseudo trajectories. The drawbacks of this situation are demonstrated for chaos based cryptosystems that use initial conditions as key. It is shown that highly correlated outputs for different initial conditions cause serious security vulnerabilities, although the statistical tests indicate otherwise. Finally, correlations in the perturbed systems are discussed.