Symbolic dynamics approach to parameter estimation without initial value

Symbolic dynamics, which partitions the infinite number of finite length trajectories into a finite number of trajectory sets, allows a simplified and 'coarse-grained' description of the dynamics of a system with a limited number of symbols. In this Letter, we will show that control parameters affect dynamical characters of symbolic sequences. To be more specific, we will analyze how control parameters affect statistical property of Skewed Tent map symbolic sequences. Besides, we will also analyze how control parameters affect ergodic property of both Logistic map and Tent map symbolic sequences. Both theoretical and experimental results show that the above mentioned effects of control parameters discourage the use of chaotic symbolic sequences in cryptography. Furthermore, we will propose an improved scheme utilizing asymptotic deterministic randomness to avoid the undesirable effects.