Assessing the security of subsampling process using modified EKF and nonlinear least squares methods

Since the theory of chaos was introduced in cryptography, the use of chaotic dynamical systems to secure communications has been widely investigated, particularly to generate chaotic pseudorandom numbers as cipher-keys. The emergent property of the ultra-weak multidimensional coupling of p one-dimensional dynamical systems lead to randomness preserving chaotic properties of continuous models in numerical simulations. This paper focuses on such families called multiparameter chaotic pseudo random number generators (M-p CPRNG) and proposes algorithm approach to test the robustness of time series generated by M-p CPRNG. First, a single one-dimensional chaotic map to construct a regular chaotic subsampling is considered. Parameters on which depends the map are estimated using only the sequences generated by this map to cipher a message. A previous study [1] using the Extended Kalman Filter (EKF) has shown that a necessary minimum shift value corresponding to a particular subsampling of a chaotic cubic map is obtained from which it is not possible to estimate the parameters. In this paper, new cipher breaking methods are considered for the same purpose: assessing the security of the time series. These methods are investigated in the same way than EKF one and compared to the results provided by EKF. The EKF was first improved by introducing a modified Gram-Schmidt method and the nonlinear least squares method was also tested. The one-dimensional cubic map was again considered and a new parameter leading to EKF oscillations is especially studied.