An Algebraic State Estimation Approach for the Recovery of Chaotically Encrypted Messages

Abstract

In this article, we use a variant of a recently introduced algebraic state estimation method obtained from a fast output signal time derivatives computation process. The fast time derivatives calculations are entirely based on the consequences of using the "algebraic approach" in linear systems description (basically; module theory and non-commutative algebra). Here, we demonstrate, through computer simulations, the effectiveness of the proposed algebraic approach in the accurate and fast (i.e. non asymptotic) estimation of the chaotic states in some of the most popular chaotic systems. The proposed state estimation method can then be used in a coding-decoding process of a secret message transmission using the message-modulated chaotic system states and the reliable transmission of the chaotic system observable output. Simulation examples, using Chen's chaotic system and the Rossler system, demonstrate the important features of the proposed fast state estimation method in the accurate extraction of a chaotically encrypted messages. In our simulation results, the proposed approach is shown to be quite robust with respect to (computer generated) transmission noise perturbations. We also propose a way to evade computational singularities associated with the local lack of observability of certain chaotic system outputs and still carry out the encrypting and decoding of secret messages in a reliable manner.

Algebraic observer for a class of switched systems with zeno phenomenon

“…For the sake of clarity, we denote by ρ (y) qdt ′ the exact derivative of HK integral which is equal to ρ (y) q anḋ ρ (y) qdt the practical derivative of HK integral which is equal to ρ (y) q f . As it is highlighted hereafter numerical differentiation introduced by Fliess and Sira-Ramirez [25] [22] gives the second solution of practical differentiation.…”

“…Thus, each signal should be equipped with a pair of filter and differentiator. In order to simplify the implementation, we use algebraic observer proposed in [4], which is based on the numerical differentiation technique proposed by Fliess et al in [25] [22]. This algebraic approach has at least the following advantages:…”

Differentiator application in altitude control for an indoor blimp robot

“…The algebraic differentiator is proposed in the paper of Sira-Ramirez and Fliess (2006). Consider a signal x(t) which is assumed to be analytic on a finite-time interval.…”

Parameter identification of a permanent magnet synchronous motor

“…1) Computing time derivatives through the algebraic derivative method: In this part, some considerations with respect to the algebraic derivative method are shown; for a comprehensive treatment of this topic, the reader is referred to [12], [14], [15], [16], and references therein. From the given assumptions, the variables i a , i b , θ are smooth with some smooth time derivatives.…”