Observability-singularity manifolds in the context of chaos based cryptography

Octaviana Datcu 1 Roger Tauleigne 2 A. Vlad 1 Jean-Pierre Barbot 3, 2
3 NON-A - Non-Asymptotic estimation for online systems
Inria Lille - Nord Europe, CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
Abstract : In the '80s Takens formulated the conditions that ensure the capability to reconstruct the dynamics of a transmitter when an observer receives one scalar output from the transmitter. In practical situations, the reconstruction of the original system is strongly influenced by the choice of the variable transmitted over the communication channel. This paper aims to analyze this influence in the context of mathematical singularities occurring in the evolution of the chaotic manifolds used in encryption. We analyze two systems having a chaotic behavior, a discrete recurrence, the Hitzl-Zele map, and a continuous system, the Colpitts oscillator. We show the existence of observability singularities in bothcases. The numerical experiments show that the dynamics of the discrete system falls in these singularities sets, but very infrequently. More surprisingly, the dynamics of the continuous system can not pass through the singularity, which is situated at ,infinity. But an exponential factor allows the chaotic dynamics to approach the vicinity of the singularity better than 10^{-7} and that, for about 30% of its duration. The noise inherent in analog signals are much higher than this value, the observation of the system is impossible in practice. For an effective application to data encryption, it will be helpful to increase the duration during which the dynamics remains in the vicinity of the singularity.
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Octaviana Datcu, Roger Tauleigne, A. Vlad, Jean-Pierre Barbot. Observability-singularity manifolds in the context of chaos based cryptography. International Conference on systems and Control (ICSC'13), Oct 2013, Alger, Algeria. ⟨hal-00923700⟩

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