Chaotic systems exhibit many interesting features which make them very attractive, and nonlinear dynamics has potential applications in several of the building blocs of a digital communication system (data compression, encryption, and modulation).
At the same time, the discovery that two chaotic systems can be synchronized and information transmitted from one location to another using a wideband chaotic signal generated a tremendous interest towards the dynamic behaviour of nonlinear systems in chaotic regime. Background : The information transmission in all its forms (mobile or indoor radio, images, video, data, ...) has achieved nowadays a really huge dimension, and concerns more and more large audience. It is obvious that the security of the information transmitted through the transmission channels against passive or active attacks is of utmost importance.
Since the cross correlations between segments of a chaotic waveform are lower than those between segments of periodic waveforms, chaotic modulation is supposed to offer better performances, proposing a potentially simple solutions. Moreover, there are many common properties between chaotic systems and cryptosystems, such as the sensitivity to initial conditions, sensitivity to small parameter variations, similar output distribution for any inputs, etc. The security is connected with a spread spectrum, which allows to "hide" the useful information inside the system’s noise. Here many practical problems arise, from the choice of the structure of the particular chaotic generator, and its parameters, to the deterioration of the chaotic properties due to the quantization noise.
Work plan, Research program :
In the practical implementation of these chaos-based algorithms a number of problems arise, coming from the uncertainties and/or the parameters mismatch, the stability of the process of synchronization of the chaotic systems, or the security issue : is the crypted message really impossible to decrypt ? In order to deal with these problems, after a bibliografic research, the Ph.D. work will deal first with the general understanding and study of the nonlinear behavior of the communication systems, completed by the analysis of bifurcations diagrams and the road wich leads to chaos, followed by a theoretical analysis of the identifiability of the system. More particularly, the concepts of discrete time nonlinear identifiability will be used to select the most suitable chaotic cryptosystem. Joint signal and system approach shall be used to select the most appropriate chaotic generator. First, the parameter and phase plane will be analyzed using the tools of the nonlinear dynamics. The problem to solve is to design the structure and to tune the parameters of the nonlinear generator, which is the most appropriate - in the sense that it will generate digital chaotic signals exhibiting high chaoticity (measured by the Lyapunov exponents and others), but also desirable statistical properties (autocorrelation, cross-correlation, ...). In addition, suitable parameters (encryption/decryption keys) selected from appropriate regions of the parameter plane, with easily specifiable boundaries (i.e. without fractal boudaries) will have to be designed. As application the Lozi weekly coupled maps shall be investigated in particular. Another crucial issue concerns the robustness : a realistic model can not assume an ideal transmission channel ; including at least an additive noise and (linear) filtering. Then, how robust the transmission will be in the (realistic) case of noise in the transmission channel ?
During this PhD project, an active collaboration is being set up with the laboratory IREENA in Nantes (group on secure communications) in the frame of an AtlanSTIC project, and also with Prof. Lozi from the University of Nice, Laboratoire J.A. Dieudonné UMR CNRS 7351 in the frame of a PEPS CNRS project (submitted in march 2012).