Titre “Chaotic dynamics: application to secure information transmission“
Laboratory : IRCCyN
Team : Control
Web site : http://www.irccyn.ec-nantes.fr/
Head of the team : F.Plestan / Ph.Chevrel
Supervisors : Ina Taralova
Michel Malabre
Emails : ina.taralova@irccyn.ec-nantes.fr
michel.malabre@irccyn.ec-nantes.fr
Scientific field (*) : Nonlinear system analysis, observers, control theory, applied mathematics
Key words: dynamical system, chaos, bifurcations, encryption
Details for the subject
Background, Context:
Chaotic systems are defined by their extreme sensitivity to small variations in the initial conditions and parameters (known as the “butterfly effect”).
It is interesting to note that chaotic behavior can be observed in basic dynamical systems such as the recurrence equation x(k+1) = x(k)^2 +c (chaos appears for some particular values of c).
Complex (e.g. chaotic) behavior can be exhibited also by some apparently simple piece-wise linear (PWL) maps, what makes them particularly attractive for different real life applications such as Secure Communications, Information Encryption or Secure Electronic Transactions.
At the same time, the PWL maps can be considered – and analyzed - as switched systems from control theory point of view. These systems when used in chaotic regime show up as perfect candidates to generate uncorrelated (independent) output signals. Under some conditions, the latter can be considered as pseudo-random sequences, and applied as chaotic pseudo-random carriers for secure information transmission.
Challenges:
The main difficulties arise from the fact that the majority of well-known chaotic maps are not naturally suitable for encryption, and most of them don’t exhibit even satisfactory properties for encryption, mainly because of their weak chaoticity.

Here many practical problems arise, from the choice of the structure and parameters of the particular chaotic generator, and its parameters, to the best coupling which satisfies the predefined criteria for security.
Research subject, work plan:
In order to deal with these problems, after a bibliographic research, the Ph.D. work will cope first with the general understanding and study of the nonlinear behavior of coupled chaotic systems, stability analysis, completed by the study of bifurcations diagrams and the roads which lead to chaos.
Different kind of chaotic maps couplings (auto- and ring- couplings), inspired by electrical – and other - circuits, will be analyzed, in order to obtain the required features for both chaotic, and pseudo-random /thus, repetitive/ behavior.
The control theory concepts of identifiability and observability for switched systems will be analyzed in order to select and design the most suitable chaotic systems. The latter will be applied as pseudo-random number generators, which may be used in various applications; here in particular for secure information transmission.
Collaborations:
During this PhD project, an active collaboration is being set up with S. El Assad from IETR
Laboratory in Rennes (group on secure communications) and R. Lozi from the CNRS
Laboratory J. A. Dieudonné, University of Nice -Sophia Antipolis.
N.B.
Successful applicants should have good background in one of the following fields: applied mathematics, dynamical systems, signal processing, control theory, communications.
Knowledge in DSP and Matlab/Simulink shall be greatly appreciated.
References:
1. G. Alvarez, S. Li , Some Basic Cryptographic Requirements for Chaos-Based
Cryptosystems . International Journal of Bifurcation and Chaos (IJBC), Vol. 16,
(2006), pp. 2129-2151.
2. I. Taralova, Chaotic orbits prediction and atypical bifurcations in a class of piecewise linear noninvertible maps , European Physical Journal - Special Topics, Vol.
165, Issue 1, (2008), pp. 45–59
3. R. Lozi, I. Taralova , From chaos to randomness via geometric undersampling .
ESAIM: Proceedings and surveys, Vol. 46, (2014), pp. 177-195
4. M. Bonilla, M. Malabre, V. Azhmyakov . An implicit systems characterization of a class of impulsive linear switched control processes. Part 1: Modeling . Nonlinear
Analysis: Hybrid Systems, Vol. 15, (2015), pp. 157–170 Ed. Elsevier
5. M. Bonilla, M. Malabre, V. Azhmyakov , An implicit systems characterization of a class of impulsive linear switched control processes.Part2: Control. Nonlinear
Analysis: Hybrid Systems, Vol. 18, (2015), pp.15–32 Ed. Elsevier

6. O. Garasym. I. Taralova, R. Lozi , Application of observer-based chaotic synchronization and identifiability to original CSK model for secure information transmission , Indian Journal of Industrial and Applied Mathematics Vol. 6, No. 1,
(2015)
7. O. Garasym, I. Taralova, R. Lozi , "New nonlinear CPRNG based on tent and logistic maps", Complex Systems and Networks - Dynamics, Controls and
Applications , 2015, Springer (book chapter) pp. 131-161.