Mathematical Treatment for Constructing a countermeasure against the one time pad attack on the Baptista Type Cryptosystem

In 1998, M.S. Baptista proposed a chaotic cryptosystem using the ergodicity property of the simple lowdimensional and chaotic logistic equation. Since then, many cryptosystems based on Baptista’s work have been proposed. However, over the years research has shown that this cryptosystem is predictable and vulnerable to attacks and is widely discussed. Among the weaknesses are the non-uniform distribution of ciphertexts and succumbing to the one-time pad attack (a type of chosen plaintext attack). In this chapter the authors give a mathematical treatment to the phenomenon such that the cryptosystem would no longer succumb to the one-time pad attack and give an example that satisfies it.

Encryption of Analog and Digital signals through Synchronized Chaotic Systems

Chaos is characterized by aperiodic, wideband, random-like, and ergodicity. Chaotic secure communication has become one of the hot topics in nonlinear dynamics since the early 1990s exploiting the technique of chaos synchronization. As distinguished by the type of information being carried, chaos-based communication systems can be categorized into analogy and digital, including four popular techniques such as Chaos Masking, Chaos Shift Keying, Chaos Modulation, and Chaos Spreading Spectrum. In this chapter, the principles of these schemes and their modifications are analyzed by theoretical analysis as well as dynamic simulation. In addition, chaos-based cryptography is a new approach to encrypt information. After analyzing the performances of chaotic sequence and designing an effective chaotic sequence generator, the authors briefly presented the principle of two classes of chaotic encryption schemes, chaotic sequence encryption and chaotic data stream encryption.

Adaptive Synchronization in Unknown Stochastic Chaotic Neural Networks with Mixed Time-Varying Delays

Neural networks (NNs) have been useful in many fields, such as pattern recognition, image processing etc. Recently, synchronization of chaotic neural networks (CNNs) has drawn increasing attention due to the high security of neural networks. In this chapter, the problem of synchronization and parameter identification for a class of chaotic neural networks with stochastic perturbation via state and output coupling, which involve both the discrete and distributed time-varying delays has been investigated. Using adaptive feedback techniques, several sufficient conditions have been derived to ensure the synchronization of stochastic chaotic neural networks. Moreover, all the connection weight matrices can be estimated while the lag synchronization and complete synchronization is achieved in mean square at the same time. The corresponding simulation results are given to show the effectiveness of the proposed method.

Synchronization of Oscillators

A simple approach is proposed in this chapter to get started on the synchronization of oscillators study. The basics are given in the beginning such that the reader can get quickly familiar with the main concepts which lead to many kinds of synchronization configurations. Chaotic synchronization is next addressed and is followed by the stability of the synchronization issue. Finally, a short introduction of the influence of noise on the synchronization process is mentioned.

Chaotic Gyros Synchronization

In this chapter, three methods for synchronizing of two chaotic gyros in the presence of uncertainties, external disturbances and dead-zone nonlinearity are studied. In the first method, there is dead-zone nonlinearity in the control input, which limits the performance of accurate control methods. The effects of this nonlinearity will be attenuated using a fuzzy parameter approximator integrated with sliding mode control method. In order to overcome the synchronization problem for a class of unknown nonlinear chaotic gyros a robust adaptive fuzzy sliding mode control scheme is proposed in the second method. In the last method, two different gyro systems have been considered and a fuzzy controller is proposed to eliminate chattering phenomena during the reaching phase of sliding mode control. Simulation results are also provided to illustrate the effectiveness of the proposed methods.

Simple Chaotic Electronic Circuits

Simple chaotic electronics circuits as diode resonator circuits, Resistor-Inductor-LED optoelectronic chaotic circuits, and Single Transistor chaotic circuits can be used as transmitters and receivers for chaotic cryptosystems. In these circuits we can change and investigate the influence of various circuit parameters to the complexity of the so generated strange attractors. Time series analysis is performed following Grassberger and Procaccia’s method while invariant parameters as correlation, and minimum embedding dimension are respectively calculated. The Kolmogorov entropy is also calculated and the RLT circuits in a critical state are examined.

Chaotic Dynamical Systems Associated with Tilings of RN

In this chapter, we consider a class of discrete dynamical systems defined on the homogeneous space associated with a regular tiling of RN, whose most familiar example is provided by the N-dimensional torus TN. It is proved that any dynamical system in this class is chaotic in the sense of Devaney, and that it admits at least one positive Lyapunov exponent. Next, a chaos-synchronization mechanism is introduced and used for masking information in a communication setup.

Unmasking Optical Chaotic Cryptosystems Based on Delayed Optoelectronic Feedback

The authors analyze the security of optical chaotic communication systems. The chaotic carrier is generated by a laser diode subject to delayed optoelectronic feedback. Transmitters with one and two fixed delay times are considered. A new type of neural networks, modular neural networks, is used to reconstruct the nonlinear dynamics of the transmitter from experimental time series in the single-delay case, and from numerical simulations in single and two-delay cases. The authors show that the complexity of the model does not increase when the delay time is increased, in spite of the very high dimension of the chaotic attractor. However, it is found that nonlinear dynamics reconstruction is more difficult when the feedback strength is increased. The extracted model is used as an unauthorized receiver to recover the message. Therefore, the authors conclude that optical chaotic cryptosystems based on optoelectronic feedback systems with several fixed time delays are vulnerable.

Digital Information Transmission using Discrete Chaotic Signal

In this work the authors thoroughly investigated a digital information transmission system using discrete chaotic signal over cable. As an example in their work the authors consider the non-autonomous 2nd order non-linear oscillator system presented in Tamaševicious, Cenys, Mycolaitis, and Namajunas (1998) which is particularly suitable for digital communications and present the experimental results regarding synchronization. The effect of noise (internal or external) on the synchronization of the drive-response system (unidirectional coupling between two identical systems) is analyzed and since in every practical implementation of a communication system, the transmitter and receiver circuits (although identical) operate under slightly different conditions the case of the mismatch between the parameters of the transmitter and the receiver is considered. Moreover, there is a study of the robustness of the system with reference to the desired security, proposing a more sophisticated approach, which combines the simplicity in the implementation of a chaotic system with an enhanced encoding scheme that will overall increase security.

Type-2 Fuzzy Sliding Mode Synchronization

This chapter presents an adaptive interval type-2 fuzzy neural network (FNN) controller to synchronize chaotic systems with training data corrupted by noise or rule uncertainties involving external disturbances. Adaptive interval type-2 FNN control scheme and sliding mode approach are incorporated to deal with the synchronization of non-identical chaotic systems. In the meantime, based on the adaptive fuzzy sliding mode control, the Laypunov stability theorem has been used to testify the asymptotic stability of the chaotic systems. The chattering phenomena in the control efforts can be reduced and the stability analysis of the proposed control scheme will be guaranteed in the sense that all the states and signals are uniformly bounded and the external disturbance on the synchronization error can be attenuated. The simulation example is included to confirm validity and performance of the advocated design methodology.