Designing new chaotic and hyperchaotic systems for chaos-based cryptography

The core of chaos-based cryptography is the selection of a good chaotic system. Most of chaotic ciphers have neglected the investigation of existence multistability in the employed chaotic systems. Meanwhile, many chaotic ciphers have applied chaotic systems with complex mathematical structure and limited chaotic behavior. Therefore, this thesis focuses on designing new chaotic and hyperchaotic systems with simple mathematical structure and complex dynamics, and discusses their performance in cryptographic applications. Furthermore, this thesis investigates the effect of existence multistability in the proposed systems from a cryptographic point of view. In the beginning, this thesis presents a new 2D discrete hyperchaotic system. Dynamic characteristics of the 2D system are investigated from the following aspects: stability, trajectory, bifurcation diagram, Lyapunov exponents and sensitivity dependence. Moreover, the complexity performance of the system is evaluated by Sample Entropy algorithm. Simulation results show that the new system has a wide hyperchaotic range with high complexity and sensitivity dependence. To investigate its performance in terms of security, a new chaos-based image encryption algorithm is also proposed. In this algorithm, the essential requirements of confusion and diffusion are accomplished, and a stochastic sequence is used to enhance the security of encrypted image. Security analysis shows that the new algorithm has good security performance.This thesis further proposed an M-dimension model as a methodological framework for producing new high-dimensional discrete hyperchaotic systems. Mathematical analysis demonstrates that the generated systems by this model have either no equilibria, or an arbitrarily large number of unstable equilibria. Moreover, numerical results show that the generated systems for certain values of parameters can produce two different behaviors: 1) single hyperchaotic attractor with high complexity and sensitivity dependence; and 2) coexistence of four attractors including single limit cycle, cluster of limit cycles, single hyperchaotic attractor, and cluster of hyperchaotic attractors, which is unusual behavior in discrete systems. However, we propose a simple feedback controller to change the chaos degradation in the multistability region from limit cycle to hyperchaotic behavior. Additionally, this thesis presents a new 4D continuous chaotic system, which is derived from Lorenz-Haken equations. Dynamics analysis, including stability of symmetric equilibria and the existence of multiple Hopf bifurcations on these equilibria, are investigated, and the coexistence of two and three different attractors is numerically revealed. Moreover, a conducted research on the complexity of the new system reveals that the complexity of a system time series can locate the parameters and initial conditions that exhibit multistability behaviors. Besides that, randomness test results demonstrate that the generated pseudo-random sequences from the multistability regions fail to pass most of the statistical tests. Finally, to choose valid pseudo-random sequences from multistability regions, this thesis constructs a new algorithm based on a new 3D multi-attribute chaotic system exhibiting extreme multistability behaviors. Unlike the existing algorithms, the proposed algorithm keeps the parameters constant with varying the initial conditions that show no non-chaotic behaviors. That means, the generated sequences are either chaotic or coexistence of chaotic attractors. Randomness test results show that the generated pseudo-random sequences by the new algorithm can pass all the statistical tests.

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