Exploiting the chaotic dynamics: novel approaches for cryptosystems, digital systems, and Lyapunov exponent estimation

Abstract

After the groundbreaking contributions of Henri Poincaré and Edward Lorenz to the field of chaos, chaotic systems have received significant attention in the scientific community. These systems exhibit characteristics like sensitivity to initial conditions and non-periodic behavior. Researchers have used these features in diverse digital applications, from pseudo-random number generation to metaheuristic optimization algorithms. However, implementing and simulating chaotic systems in the digital realm is a complex task, necessitating careful consideration of many issues, such as number representation, finite precision, and dynamical degradation of their chaotic properties. This thesis shows that the orbits of chaotic systems can vary across distinct hardware and software setups, negatively impacting applications like chaos-based encryption schemes. This divergence extends to mathematically equivalent chaotic system models when the order to perform their arithmetic operations differs. Furthermore, while substantial efforts have been devoted to digitally representing chaotic systems and mitigating dynamical degradation, limited attention has been given to developing efficient digital circuits for these systems. Thus, this thesis uses these insights to exploit the chaotic dynamics through three distinct approaches. First, a novel chaotic image encryption algorithm based on finite fields is introduced. By adapting the logistic map using Galois field to generate pseudo-random numbers, this approach ensures that encryption results are consistent across different digital devices and guarantees security against cyberattacks. Second, a new method to estimate the positive largest Lyapunov exponent of chaotic systems using SPICE-like programs is proposed. This approach simplifies the estimation by directly analyzing systems represented in circuit diagrams, inspired by interval extensions and pseudo-orbits of chaotic systems. Thus, users with basic SPICE-like program skills can readily employ this method. Third, utilizing FPGA as a digital platform and using 1's complement fixed-point format, a hardware-efficient solution for implementing the tent map and also a hardware perturbation method to reduce its dynamical degradation are presented. The resulting digital circuit maintains chaotic properties and serves as a pseudo-random number generator for particle swarm optimization. In summary, this thesis not only explores chaotic systems in digital environments but also offers practical solutions and insights with implications for diverse applications.