Digital Modified-Logistic Map Based Systems for Secure Communications

Abstract

The orbit of a chaotic system is irregular, aperiodic, unpredictable, and sensitive to initial conditions. These characteristics coincide with the confusion and diffusion properties in cryptography. In recent years, chaotic systems have been studied for secure communications. However, when a chaotic system is digitalized, it results in some unexpected behaviors due to limited precision. It is known as dynamical degradation. An obvious phenomenon is that an orbit enters a cycle with unpredictable length. The orbit with short cycle length has poor quality of randomness because it can be easily analyzed from statistical point of view. In this dissertation, we focus on improving quality of randomness for digitalized logistic maps. New modified logistic map and techniques are proposed to improve the degree of complexity for secure communications and pseudo random number generation. The main achievements of this dissertation are as follows. First, we propose a Robust Logistic Map (RLM) which has a larger parameter space than classical logistic map. Moreover, there are no windows with short period-length in the parameter space. Based on RLM, a Robust Hyper-Chaotic System (RHCS) is constructed for secure-communication systems with large parameter space. Second, we propose a Variational Logistic Map (VLM) to significantly increase the throughout and quality of randomness of RLM. Moreover, a Multiple Variational Logistic Map (MVLM) is proposed for fast chaotic sequence generator. Because of the regular architecture of MVLM, it is easy to scale up the system degree to provide long output sequence with high degree of complexity and large key space for secure communications. Pseudo Random Number Generators (PRNGs) are often an important component in secure communications. In the third part of this dissertation, we propose a PRNG based on a Digitalized Modified Logistic Map (DMLM). Two techniques, constant parameter selection and output scrambling are employed to reduce the computation cost and to increase the complexity of the PRNG. Compared to previous digitalized chaotic systems based PRNGs, our DMLM-PRNG has better quality of randomness and lower hardware cost. Each of our system mentioned above has been implemented. Comparisons between our systems and previous work are conducted in terms of hardware cost and throughput. Moreover, the quality of randomness is demonstrated by statistical analysis.