This thesis presents a computationally efficient cryptosystem based on chaotic continuous-interval cellular automata (CCA). This cryptosystem increases data protection as demonstrated by its flexibility to encrypt/decrypt information from distinct sources (e.g., text, sound, and images). This cryptosystem has the following enhancements over the previous chaos-based cryptosystems: (i) a mathematical model based on a new chaotic CCA strange attractor, (ii) integration of modules containing dynamical systems to generate complex sequences, (iii) generation of an unlimited number of keys due to the features of chaotic phenomena obtained through CCA, which is an improvement over previous symmetric cryptosystems, and (iv) a high-quality concealment of the cryptosystem strange attractor. Instead of using differential equations, a process of mixing chaotic sequences obtained from CCA is also introduced. As compared to other recent approaches, this mixing process provides a basis to achieve higher security by using a higher degree of complexity for the encryption/decryption processes. This cryptosystem is tested through the following three methods: (i) a stationarity test based on the invariance of the first ten statistical moments, (ii) a polyscale test based on the variance fractal dimension trajectory (VFDT) and the spectral fractal dimension (SFD), and (iii) a surrogate data test. This cryptosystem secures data from distinct sources, while leaving no patterns in the ciphertexts. This cryptosystem is robust in terms of resisting attacks that: (i) identify a chaotic system in the time domain, (ii) reconstruct the chaotic attractor by monitoring the system state variables, (iii) search the system synchronization parameters, (iv) statistical cryptanalysis, and (v) polyscale cryptanalysis.

In this thesis, we provide a summary of fully homomorphic encryption, and in particular, look at the BGV encryption scheme by Brakerski, Gentry, and Vaikuntanathan; as well the DGHV encryption scheme by van Dijk, Gentry, Halevi, and Vaikuntanathan. We explain the mechanisms developed by Gentry in his breakthrough work, and show examples of how they are used. While looking at the BGV encryption scheme, we make improvements to the underlying lemmas dealing with modulus switching and noise management, and show that the lemmas as currently stated are false. We then examine a lower bound on the hardness of the Learning With Errors lattice problem, and use this to develop specific parameters for the BGV encryption scheme at a variety of security levels. We then study the DGHV encryption scheme, and show how the somewhat homomorphic encryption scheme can be implemented as both a fully homomorphic encryption scheme with bootstrapping, as well as a leveled fully homomorphic encryption scheme using the techniques from the BGV encryption scheme. We then extend the parameters from the optimized version of this scheme to higher security levels, and describe a more straightforward way of arriving at these parameters.

Thesis (PhDElectronicEngineering)--University of South Australia, 2005.

Thesis (Ph. D.)--Oregon State University, 2008. / Printout. Includes bibliographical references (leaves 103-110). Also available on the World Wide Web.

Thesis (M.S.)--Rochester Institute of Technology, 2007. / Typescript. Includes bibliographical references (p. 64-66).

Thesis (Ph. D.)--Ohio University, November, 2003. / Title from PDF t.p. Includes bibliographical references (leaves 84-85).

Thesis (Ph.D.)--University of Wollongong, 2004. / Typescript. Includes bibliographical references: leaf 195-202.

Thesis (M. Eng. (Computer Engineering)) -- Universiteit van Pretoria, 2005. / Includes bibliographical references.

Thesis (M.C.S.) Carleton University, 2005. / Includes bibliographical references (p. 106-121). Also available in electronic format on the Internet.

Report (M.S.)--Virginia Polytechnic Institute and State University, 1994. / Abstract. Includes bibliographical references (leaf 32). Also available via the Internet.