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by Wai-Chiu Cheung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 117-119). / Introduction --- p.1 / Chapter Chapter 1 --- Topological Riesz Spaces --- p.5 / Chapter 1.1 --- Locally convex spaces --- p.5 / Chapter 1.2 --- Ordered vector spaces and Riesz spaces --- p.10 / Chapter 1.3 --- Locally convex Riesz spaces --- p.16 / Chapter 1.4 --- Banach lattices --- p.23 / Chapter Chapter 2 --- Operator Modules and Ideal Cones --- p.32 / Chapter 2.1 --- Operator Modules and Ideal Cones on Banach lattices --- p.32 / Chapter 2.2 --- Half-injective hull and half-surjective hull of operator modules and ideal cones --- p.38 / Chapter 2.3 --- Topologies determined by operator modules and ideal cones --- p.49 / Chapter 2.4 --- Bornologies determined by operator modules and ideal cones --- p.56 / Chapter Chapter 3 --- Banach lattices of operators --- p.63 / Chapter 3.1 --- Cone absolutely summing maps --- p.64 / Chapter 3.2 --- Compact operators on Banach lattices --- p.72 / Chapter 3.3 --- PL-compact operators and locally order precompact operators --- p.85 / Chapter 3.4 --- Almost order bounded sets and semicompact operators --- p.100 / Ref erences --- p.117

The purpose of this thesis is to give a classification of anisotropic algebraic groups over number fields of higher real rank. This will complete the classification of algebraic groups over number fields of higher real rank, which was begun by V. Chernousov, L. Lifschitz and D.W. Morris in their paper "Almost-Minimal Non-Uniform Lattices of Higher Rank''. The classification of anisotropic groups of higher real rank is also used to provide a classification of uniform lattices of higher rank contained in semisimple Lie groups with no compact factors. In particular, it is shown that all such lattices sit inside Lie groups of type An. This thesis proceeds as follows: The first chapter provides motivation for the classification and introduces all the main results of the thesis. The second chapter provides relevant definitions and background material for the proof. The next chapters provide a proof of the classification theorem, with chapters 3-5 examining the absolutely simple groups and the final chapter examining the simple groups which are not absolutely simple. / Mathematics

Thesis--M. Phil., University of Hong Kong, 1985.

Pseudorandom functions (PRFs) are the building blocks of symmetric-key cryptography. Almost all central goals of symmetric cryptography (e.g., encryption, authentication, identification) have simple solutions that make efficient use of a PRF. Most existing constructions of these objects are either (a) extremely fast in practice but without provable security guarantees based on hard mathematical problems [AES, Blowfish etc.], or (b) provably secure under assumptions like the hardness of factoring, but extremely inefficient in practice. Lattice-based constructions enjoy strong security guarantees based on natural mathematical problems, are asymptotically and practically efficient, and have thus far even withstood attacks by quantum algorithms. However, most recent lattice-based constructions are of public-key objects, and it's natural to ask whether these advantages can be brought to the world of symmetric-key constructions. In this thesis, we construct asymptotically fast and parallel pseudorandom functions basing their security on a well known hard lattice problem called the learning with errors problem. We provide several types of constructions that have their respective efficiency and security advantages. In addition to this, we also provide improved constructions of key-homomorphic PRFs that achieve almost optimal quasi-linear magnitudes of public parameters, key sizes and incremental run times. We also propose a new cryptographic primitive, constrained key-homomorphic PRFs, provide secure candidate constructions and applications. Lastly, we detail an implementation in software of a candidate PRF and analyze its efficiency and security.

published_or_final_version / Physics / Master / Master of Philosophy

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