Design of tch-type sequences for communications

This thesis deals with the design of a class of cyclic codes inspired by TCH codewords. Since TCH codes are linked to finite fields the fundamental concepts and facts about abstract algebra, namely group theory and number theory, constitute the first part of the thesis. By exploring group geometric properties and identifying an equivalence between some operations on codes and the symmetries of the dihedral group we were able to simplify the generation of codewords thus saving on the necessary number of computations. Moreover, we also presented an algebraic method to obtain binary generalized TCH codewords of length N = 2k, k = 1,2, . . . , 16. By exploring Zech logarithm’s properties as well as a group theoretic isomorphism we developed a method that is both faster and less complex than what was proposed before. In addition, it is valid for all relevant cases relating the codeword length N and not only those resulting from N = p

Discrete Harmonic Analysis. Representations, Number Theory, Expanders and the Fourier Transform

This self-contained book introduces readers to discrete harmonic analysis with an emphasis on the Discrete Fourier Transform and the Fast Fourier Transform on finite groups and finite fields, as well as their noncommutative versions. It also features applications to number theory, graph theory, and representation theory of finite groups. Beginning with elementary material on algebra and number theory, the book then delves into advanced topics from the frontiers of current research, including spectral analysis of the DFT, spectral graph theory and expanders, representation theory of finite groups and multiplicity-free triples, Tao's uncertainty principle for cyclic groups, harmonic analysis on GL(2,Fq), and applications of the Heisenberg group to DFT and FFT. With numerous examples, figures, and over 160 exercises to aid understanding, this book will be a valuable reference for graduate students and researchers in mathematics, engineering, and computer science

Sequences design for OFDM and CDMA systems

With the emergence of multi-carrier (MC) orthogonal frequency division multiplexing (OFDM) scheme in the current WLAN standards and next generation wireless broadband standards, the necessitation to acquire a method for combating high peak to average power ratio (PMEPR) becomes imminent. In this thesis, we will explore various sequences to determine their PMEPR behaviours, in hopes to find some sequences which could potentially be suitable for PMEPR reduction control under MC system settings. These sequences include $m$ sequences, Sidelnikov sequences, new sequences, Golay sequences, FZC sequences and Legendre sequences. We will also examine the merit factor properties of these sequences, and then we will derive a bound between PMEPR and merit factor. Moreover, in the design of code division multiple access (CDMA) spreading sequence sets, it is critical that each sequence in the set has low autocorrelations and low cross-correlation with other sequences in the same set. In the thesis, we will present a class of GDJ Golay sequences which contains a large zero autocorrelation zone (ZACZ), which could satisfy the low autocorrelation requirement. This class of Golay sequences could potentially be used to construct new CDMA spreading sequence sets

Formal duality

We provide an overview of formal duality with an emphasis on the authors contributions. Every formally dual set can be obtained from a primitive formally dual set or, more generally from an irreducible formally dual set.Using several methods, including even set theory and the field-descent method, it is possible to obtain examples of primitive/irreducible formally dual sets as well as non-existence results. A graph search algorithm can be used for further investigation. Overall, primitive formally dual sets seem rare in cyclic groups, but occasionally exist in finite abelian groups.In dieser Dissertation geben wir einen Überblick über formale Dualität. Jede formal duale Menge kann von einer primitiven, oder allgemeiner von einer irreduziblen, formal dualen Menge, konstruiert werden. Methoden wie die 'even set' Theorie oder die 'field-descent' Methode, können genutzt werden, um Beispiele für primitive/irreduzible formal duale Mengen sowie nicht-Existenz Resultate zu erhalten. Weiterhin kann ein Suchalgorithmus genutzt werden. Primitive formal duale Mengen scheinen in zyklischen Gruppen selten zu sein, kommen aber gelegentlich in endlichen abelschen Gruppen vor

Correlated Pseudorandomness from the Hardness of Quasi-Abelian Decoding

Secure computation often benefits from the use of correlated randomness to achieve fast, non-cryptographic online protocols. A recent paradigm put forth by Boyle $\textit{et al.}$ (CCS 2018, Crypto 2019) showed how pseudorandom correlation generators (PCG) can be used to generate large amounts of useful forms of correlated (pseudo)randomness, using minimal interactions followed solely by local computations, yielding silent secure two-party computation protocols (protocols where the preprocessing phase requires almost no communication). An additional property called programmability allows to extend this to build N-party protocols. However, known constructions for programmable PCG's can only produce OLE's over large fields, and use rather new splittable Ring-LPN assumption. In this work, we overcome both limitations. To this end, we introduce the quasi-abelian syndrome decoding problem (QA-SD), a family of assumptions which generalises the well-established quasi-cyclic syndrome decoding assumption. Building upon QA-SD, we construct new programmable PCG's for OLE's over any field $\mathbb{F}_q$ with $q>2$. Our analysis also sheds light on the security of the ring-LPN assumption used in Boyle $\textit{et al.}$ (Crypto 2020). Using our new PCG's, we obtain the first efficient N-party silent secure computation protocols for computing general arithmetic circuit over $\mathbb{F}_q$ for any $q>2$.Comment: This is a long version of a paper accepted at CRYPTO'2

Orthogonal transforms and their application to image coding

Imperial Users onl

Designing topological quantum matter in and out of equilibrium

Recent advances in experimental condensed matter physics suggest a powerful new paradigm for the realization of exotic phases of quantum matter in the laboratory. Rather than conducting an exhaustive search for materials that realize these phases at low temperatures, it may be possible to design quantum systems that exhibit the desired properties. With the numerous advances made recently in the fields of cold atomic gases, superconducting qubits, trapped ions, and nitrogen-vacancy centers in diamond, it appears that we will soon have a host of platforms that can be used to put exotic theoretical predictions to the test. In this dissertation, I will highlight two ways in which theorists can interact productively with this fast-emerging field. First, there is a growing interest in driving quantum systems out of equilibrium in order to induce novel topological phases where they would otherwise never appear. In particular, systems driven by time-periodic perturbations—known as “Floquet systems”—offer fertile ground for theoretical investigation. This approach to designer quantum matter brings its own unique set of challenges. In particular, Floquet systems explicitly violate conservation of energy, providing no notion of a ground state. In the first part of my dissertation, I will present research that addresses this problem in two ways. First, I will present studies of open Floquet systems, where coupling to an external reservoir drives the system into a steady state at long times. Second, I will discuss examples of isolated quantum systems that exhibit signatures of topological properties in their finite-time dynamics. The second part of this dissertation presents another way in which theorists can benefit from the designer approach to quantum matter; in particular, one can design analytically tractable theories of exotic phases. I will present an exemplar of this philosophy in the form of coupled-wire constructions. In this approach, one builds a topological state of matter from the ground up by coupling together an array of one-dimensional quantum wires with local interactions. I will demonstrate the power of this technique by showing how to build both Abelian and non-Abelian topological phases in three dimensions by coupling together an array of quantum wires