Group law computations on Jacobians of hyperelliptic Curves

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form.

An exploration of affine group laws for elliptic curves

Several forms of elliptic curves are suggested for an efficient implementation of Elliptic Curve Cryptography. However, a complete description of the group law has not appeared in the literature for most popular forms. This paper presents group law in affine coordinates for three forms of elliptic curves. With the existence of the proposed affine group laws, stating the projective group law for each form becomes trivial. This work also describes an automated framework for studying elliptic curve group law, which is applied internally when preparing this work.

Mutually unbiased bases as submodules and subspaces

Mutually unbiased bases (MUBs) have been used in several cryptographic and communications applications. There has been much speculation regarding connections between MUBs and finite geometries. Most of which has focused on a connection with projective and affine planes. We propose a connection with higher dimensional projective geometries and projective Hjelmslev geometries. We show that this proposed geometric structure is present in several constructions of MUBs.

Distributed low-power image processing in wireless sensor networks for intelligent video surveillance applications

Distributed Wireless Smart Camera (DWSC) network is a special type of Wireless Sensor Network (WSN) that processes captured images in a distributed manner. While image processing on DWSCs sees a great potential for growth, with its applications possessing a vast practical application domain such as security surveillance and health care, it suffers from tremendous constraints. In addition to the limitations of conventional WSNs, image processing on DWSCs requires more computational power, bandwidth and energy that presents significant challenges for large scale deployments. This dissertation has developed a number of algorithms that are highly scalable, portable, energy efficient and performance efficient, with considerations of practical constraints imposed by the hardware and the nature of WSN. More specifically, these algorithms tackle the problems of multi-object tracking and localisation in distributed wireless smart camera net- works and optimal camera configuration determination. Addressing the first problem of multi-object tracking and localisation requires solving a large array of sub-problems. The sub-problems that are discussed in this dissertation are calibration of internal parameters, multi-camera calibration for localisation and object handover for tracking. These topics have been covered extensively in computer vision literatures, however new algorithms must be invented to accommodate the various constraints introduced and required by the DWSC platform. A technique has been developed for the automatic calibration of low-cost cameras which are assumed to be restricted in their freedom of movement to either pan or tilt movements. Camera internal parameters, including focal length, principal point, lens distortion parameter and the angle and axis of rotation, can be recovered from a minimum set of two images of the camera, provided that the axis of rotation between the two images goes through the camera's optical centre and is parallel to either the vertical (panning) or horizontal (tilting) axis of the image. For object localisation, a novel approach has been developed for the calibration of a network of non-overlapping DWSCs in terms of their ground plane homographies, which can then be used for localising objects. In the proposed approach, a robot travels through the camera network while updating its position in a global coordinate frame, which it broadcasts to the cameras. The cameras use this, along with the image plane location of the robot, to compute a mapping from their image planes to the global coordinate frame. This is combined with an occupancy map generated by the robot during the mapping process to localised objects moving within the network. In addition, to deal with the problem of object handover between DWSCs of non-overlapping fields of view, a highly-scalable, distributed protocol has been designed. Cameras that follow the proposed protocol transmit object descriptions to a selected set of neighbours that are determined using a predictive forwarding strategy. The received descriptions are then matched at the subsequent camera on the object's path using a probability maximisation process with locally generated descriptions. The second problem of camera placement emerges naturally when these pervasive devices are put into real use. The locations, orientations, lens types etc. of the cameras must be chosen in a way that the utility of the network is maximised (e.g. maximum coverage) while user requirements are met. To deal with this, a statistical formulation of the problem of determining optimal camera configurations has been introduced and a Trans-Dimensional Simulated Annealing (TDSA) algorithm has been proposed to effectively solve the problem.

Video analytics for the detection of near-miss incidents on approach to railway level crossings

Recent modelling of socio-economic costs by the Australian railway industry in 2010 has estimated the cost of level crossing accidents to exceed AU$116 million annually. To better understand causal factors that contribute to these accidents, the Cooperative Research Centre for Rail Innovation is running a project entitled Baseline Level Crossing Video. The project aims to improve the recording of level crossing safety data by developing an intelligent system capable of detecting near-miss incidents and capturing quantitative data around these incidents. To detect near-miss events at railway level crossings a video analytics module is being developed to analyse video footage obtained from forward-facing cameras installed on trains. This paper presents a vision base approach for the detection of these near-miss events. The video analytics module is comprised of object detectors and a rail detection algorithm, allowing the distance between a detected object and the rail to be determined. An existing publicly available Histograms of Oriented Gradients (HOG) based object detector algorithm is used to detect various types of vehicles in each video frame. As vehicles are usually seen from a sideway view from the cabin’s perspective, the results of the vehicle detector are verified using an algorithm that can detect the wheels of each detected vehicle. Rail detection is facilitated using a projective transformation of the video, such that the forward-facing view becomes a bird’s eye view. Line Segment Detector is employed as the feature extractor and a sliding window approach is developed to track a pair of rails. Localisation of the vehicles is done by projecting the results of the vehicle and rail detectors on the ground plane allowing the distance between the vehicle and rail to be calculated. The resultant vehicle positions and distance are logged to a database for further analysis. We present preliminary results regarding the performance of a prototype video analytics module on a data set of videos containing more than 30 different railway level crossings. The video data is captured from a journey of a train that has passed through these level crossings.

An algorithm for constructing Hjelmslev planes

Projective Hjelmslev planes and affine Hjelmslev planes are generalisations of projective planes and affine planes. We present an algorithm for constructing projective Hjelmslev planes and affine Hjelmslev planes that uses projective planes, affine planes and orthogonal arrays. We show that all 2-uniform projective Hjelmslev planes, and all 2-uniform affine Hjelmslev planes can be constructed in this way. As a corollary it is shown that all $2$-uniform affine Hjelmslev planes are sub-geometries of $2$-uniform projective Hjelmslev planes.

Applying POVM to model non-orthogonality in quantum cognition

Much of the work currently occurring in the field of Quantum Interaction (QI) relies upon Projective Measurement. This is perhaps not optimal, cognitive states are not nearly as well behaved as standard quantum mechanical systems; they exhibit violations of repeatability, and the operators that we use to describe measurements do not appear to be naturally orthogonal in cognitive systems. Here we attempt to map the formalism of Positive Operator Valued Measure (POVM) theory into the domain of semantic memory, showing how it might be used to construct Bell-type inequalities.

Contributions to the Theory of Finite-State Based Grammars

This dissertation is a theoretical study of finite-state based grammars used in natural language processing. The study is concerned with certain varieties of finite-state intersection grammars (FSIG) whose parsers define regular relations between surface strings and annotated surface strings. The study focuses on the following three aspects of FSIGs: (i) Computational complexity of grammars under limiting parameters In the study, the computational complexity in practical natural language processing is approached through performance-motivated parameters on structural complexity. Each parameter splits some grammars in the Chomsky hierarchy into an infinite set of subset approximations. When the approximations are regular, they seem to fall into the logarithmic-time hierarchyand the dot-depth hierarchy of star-free regular languages. This theoretical result is important and possibly relevant to grammar induction. (ii) Linguistically applicable structural representations Related to the linguistically applicable representations of syntactic entities, the study contains new bracketing schemes that cope with dependency links, left- and right branching, crossing dependencies and spurious ambiguity. New grammar representations that resemble the Chomsky-Schützenberger representation of context-free languages are presented in the study, and they include, in particular, representations for mildly context-sensitive non-projective dependency grammars whose performance-motivated approximations are linear time parseable. (iii) Compilation and simplification of linguistic constraints Efficient compilation methods for certain regular operations such as generalized restriction are presented. These include an elegant algorithm that has already been adopted as the approach in a proprietary finite-state tool. In addition to the compilation methods, an approach to on-the-fly simplifications of finite-state representations for parse forests is sketched. These findings are tightly coupled with each other under the theme of locality. I argue that the findings help us to develop better, linguistically oriented formalisms for finite-state parsing and to develop more efficient parsers for natural language processing. Avainsanat: syntactic parsing, finite-state automata, dependency grammar, first-order logic, linguistic performance, star-free regular approximations, mildly context-sensitive grammars

On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Generalised quaternion methods in conformal geometry

A new approach to Penrose's twistor algebra is given. It is based on the use of a generalised quaternion algebra for the translation of statements in projective five-space into equivalent statements in twistor (conformal spinor) space. The formalism leads toSO(4, 2)-covariant formulations of the Pauli-Kofink and Fierz relations among Dirac bilinears, and generalisations of these relations.

The Noether-Lefschetz theorem for the divisor class group

Let X be a normal projective threefold over a field of characteristic zero and vertical bar L vertical bar be a base-point free, ample linear system on X. Under suitable hypotheses on (X, vertical bar L vertical bar), we prove that for a very general member Y is an element of vertical bar L vertical bar, the restriction map on divisor class groups Cl(X) -> Cl(Y) is an isomorphism. In particular, we are able to recover the classical Noether-Lefschetz theorem, that a very general hypersurface X subset of P-C(3) of degree >= 4 has Pic(X) congruent to Z.

Combinatorial Design of Secure Lightweight Data Dispersal Schemes (Work in Progress)

Dispersing a data object into a set of data shares is an elemental stage in distributed communication and storage systems. In comparison to data replication, data dispersal with redundancy saves space and bandwidth. Moreover, dispersing a data object to distinct communication links or storage sites limits adversarial access to whole data and tolerates loss of a part of data shares. Existing data dispersal schemes have been proposed mostly based on various mathematical transformations on the data which induce high computation overhead. This paper presents a novel data dispersal scheme where each part of a data object is replicated, without encoding, into a subset of data shares according to combinatorial design theory. Particularly, data parts are mapped to points and data shares are mapped to lines of a projective plane. Data parts are then distributed to data shares using the point and line incidence relations in the plane so that certain subsets of data shares collectively possess all data parts. The presented scheme incorporates combinatorial design theory with inseparability transformation to achieve secure data dispersal at reduced computation, communication and storage costs. Rigorous formal analysis and experimental study demonstrate significant cost-benefits of the presented scheme in comparison to existing methods.

A Combinatorial Construct for Keyless Message Dispersal (Work in Progress)

We propose a keyless and lightweight message transformation scheme based on the combinatorial design theory for the confidentiality of a message transmitted in multiple parts through a network with multiple independent paths, or for data stored in multiple parts by a set of independent storage services such as the cloud providers. Our combinatorial scheme disperses a message into v output parts so that (k-1) or less parts do not reveal any information about any message part, and the message can only be recovered by the party who possesses all v output parts. Combinatorial scheme generates an xor transformation structure to disperse the message into v output parts. Inversion is done by applying the same xor transformation structure on output parts. The structure is generated using generalized quadrangles from design theory which represents symmetric point and line incidence relations in a projective plane. We randomize our solution by adding a random salt value and dispersing it together with the message. We show that a passive adversary with capability of accessing (k-1) communication links or storage services has no advantage so that the scheme is indistinguishable under adaptive chosen ciphertext attack (IND-CCA2).

Constructing commutative semifields of square order

The projection construction has been used to construct semifields of odd characteristic using a field and a twisted semifield [Commutative semi-fields from projection mappings, Designs, Codes and Cryptography, 61 (2011), 187{196]. We generalize this idea to a projection construction using two twisted semifields to construct semifields of odd characteristic. Planar functions and semifields have a strong connection so this also constructs new planar functions.

Real Theta Characteristics And Automorphisms Of A Real Curve

Let X be a geometrically irreductble smooth projective cruve defined over R. of genus at least 2. that admits a nontrivial automorphism, sigma. Assume that X does not have any real points. Let tau be the antiholomorphic involution of the complexification lambda(C) of X. We show that if the action of sigma on the set S(X) of all real theta characteristics of X is trivial. then the order of sigma is even, say 2k and the automorphism tau o (sigma) over cap (lambda) of X-C has a fixed point, where (sigma) over cap is the automorphism of X x C-R defined by sigma We then show that there exists X with a real point and admitting a nontrivial automorphism sigma, such that the action of sigma on S(X) is trivial, while X/ not equal P-R(1) We also give an example of X with no real points and admitting a nontrivial automorphisim sigma such that the automorphism tau o (sigma) over cap (lambda) has a fixed point, the action of sigma on S(X) is trivial, and X/ not equal P-R(1)