Elliptic curves, group law, and efficient computation

This thesis is about the derivation of the addition law on an arbitrary elliptic curve and efficiently adding points on this elliptic curve using the derived addition law. The outcomes of this research guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology. Mastered by the 19th century mathematicians, the study of the theory of elliptic curves has been active for decades. Elliptic curves over finite fields made their way into public key cryptography in late 1980’s with independent proposals by Miller [Mil86] and Koblitz [Kob87]. Elliptic Curve Cryptography (ECC), following Miller’s and Koblitz’s proposals, employs the group of rational points on an elliptic curve in building discrete logarithm based public key cryptosystems. Starting from late 1990’s, the emergence of the ECC market has boosted the research in computational aspects of elliptic curves. This thesis falls into this same area of research where the main aim is to speed up the additions of rational points on an arbitrary elliptic curve (over a field of large characteristic). The outcomes of this work can be used to speed up applications which are based on elliptic curves, including cryptographic applications in ECC. The aforementioned goals of this thesis are achieved in five main steps. As the first step, this thesis brings together several algebraic tools in order to derive the unique group law of an elliptic curve. This step also includes an investigation of recent computer algebra packages relating to their capabilities. Although the group law is unique, its evaluation can be performed using abundant (in fact infinitely many) formulae. As the second step, this thesis progresses the finding of the best formulae for efficient addition of points. In the third step, the group law is stated explicitly by handling all possible summands. The fourth step presents the algorithms to be used for efficient point additions. In the fifth and final step, optimized software implementations of the proposed algorithms are presented in order to show that theoretical speedups of step four can be practically obtained. In each of the five steps, this thesis focuses on five forms of elliptic curves over finite fields of large characteristic. A list of these forms and their defining equations are given as follows: (a) Short Weierstrass form, y2 = x3 + ax + b, (b) Extended Jacobi quartic form, y2 = dx4 + 2ax2 + 1, (c) Twisted Hessian form, ax3 + y3 + 1 = dxy, (d) Twisted Edwards form, ax2 + y2 = 1 + dx2y2, (e) Twisted Jacobi intersection form, bs2 + c2 = 1, as2 + d2 = 1, These forms are the most promising candidates for efficient computations and thus considered in this work. Nevertheless, the methods employed in this thesis are capable of handling arbitrary elliptic curves. From a high level point of view, the following outcomes are achieved in this thesis. - Related literature results are brought together and further revisited. For most of the cases several missed formulae, algorithms, and efficient point representations are discovered. - Analogies are made among all studied forms. For instance, it is shown that two sets of affine addition formulae are sufficient to cover all possible affine inputs as long as the output is also an affine point in any of these forms. In the literature, many special cases, especially interactions with points at infinity were omitted from discussion. This thesis handles all of the possibilities. - Several new point doubling/addition formulae and algorithms are introduced, which are more efficient than the existing alternatives in the literature. Most notably, the speed of extended Jacobi quartic, twisted Edwards, and Jacobi intersection forms are improved. New unified addition formulae are proposed for short Weierstrass form. New coordinate systems are studied for the first time. - An optimized implementation is developed using a combination of generic x86-64 assembly instructions and the plain C language. The practical advantages of the proposed algorithms are supported by computer experiments. - All formulae, presented in the body of this thesis, are checked for correctness using computer algebra scripts together with details on register allocations.

Fixed argument pairings

A common scenario in many pairing-based cryptographic protocols is that one argument in the pairing is fixed as a long term secret key or a constant parameter in the system. In these situations, the runtime of Miller's algorithm can be significantly reduced by storing precomputed values that depend on the fixed argument, prior to the input or existence of the second argument. In light of recent developments in pairing computation, we show that the computation of the Miller loop can be sped up by up to 37 if precomputation is employed, with our method being up to 19.5 faster than the previous precomputation techniques.

Certificateless key agreement in the standard model

We show how to construct a certificateless key agreement protocol from the certificateless key encapsulation mechanism introduced by \cite{lippold-ICISC_2009} in ICISC 2009 using the \cite{DBLP:conf/acisp/BoydCNP08} protocol from ACISP 2008. We introduce the Canetti-Krawczyk (CK) model for certificateless cryptography, give security notions for Type I and Type II adversaries in the CK model, and highlight the differences to the existing e$^2$CK model discussed by \cite{DBLP:conf/pairing/LippoldBN09}. The resulting CK model is more relaxed thus giving more power to the adversary than the original CK model.

NAPLAN, a high priority or . . . ? Statements about its purpose in Queensland schools

This paper reports on statements from Professional Development participants who were asked to comment on NAPLAN. The participants were involved in a project designed by the YuMi Deadly Centre (YDC) for implementation into 25 Queensland School to enhance the teaching and learning of mathematics to Aboriginal and Torres Strait Islander students and low SES students. Using an action research framework and a survey questionnaire, the preliminary data obtained from participating principals is mixed, with statements indicating that NAPLAN is a high priority for some schools while others indicated that it does not “tell” the whole story of student learning.

Contextualising the teaching and learning of measurement within Torres Strait Islander schools

A one year mathematics project that focused on measurement was conducted with six Torres Strait Islander schools and communities. Its key focus was to contextualise the teaching and learning of measurement within the students’ culture, communities and home languages. There were six teachers and two teacher aides who participated in the project. This paper reports on the findings from the teachers’ and teacher aides’ survey questionnaire used in the first Professional Development session to identify: a) teachers’ experience of teaching in Torres Strait Islands, b) teachers’ beliefs about effective ways to teach Torres Strait Islander students, and c) contexualising measurement within Torres Strait Islander culture, Communities and home languages. A wide range of differing levels of knowledge and understanding about how to contextualise measurement to support student learning were identified and analysed. For example, an Indigenous teacher claimed that mathematics and the environment are relational, that is, they are not discrete and in isolation from one another, rather they interconnect with mathematical ideas emerging from the environment of the Torres Strait Communities.

A national framework for disaster health education in Australia

INTRODUCTION: Recent events have heightened awareness of disaster health issues and the need to prepare the health workforce to plan for and respond to major incidents. This has been reinforced at an international level by the World Association for Disaster and Emergency Medicine, which has proposed an international educational framework. ----------- OBJECTIVE: The aim of this paper is to outline the development of a national educational framework for disaster health in Australia. ----------- METHODS: The framework was developed on the basis of the literature and the previous experience of members of a National Collaborative for Disaster Health Education and Research. The Collaborative was brought together in a series of workshops and teleconferences, utilizing a modified Delphi technique to finalize the content at each level of the framework and to assign a value to the inclusion of that content at the various levels. ----------- FRAMEWORK: The framework identifies seven educational levels along with educational outcomes for each level. The framework also identifies the recommended contents at each level and assigns a rating of depth for each component. The framework is not intended as a detailed curriculum, but rather as a guide for educationalists to develop specific programs at each level. ----------- CONCLUSIONS: This educational framework will provide an infrastructure around which future educational programs in Disaster Health in Australia may be designed and delivered. It will permit improved articulation for students between the various levels and greater consistency between programs so that operational responders may have a consistent language and operational approach to the management of major events.

Music is a Wordless Knowing of Others : Resilience in Virtual Ensembles

Music making affects relationships with self and others by generating a sense of belonging to a culture or ideology (Bamford, 2006; Barovick, 2001; Dillon & Stewart, 2006; Fiske, 2000; Hallam, 2001). Whilst studies from arts education research present compelling examples of these relationships, others argue that they do not present sufficiently validated evidence of a causal link between music making experiences and cognitive or social change (Winner & Cooper, 2000; Winner & Hetland, 2000a, 2000b, 2001). I have suggested elsewhere that this disconnection between compelling evidence and observations of the effects of music making are in part due to the lack of rigor in research and the incapacity of many methods to capture these experiences in meaningful ways (Dillon, 2006). Part of the answer to these questions about rigor and causality lay in the creative use of new media technologies that capture the results of relationships in music artefacts. Crucially, it is the effective management of these artefacts within computer systems that allows researchers and practitioners to collect, organize, analyse and then theorise such music making experiences.

Settlement geography of African refugee communities in Southeast Queensland : an analysis of residential distribution and secondary migration

Before 2001, most Africans immigrating to Australia were white South Africans and Zimbabweans who arrived as economic and family-reunion migrants (Cox, Cooper & Adepoju, 1999). Black African communities are a more recent addition to the Australian landscape, with most entering Australia as refugees after 2001. African refugees are a particularly disadvantaged immigrant group, which the Department of Immigration and Multicultural Affairs (in the Community Relations Commission of New South Wales, 2006) suggests require high levels of settlement support (p.23). Decision makers and settlement service providers need to have settlement data on the communities so that they can be effective in planning, budgeting and delivering support where it is most needed. Settlement data are also useful for determining the challenges that these communities face in trying to establish themselves in resettlement. There has been no verification of existing secondary data sources, however, or previous formal study of African refugee settlement geography in Southeast Queensland. This research addresses the knowledge gap by using a mixed-method approach to identify and describe the distribution and population size of eight African communities in Southeast Queensland, examine secondary migration patterns in these communities and assess the relationship between these geographic features and housing, a critical factor in successful settlement. Significant discrepancies exist between the primary data gathered in the study and existing secondary data relating to population size and distribution of the communities. Results also reveal a tension between the socio-cultural forces and the housing and economic imperatives driving secondary migration in the communities, and a general lack of engagement by African refugees with structured support networks. These findings have a wide range of implications for policy and for groups that provide settlement support to these communities.

Estimating the cost of health care–associated infections : mind your p’s and q’s

Monetary valuations of the economic cost of health care–associated infections (HAIs) are important for decision making and should be estimated accurately. Erroneously high estimates of costs, designed to jolt decision makers into action, may do more harm than good in the struggle to attract funding for infection control. Expectations among policy makers might be raised, and then they are disappointed when the reduction in the number of HAIs does not yield the anticipated cost saving. For this article, we critically review the field and discuss 3 questions. Why measure the cost of an HAI? What outcome should be used to measure the cost of an HAI? What is the best method for making this measurement? The aim is to encourage researchers to collect and then disseminate information that accurately guides decisions about the economic value of expanding or changing current infection control activities.

Years 2 to 6 students' ability to generalise : models, representations and theory for teaching and learning

Over the last three years, in our Early Algebra Thinking Project, we have been studying Years 3 to 5 students’ ability to generalise in a variety of situations, namely, compensation principles in computation, the balance principle in equivalence and equations, change and inverse change rules with function machines, and pattern rules with growing patterns. In these studies, we have attempted to involve a variety of models and representations and to build students’ abilities to switch between them (in line with the theories of Dreyfus, 1991, and Duval, 1999). The results have shown the negative effect of closure on generalisation in symbolic representations, the predominance of single variance generalisation over covariant generalisation in tabular representations, and the reduced ability to readily identify commonalities and relationships in enactive and iconic representations. This chapter uses the results to explore the interrelation between generalisation and verbal and visual comprehension of context. The studies evidence the importance of understanding and communicating aspects of representational forms which allowed commonalities to be seen across or between representations. Finally the chapter explores the implications of the studies for a theory that describes a growth in integration of models and representations that leads to generalisation.

Processing load and the use of concrete representations and strategies for solving linear equations

This paper is a report of students' responses to instruction which was based on the use of concrete representations to solve linear equations. The sample consisted of 21 Grade 8 students from a middle-class suburban state secondary school with a reputation for high academic standards and innovative mathematics teaching. The students were interviewed before and after instruction. Interviews and classroom interactions were observed and videotaped. A qualitative analysis of the responses revealed that students did not use the materials in solving problems. The increased processing load caused by concrete representations is hypothesised as a reason.

Multiple human single-stranded DNA binding proteins function in genome maintenance : structural, biochemical and functional analysis

DNA exists predominantly in a duplex form that is preserved via specific base pairing. This base pairing affords a considerable degree of protection against chemical or physical damage and preserves coding potential. However, there are many situations, e.g. during DNA damage and programmed cellular processes such as DNA replication and transcription, in which the DNA duplex is separated into two singlestranded DNA (ssDNA) strands. This ssDNA is vulnerable to attack by nucleases, binding by inappropriate proteins and chemical attack. It is very important to control the generation of ssDNA and protect it when it forms, and for this reason all cellular organisms and many viruses encode a ssDNA binding protein (SSB). All known SSBs use an oligosaccharide/oligonucleotide binding (OB)-fold domain for DNA binding. SSBs have multiple roles in binding and sequestering ssDNA, detecting DNA damage, stimulating strand-exchange proteins and helicases, and mediation of protein–protein interactions. Recently two additional human SSBs have been identified that are more closely related to bacterial and archaeal SSBs. Prior to this it was believed that replication protein A, RPA, was the only human equivalent of bacterial SSB. RPA is thought to be required for most aspects of DNA metabolism including DNA replication, recombination and repair. This review will discuss in further detail the biological pathways in which human SSBs function.

A cell migration device that maintains a defined surface with no cellular damage during wound edge generation

Studying the rate of cell migration provides insight into fundamental cell biology as well as a tool to assess the functionality of synthetic surfaces and soluble environments used in tissue engineering. The traditional tools used to study cell migration include the fence and wound healing assays. In this paper we describe the development of a microchannel based device for the study of cell migration on defined surfaces. We demonstrate that this device provides a superior tool, relative to the previously mentioned assays, for assessing the propagation rate of cell wave fronts. The significant advantage provided by this technology is the ability to maintain a virgin surface prior to the commencement of the cell migration assay. Here, the device is used to assess rates of mouse fibroblasts (NIH 3T3) and human osteosarcoma (SaOS2) cell migration on surfaces functionalized with various extracellular matrix proteins as a demonstration that confining cell migration within a microchannel produces consistent and robust data. The device design enables rapid and simplistic assessment of multiple repeats on a single chip, where surfaces have not been previously exposed to cells or cellular secretions.