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.

Exponential asymptotics with multiple stokes lines

When asymptotic series methods are applied in order to solve problems that arise in applied mathematics in the limit that some parameter becomes small, they are unable to demonstrate behaviour that occurs on a scale that is exponentially small compared to the algebraic terms of the asymptotic series. There are many examples of physical systems where behaviour on this scale has important effects and, as such, a range of techniques known as exponential asymptotic techniques were developed that may be used to examinine behaviour on this exponentially small scale. Many problems in applied mathematics may be represented by behaviour within the complex plane, which may subsequently be examined using asymptotic methods. These problems frequently demonstrate behaviour known as Stokes phenomenon, which involves the rapid switches of behaviour on an exponentially small scale in the neighbourhood of some curve known as a Stokes line. Exponential asymptotic techniques have been applied in order to obtain an expression for this exponentially small switching behaviour in the solutions to orginary and partial differential equations. The problem of potential flow over a submerged obstacle has been previously considered in this manner by Chapman & Vanden-Broeck (2006). By representing the problem in the complex plane and applying an exponential asymptotic technique, they were able to detect the switching, and subsequent behaviour, of exponentially small waves on the free surface of the flow in the limit of small Froude number, specifically considering the case of flow over a step with one Stokes line present in the complex plane. We consider an extension of this work to flow configurations with multiple Stokes lines, such as flow over an inclined step, or flow over a bump or trench. The resultant expressions are analysed, and demonstrate interesting implications, such as the presence of exponentially sub-subdominant intermediate waves and the possibility of trapped surface waves for flow over a bump or trench. We then consider the effect of multiple Stokes lines in higher order equations, particu- larly investigating the behaviour of higher-order Stokes lines in the solutions to partial differential equations. These higher-order Stokes lines switch off the ordinary Stokes lines themselves, adding a layer of complexity to the overall Stokes structure of the solution. Specifically, we consider the different approaches taken by Howls et al. (2004) and Chap- man & Mortimer (2005) in applying exponential asymptotic techniques to determine the higher-order Stokes phenomenon behaviour in the solution to a particular partial differ- ential equation.

Implications of faecal indicator bacteria for the microbiological assessment of roof harvested rainwater quality in Southeast Queensland, Australia

The study aimed to evaluate the suitability of Escherichia coli, enterococci and C. perfringens to assess the microbiological quality of roof harvested rainwater, and to assess whether the concentrations of these faecal indicators can be used to predict the presence or absence of specific zoonotic bacterial or protozoan pathogens. From a total of 100 samples tested, respectively 58%, 83% and 46% of samples were found to be positive for E. coli, enterococci and C. perfringens spores, as determined by traditional culture based methods. Additionally, in the samples tested, 7%, 19%, 1%, 8%, 17%, and 15% were PCR positive for A. hydrophila lip, C. coli ceuE, C. jejuni mapA, L. pneumophila mip, Salmonella invA, and G. lamblia β-giardin genes. However, none of the samples was positive for E. coli O157 LPS, VT1, VT2 and C. parvum COWP genes. The presence or absence of these potential pathogens did not correlate with any of the faecal indicator bacterial concentrations as determined by a binary logistic regression model. The roof-harvested rainwater samples tested in this study appear to be of poor microbiological quality and no significant correlation was found between the concentration of faecal indicators and pathogenic microorganisms. The use of faecal indicator bacteria raises questions regarding their reliability in assessing the microbiological quality of water and particularly their poor correlation with pathogenic microorganisms. The presence of one or more zoonotic pathogens suggests that the microbiological analysis of water should be performed, and appropriate treatment measures should be undertaken especially in tanks where the water is used for drinking.

Appetite sensations and satiety quotient : predictors of energy intake and weight loss

PURPOSE: The aim of this study was to further evaluate the validity and clinical meaningfulness of appetite sensations to predict overall energy intake as well as body weight loss. METHODS: Men (n=176) and women (n=139) involved in six weight loss studies were selected to participate in this study. Visual analogue scales were used to measure appetite sensations before and after a fixed test meal. Fasting appetite sensations, 1 h post-prandial area under the curve (AUC) and the satiety quotient (SQ) were used as predictors of energy intake and body weight loss. Two separate measures of energy intake were used: a buffet style ad libitum test lunch and a three-day self-report dietary record. RESULTS: One-hour post-prandial AUC for all appetite sensations represented the strongest predictors of ad libitum test lunch energy intake (p0.001). These associations were more consistent and pronounced for women than men. Only SQ for fullness was associated with ad libitum test lunch energy intake in women. Similar but weaker relationships were found between appetite sensations and the 3-day self-reported energy intake. Weight loss was associated with changes in appetite sensations (p0.01) and the best predictors of body weight loss were fasting desire to eat; hunger; and PFC (p0.01). CONCLUSIONS: These results demonstrate that appetite sensations are relatively useful predictors of spontaneous energy intake, free-living total energy intake and body weight loss. They also confirm that SQ for fullness predicts energy intake, at least in women.

Patterns, correlates, and prognostic significance of quality of life following breast cancer

Objective To describe quality of life (QOL) over a 12-month period among women with breast cancer, consider the association between QOL and overall survival (OS), and explore characteristics associated with QOL declines. Methods A population-based sample of Australian women (n=287) with invasive, unilateral breast cancer (Stage I+), was observed prospectively for a median of 6.6 years. QOL was assessed at six, 12 and 18 months post-diagnosis, using the Functional Assessment of Cancer Therapy, Breast (FACT-B+4) questionnaire. Raw scores for the FACT-B+4 and subscales were computed and individuals were categorized according to whether QOL declined, remained stable or improved between six and 18 months. Kaplan-Meier and Cox proportional hazards survival methods were used to estimate OS and its associations with QOL. Logistic regression models identified factors associated with QOL decline. Results Within FACT-B+4 sub-scales, between 10% and 23% of women showed declines in QOL. Following adjustment for established prognostic factors, emotional wellbeing and FACT-B+4 scores at six months post-diagnosis were associated with OS (p<0.05). Declines in physical (p<0.01) or functional (p=0.02) well-being between six and 18 months post-diagnosis were also associated significantly with OS. Receiving multiple forms of adjuvant treatment, a perception of not handling stress well and reporting one or more other major life events at six months post-diagnosis were factors associated with declines in QOL in multivariable analyses. Conclusions Interventions targeted at preventing QOL declines may ultimately improve quantity as well as quality of life following breast cancer.

The effect of friction on indenter force and pile-up in numerical simulations of bone nanoindentation

Nanoindentation is a useful technique for probing the mechanical properties of bone, and finite element (FE) modeling of the indentation allows inverse determination of elasto-plastic constitutive properties. However, FE simulations to date have assumed frictionless contact between indenter and bone. The aim of this study was to explore the effect of friction in simulations of bone nanoindentation. Two dimensional axisymmetric FE simulations were performed using a spheroconical indenter of tip radius 0.6m and angle 90°. The coefficient of friction between indenter and bone was varied between 0.0 (frictionless) and 0.3. Isotropic linear elasticity was used in all simulations, with bone elastic modulus E=13.56GPa and Poisson’s ratio =0.3. Plasticity was incorporated using both Drucker-Prager and von Mises yield surfaces. Friction had a modest effect on the predicted force-indentation curve for both von Mises and Drucker-Prager plasticity, reducing maximum indenter displacement by 10% and 20% respectively as friction coefficient was increased from zero to 0.3 (at a maximum indenter force of 5mN). However, friction has a much greater effect on predicted pile-up after indentation, reducing predicted pile-up from 0.27m to 0.11m with a von Mises model, and from 0.09m to 0.02m with Drucker-Prager plasticity. We conclude that it is important to include friction in nanoindentation simulations of bone.

Structural equation model of construction contract dispute potential

This paper presents the results of a structural equation model (SEM) for describing and quantifying the fundamental factors that affect contract disputes between owners and contractors in the construction industry. Through this example, the potential impact of SEM analysis in construction engineering and management research is illustrated. The purpose of the specific model developed in this research is to explain how and why contract related construction problems occur. This study builds upon earlier work, which developed a disputes potential index, and the likelihood of construction disputes was modeled using logistic regression. In this earlier study, questionnaires were completed on 159 construction projects, which measured both qualitative and quantitative aspects of contract disputes, management ability, financial planning, risk allocation, and project scope definition for both owners and contractors. The SEM approach offers several advantages over the previously employed logistic regression methodology. The final set of structural equations provides insight into the interaction of the variables that was not apparent in the original logistic regression modeling methodology.

Vibrational spectroscopic studies of wool

Fourier transfonn (FT) Raman, Raman microspectroscopy and Fourier transform infrared (FTIR) spectroscopy have been used for the structural analysis and characterisation of untreated and chemically treated wool fibres. For FT -Raman spectroscopy novel methods of sample presentation have been developed and optimised for the analysis of wool. No significant fluorescence was observed and the spectra could be obtained routinely. The stability of wool keratin to the laser source was investigated and the visual and spectroscopic signs of sample damage were established. Wool keratin was found to be extremely robust with no signs of sample degradation observed for laser powers of up to 600 m W and for exposure times of up to seven and half hours. Due to improvements in band resolution and signal-to-noise ratio, several previously unobserved spectral features have become apparent. The assignment of the Raman active vibrational modes of wool have been reviewed and updated to include these features. The infrared spectroscopic techniques of attenuated total reflectance (ATR) and photoacoustic (P A) have been used to examine shrinkproofed and mothproofed wool samples. Shrinkproofing is an oxidative chemical treatment used to selectively modifY the surface of a wool fibre. Mothproofing is a chemical treatment applied to wool for the prevention of insect attack. The ability of PAS and A TR to vary the penetration depth by varying certain instrumental parameters was used to obtain spectra of the near surface regions of these chemically treated samples. These spectra were compared with those taken with a greater penetration depth, which therefore represent more of the bulk wool sample. The PA and ATR spectra demonstrated that oxidation was restricted to the near-surface layer of wool. Extensive curve fitting of ATR spectra of untreated wool indicated that cuticle was composed of a mixed protein conformation, but was predominately that of an a.-helix. The cortex was proposed to be a mixture of both a.helical and ~-pleated sheet protein conformations. These findings were supported by PAS depth profiling results. Raman microspectroscopy was used in an extensive investigation of the molecular structure of the wool fibre. This included determining the orientation of certain functional groups within the wool fibre and the symmetry of particular vibrations. The orientation ofbonds within the wool fibre was investigated by orientating the wool fibre axis parallel and then perpendicular to the plane of polarisation of the electric vector of the incident radiation. It was experimentally determined that the majority of C=O and N-H bonds of the peptide bond of wool lie parallel to the fibre axis. Additionally, a number of the important vibrations associated with the a-helix were also found to lie parallel to the fibre axis. Further investigation into the molecular structure of wool involved determining what effect stretching the wool fibre had on bond orientation. Raman spectra of stretched and unstretched wool fibres indicated that extension altered the orientation ofthe aromatic rings, the CH2 and CH3 groups of the amino acids. Curve fitting results revealed that extension resulted in significant destruction of the a-helix structure a substantial increase in the P-pleated sheet structure. Finally, depolarisation ratios were calculated for Raman spectra. The vibrations associated with the aromatic rings of amino acids had very low ratios which indicated that the vibrations were highly symmetrical.

Hierarchical models for 2D presence/absence data having ambiguous zeroes: With a biogeographical case study on dingo behaviour

This dissertation is primarily an applied statistical modelling investigation, motivated by a case study comprising real data and real questions. Theoretical questions on modelling and computation of normalization constants arose from pursuit of these data analytic questions. The essence of the thesis can be described as follows. Consider binary data observed on a two-dimensional lattice. A common problem with such data is the ambiguity of zeroes recorded. These may represent zero response given some threshold (presence) or that the threshold has not been triggered (absence). Suppose that the researcher wishes to estimate the effects of covariates on the binary responses, whilst taking into account underlying spatial variation, which is itself of some interest. This situation arises in many contexts and the dingo, cypress and toad case studies described in the motivation chapter are examples of this. Two main approaches to modelling and inference are investigated in this thesis. The first is frequentist and based on generalized linear models, with spatial variation modelled by using a block structure or by smoothing the residuals spatially. The EM algorithm can be used to obtain point estimates, coupled with bootstrapping or asymptotic MLE estimates for standard errors. The second approach is Bayesian and based on a three- or four-tier hierarchical model, comprising a logistic regression with covariates for the data layer, a binary Markov Random field (MRF) for the underlying spatial process, and suitable priors for parameters in these main models. The three-parameter autologistic model is a particular MRF of interest. Markov chain Monte Carlo (MCMC) methods comprising hybrid Metropolis/Gibbs samplers is suitable for computation in this situation. Model performance can be gauged by MCMC diagnostics. Model choice can be assessed by incorporating another tier in the modelling hierarchy. This requires evaluation of a normalization constant, a notoriously difficult problem. Difficulty with estimating the normalization constant for the MRF can be overcome by using a path integral approach, although this is a highly computationally intensive method. Different methods of estimating ratios of normalization constants (N Cs) are investigated, including importance sampling Monte Carlo (ISMC), dependent Monte Carlo based on MCMC simulations (MCMC), and reverse logistic regression (RLR). I develop an idea present though not fully developed in the literature, and propose the Integrated mean canonical statistic (IMCS) method for estimating log NC ratios for binary MRFs. The IMCS method falls within the framework of the newly identified path sampling methods of Gelman & Meng (1998) and outperforms ISMC, MCMC and RLR. It also does not rely on simplifying assumptions, such as ignoring spatio-temporal dependence in the process. A thorough investigation is made of the application of IMCS to the three-parameter Autologistic model. This work introduces background computations required for the full implementation of the four-tier model in Chapter 7. Two different extensions of the three-tier model to a four-tier version are investigated. The first extension incorporates temporal dependence in the underlying spatio-temporal process. The second extensions allows the successes and failures in the data layer to depend on time. The MCMC computational method is extended to incorporate the extra layer. A major contribution of the thesis is the development of a fully Bayesian approach to inference for these hierarchical models for the first time. Note: The author of this thesis has agreed to make it open access but invites people downloading the thesis to send her an email via the 'Contact Author' function.

Wandering and the physical environment

Background/Rationale Guided by the need-driven dementia-compromised behavior (NDB) model, this study examined influences of the physical environment on wandering behavior. Methods Using a descriptive, cross-sectional design, 122 wanderers from 28 long-term care (LTC) facilities were videotaped 10 to 12 times; data on wandering, light, sound, temperature and humidity levels, location, ambiance, and crowding were obtained. Associations between environmental variables and wandering were evaluated with chi-square and t tests; the model was evaluated using logistic regression. Results In all, 80% of wandering occurred in the resident’s own room, dayrooms, hallways, or dining rooms. When observed in other residents’ rooms, hallways, shower/baths, or off-unit locations, wanderers were likely (60%-92% of observations) to wander. The data were a good fit to the model overall (LR [logistic regression] χ2 (5) = 50.38, P < .0001) and by wandering type. Conclusions Location, light, sound, proximity of others, and ambiance are associated with wandering and may serve to inform environmental designs and care practices.

Safety-restraint use rate as function of law enforcement and other factors : preliminary analysis

Persistent use of safety restraints prevents deaths and reduces the severity and number of injuries resulting from motor vehicle crashes. However, safety-restraint use rates in the United States have been below those of other nations with safety-restraint enforcement laws. With a better understanding of the relationship between safety-restraint law enforcement and safety-restraint use, programs can be implemented to decrease the number of deaths and injuries resulting from motor vehicle crashes. Does safety-restraint use increase as enforcement increases? Do motorists increase their safety-restraint use in response to the general presence of law enforcement or to targeted law enforcement efforts? Does a relationship between enforcement and restraint use exist at the countywide level? A logistic regression model was estimated by using county-level safety-restraint use data and traffic citation statistics collected in 13 counties within the state of Florida in 1997. The model results suggest that safety-restraint use is positively correlated with enforcement intensity, is negatively correlated with safety-restraint enforcement coverage (in lanemiles of enforcement coverage), and is greater in urban than rural areas. The quantification of these relationships may assist Florida and other law enforcement agencies in raising safety-restraint use rates by allocating limited funds more efficiently either by allocating additional time for enforcement activities of the existing force or by increasing enforcement staff. In addition, the research supports a commonsense notion that enforcement activities do result in behavioral response.

The effectiveness of a short-term group music therapy intervention for parents who have a child with a disability

The relationship between the quality of parent-child interactions and positive child developmental trajectories is well established (Guralnick, 2006; Shonkoff & Meissels, 2000; Zubrick et al., 2008). However, a range of parental, family, and socio-economic factors can pose risks to parents’ capacity to participate in quality interactions with their children. In particular, families with a child with a disability have been found to have higher levels of parenting stress, and are more likely to experience economic disadvantage, as well as social isolation. The importance of early interventions to promote positive parenting and child development for these families is widely recognised (Shonkoff & Meissels, 2000). However, to date, there is a lack of evidence about the effectiveness of early parenting programs for families who have a young child with a disability. This thesis investigates the impact of a music therapy parenting program, Sing & Grow, on 201 parent-child dyads who attended programs specifically targeted to parents who had a young child with a disability. Sing & Grow is an Australian national early parenting intervention funded by the Australian Government Department of Families, Housing, Community Services and Indigenous Affairs and delivered by Playgroup Queensland. It is designed and delivered by Registered Music Therapists for families with children aged from birth to three years. It aims to improve parenting skills and confidence, improve family functioning (positive parent-child interactions), enhance child development, and provide social networking opportunities to socially isolated families. The intervention targets a range of families in circumstances that have the potential to impact negatively on family functioning. This thesis uses data from the National Evaluation Study of Sing & Grow from programs which were targeted at families who had a young child with a disability. Three studies were conducted to address the objectives of this thesis. Study 1 examines the effects of the Sing & Grow intervention on parent reported pre and post parent mental health, parenting confidence, parenting skills, and child development, and other parent reported outcomes including social support, use of intervention resources, satisfaction with the intervention and perceived benefits of and barriers to participation. Significant improvements from pre to post were found for parent mental health and parent reported child communication and social skills, along with evidence that parents were very satisfied with the program and that it brought social benefits to families. Study 2 explored the pre to post effects of the intervention on children’s developmental skills and parent-child interactions using observational ratings made by clinicians. Significant pre to post improvements were found for parenting sensitivity, parental engagement with child and acceptance of child as well as for child responsiveness to parent, interest, and participation in the intervention, and social skills. Study 3 examined the nature of child and family characteristics that predicted better outcomes for families while taking account of the level of participation in the program. An overall outcome index was calculated and served as the dependent variable in a logistic regression analysis. Families who attended six or more sessions and mothers who had not completed high school were more likely to have higher outcome scores at post intervention than those who attended fewer sessions and those with more educated mothers respectively. The findings of this research indicate that the intervention had a positive impact on participants’ mental health, parenting behaviours and child development and that level of attendance was associated with better outcomes. There was also evidence that the program reached its target of high risk families (i.e., families in which mothers had lower educational levels) and that for these families better outcomes were achieved. There were also indications that the program was accessible and highly regarded by families and that it promoted social connections for participants. A theoretical model of how the intervention is currently working for families is proposed to explain the connections between early parenting, child development and maternal wellbeing. However, more research is required to further elucidate the mechanisms by which the intervention creates change for families. This research presents promising evidence that a short term group music therapy program can elicit important therapeutic benefits for families who have a child with a disability.