Fast formulas for computing cryptographic pairings

The most powerful known primitive in public-key cryptography is undoubtedly elliptic curve pairings. Upon their introduction just over ten years ago the computation of pairings was far too slow for them to be considered a practical option. This resulted in a vast amount of research from many mathematicians and computer scientists around the globe aiming to improve this computation speed. From the use of modern results in algebraic and arithmetic geometry to the application of foundational number theory that dates back to the days of Gauss and Euler, cryptographic pairings have since experienced a great deal of improvement. As a result, what was an extremely expensive computation that took several minutes is now a high-speed operation that takes less than a millisecond. This thesis presents a range of optimisations to the state-of-the-art in cryptographic pairing computation. Both through extending prior techniques, and introducing several novel ideas of our own, our work has contributed to recordbreaking pairing implementations.

Zernike radial slope polynomials for wavefront reconstruction and refraction

Ophthalmic wavefront sensors typically measure wavefront slope, from which wavefront phase is reconstructed. We show that ophthalmic prescriptions (in power-vector format) can be obtained directly from slope measurements without wavefront reconstruction. This is achieved by fitting the measurement data with a new set of orthonormal basis functions called Zernike radial slope polynomials. Coefficients of this expansion can be used to specify the ophthalmic power vector using explicit formulas derived by a variety of methods. Zernike coefficients for wavefront error can be recovered from the coefficients of radial slope polynomials, thereby offering an alternative way to perform wavefront reconstruction.

Avoiding full extension field arithmetic in pairing computations

The most costly operations encountered in pairing computations are those that take place in the full extension field Fpk . At high levels of security, the complexity of operations in Fpk dominates the complexity of the operations that occur in the lower degree subfields. Consequently, full extension field operations have the greatest effect on the runtime of Miller’s algorithm. Many recent optimizations in the literature have focussed on improving the overall operation count by presenting new explicit formulas that reduce the number of subfield operations encountered throughout an iteration of Miller’s algorithm. Unfortunately, almost all of these improvements tend to suffer for larger embedding degrees where the expensive extension field operations far outweigh the operations in the smaller subfields. In this paper, we propose a new way of carrying out Miller’s algorithm that involves new explicit formulas which reduce the number of full extension field operations that occur in an iteration of the Miller loop, resulting in significant speed ups in most practical situations of between 5 and 30 percent.

Faster pairing computations on curves with high-degree twists

Research on efficient pairing implementation has focussed on reducing the loop length and on using high-degree twists. Existence of twists of degree larger than 2 is a very restrictive criterion but luckily constructions for pairing-friendly elliptic curves with such twists exist. In fact, Freeman, Scott and Teske showed in their overview paper that often the best known methods of constructing pairing-friendly elliptic curves over fields of large prime characteristic produce curves that admit twists of degree 3, 4 or 6. A few papers have presented explicit formulas for the doubling and the addition step in Miller’s algorithm, but the optimizations were all done for the Tate pairing with degree-2 twists, so the main usage of the high- degree twists remained incompatible with more efficient formulas. In this paper we present efficient formulas for curves with twists of degree 2, 3, 4 or 6. These formulas are significantly faster than their predecessors. We show how these faster formulas can be applied to Tate and ate pairing variants, thereby speeding up all practical suggestions for efficient pairing implementations over fields of large characteristic.

Exploring the influence of a science content course incorporating explicit nature of science and argumentation instruction on preservice primary teachers' views of nature of science

There exists a general consensus in the science education literature around the goal of enhancing students. and teachers. views of nature of science (NOS). An emerging area of research in science education explores NOS and argumentation, and the aim of this study was to explore the effectiveness of a science content course incorporating explicit NOS and argumentation instruction on preservice primary teachers. views of NOS. A constructivist perspective guided the study, and the research strategy employed was case study research. Five preservice primary teachers were selected for intensive investigation in the study, which incorporated explicit NOS and argumentation instruction, and utilised scientific and socioscientific contexts for argumentation to provide opportunities for participants to apply their NOS understandings to their arguments. Four primary sources of data were used to provide evidence for the interpretations, recommendations, and implications that emerged from the study. These data sources included questionnaires and surveys, interviews, audio- and video-taped class sessions, and written artefacts. Data analysis involved the formation of various assertions that informed the major findings of the study, and a variety of validity and ethical protocols were considered during the analysis to ensure the findings and interpretations emerging from the data were valid. Results indicated that the science content course was effective in enabling four of the five participants. views of NOS to be changed. All of the participants expressed predominantly limited views of the majority of the examined NOS aspects at the commencement of the study. Many positive changes were evident at the end of the study with four of the five participants expressing partially informed and/or informed views of the majority of the examined NOS aspects. A critical analysis of the effectiveness of the various course components designed to facilitate the development of participants‟ views of NOS in the study, led to the identification of three factors that mediated the development of participants‟ NOS views: (a) contextual factors (including context of argumentation, and mode of argumentation), (b) task-specific factors (including argumentation scaffolds, epistemological probes, and consideration of alternative data and explanations), and (c) personal factors (including perceived previous knowledge about NOS, appreciation of the importance and utility value of NOS, and durability and persistence of pre-existing beliefs). A consideration of the above factors informs recommendations for future studies that seek to incorporate explicit NOS and argumentation instruction as a context for learning about NOS.

The explicit hazard model – part 1: theoretical development

Modern Engineering Asset Management (EAM) requires the accurate assessment of current and the prediction of future asset health condition. Appropriate mathematical models that are capable of estimating times to failures and the probability of failures in the future are essential in EAM. In most real-life situations, the lifetime of an engineering asset is influenced and/or indicated by different factors that are termed as covariates. Hazard prediction with covariates is an elemental notion in the reliability theory to estimate the tendency of an engineering asset failing instantaneously beyond the current time assumed that it has already survived up to the current time. A number of statistical covariate-based hazard models have been developed. However, none of them has explicitly incorporated both external and internal covariates into one model. This paper introduces a novel covariate-based hazard model to address this concern. This model is named as Explicit Hazard Model (EHM). Both the semi-parametric and non-parametric forms of this model are presented in the paper. The major purpose of this paper is to illustrate the theoretical development of EHM. Due to page limitation, a case study with the reliability field data is presented in the applications part of this study.

The Explicit Hazard Model – Part 2 : Applications

Hazard and reliability prediction of an engineering asset is one of the significant fields of research in Engineering Asset Health Management (EAHM). In real-life situations where an engineering asset operates under dynamic operational and environmental conditions, the lifetime of an engineering asset can be influenced and/or indicated by different factors that are termed as covariates. The Explicit Hazard Model (EHM) as a covariate-based hazard model is a new approach for hazard prediction which explicitly incorporates both internal and external covariates into one model. EHM is an appropriate model to use in the analysis of lifetime data in presence of both internal and external covariates in the reliability field. This paper presents applications of the methodology which is introduced and illustrated in the theory part of this study. In this paper, the semi-parametric EHM is applied to a case study so as to predict the hazard and reliability of resistance elements on a Resistance Corrosion Sensor Board (RCSB).

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.

Making the processes of designing explicit within an information technology environment

In this paper, technology is described as involving processes whereby resources are utilised to satisfy human needs or to take advantage of opportunities, to develop practical solutions to problems. This study, set within one type of technology context, information technology, investigated how, through a one semester undergraduate university course, elements of technological processes were made explicit to students. While it was acknowledged in the development and implementation of this course that students needed to learn technical skills, technological skills and knowledge, including design, were seen as vital also, to enable students to think about information technology from a perspective that was not confined and limited to `technology as hardware and software'. This paper describes how the course, set within a three year program of study, was aimed at helping students to develop their thinking and their knowledge about design processes in an explicit way. An interpretive research approach was used and data sources included a repertory grid `survey'; student interviews; video recordings of classroom interactions, audio recordings of lectures, observations of classroom interactions made by researchers; and artefacts which included students' journals and portfolios. The development of students' knowledge about design practices is discussed and reflections upon student knowledge development in conjunction with their learning experiences are made. Implications for ensuring explicitness of design practice within information technology contexts are presented, and the need to identify what constitutes design knowledge is argued.

Reliability prediction using the non-parametric explicit hazard model : a case study

Survival probability prediction using covariate-based hazard approach is a known statistical methodology in engineering asset health management. We have previously reported the semi-parametric Explicit Hazard Model (EHM) which incorporates three types of information: population characteristics; condition indicators; and operating environment indicators for hazard prediction. This model assumes the baseline hazard has the form of the Weibull distribution. To avoid this assumption, this paper presents the non-parametric EHM which is a distribution-free covariate-based hazard model. In this paper, an application of the non-parametric EHM is demonstrated via a case study. In this case study, survival probabilities of a set of resistance elements using the non-parametric EHM are compared with the Weibull proportional hazard model and traditional Weibull model. The results show that the non-parametric EHM can effectively predict asset life using the condition indicator, operating environment indicator, and failure history.

Weighted fairness with MCR guarantee and a general-purpose explicit rate allocation algorithm for ATM ABR service

In this paper, weighted fair rate allocation for ATM available bit rate (ABR) service is discussed with the concern of the minimum cell rate (MCR). Weighted fairness with MCR guarantee has been discussed recently in the literature. In those studies, each ABR virtual connection (VC) is first allocated its MCR, then the remaining available bandwidth is further shared among ABR VCs according to their weights. For the weighted fairness defined in this paper, the bandwidth is first allocated according to each VC's weight; if a VC's weighted share is less than its MCR, it should be allocated its MCR instead of the weighted share. This weighted fairness with MCR guarantee is referred to as extended weighted (EXW) fairness. Certain theoretical issues related to EXW, such as its global solution and bottleneck structure, are first discussed in the paper. A distributed explicit rate allocation algorithm is then proposed to achieve EXW fairness in ATM networks. The algorithm is a general-purpose explicit rate algorithm in the sense that it can realise almost all the fairness principles proposed for ABR so far whilst only minor modifications may be needed.