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.

The soft skills of project management : a view from diploma graduates

A research study was conducted in a key area of project management: stakeholder and relationship management through communication - ‘the soft skills’. It was conducted with Diploma of Project Management graduates from one Australian Registered Training Organisation (RTO), the Australian College of Project Management (ACPM). The study was designed to initially identify the qualifications and project management experience of the participants. Further, it identified the respondents’ understanding of and attitude to commonly held principles and literature within the project management field as it relates to the soft skills of projects. This is specifically connected to their project experience and knowledge, approach to project communications, and the stakeholder’s needs. Some of the literature showed that through the management and application of the project soft skills by project managers may actually be a recipe for project success. Hence, an important underpinning of this study was that the project manager can enhance project success (or reduce the impact of failure) by identifying and prioritising stakeholders, developing and implementing strategies for engaging and communicating with them. The use of a positivist approach to this research study allowed for the evaluation and understanding of respondents to the emergent theories of successful projects being delivered through the management of stakeholders, communications, and relationships. Consequently, a quantitative approach to this study was undertaken. The participants were drawn from graduates who completed (graduated) from the ACPM with the Diploma of Project Management between January 2004 and December 2007 only. A list of graduates was collated from this period indicating that a total of 656 graduates have completed and graduated with the qualification. The data collection for this study was done in one phase only. The questionnaire was emailed individually by the researcher directly to the selected potential respondents. Subsequently, a total of 44 responses were received, providing an overall response rate of 43%. Two key factors emerged from the survey questionnaire. Firstly, the need for the soft skills to be incorporated in project management curriculum and education programs, and secondly, that successful projects are delivered through the management and application of the project soft skills. It is expected that the findings of this study be provided across various forums (such as vocational education and training, and project management conferences) and via project management bodies such as the Australian Institute of Project Management (AIPM) to inform learning and provide greater insight into the soft skills of project management. It is the contention of the researcher that this quantitative study of Diploma of Project Management graduates’ views and attitudes highlights the importance of project soft skills and its importance in the delivery of successful projects as well as being part of the competencies of a successful project manager. This study also revealed the value of project experience and knowledge as it pertains to the management and application of the project soft skills.

Graduate research capabilities : a new agenda for research supervisors

There has been a conversation about university graduate employability within the Higher Education literature for some time (Cryer, 1997; Barrie, 2004, 2006, 2007; Murray, 2000; McAlpine, 2005). Within this, and often under the banner of questioning the relevance of the PhD (Murray, 2000), there have been discussions about the employability of research postgraduates. Both the broad discussion of graduate employment and the specific discussion of research degree graduate employment have produced an agenda of graduate research capabilities. Traditionally, assisting research higher degree (RHD) students with their career development has not been an articulated part of the research supervision process. However, the graduate research capabilities agenda has added a new element to the practices of research supervision, in that it brings with it a mandate for research graduates to be aware of the range of capabilities they have acquired through their research degree candidature and how these apply in the workforce. Additionally, there is an emphasis on preparing students for varied career paths rather than a traditional academic route (e.g., in industry or government). Supervisors have a vital role to play in assisting students with these important career development tasks. In this practice application brief we report on a strategy recently used at Queensland University of Technology (QUT) to assist supervisors understand their role in a student’s career development.

Description and evaluation of a Newton-based electronic appetite rating system for temporal tracking of appetite in human subjects

This study assessed the reliability and validity of a palm-top-based electronic appetite rating system (EARS) in relation to the traditional paper and pen method. Twenty healthy subjects [10 male (M) and 10 female (F)] — mean age M=31 years (S.D.=8), F=27 years (S.D.=5); mean BMI M=24 (S.D.=2), F=21 (S.D.=5) — participated in a 4-day protocol. Measurements were made on days 1 and 4. Subjects were given paper and an EARS to log hourly subjective motivation to eat during waking hours. Food intake and meal times were fixed. Subjects were given a maintenance diet (comprising 40% fat, 47% carbohydrate and 13% protein by energy) calculated at 1.6×Resting Metabolic Rate (RMR), as three isoenergetic meals. Bland and Altman's test for bias between two measurement techniques found significant differences between EARS and paper and pen for two of eight responses (hunger and fullness). Regression analysis confirmed that there were no day, sex or order effects between ratings obtained using either technique. For 15 subjects, there was no significant difference between results, with a linear relationship between the two methods that explained most of the variance (r2 ranged from 62.6 to 98.6). The slope for all subjects was less than 1, which was partly explained by a tendency for bias at the extreme end of results on the EARS technique. These data suggest that the EARS is a useful and reliable technique for real-time data collection in appetite research but that it should not be used interchangeably with paper and pen techniques.

Design and evaluation of an MRI compatible axial compression device for 3D assessment of spinal deformity and flexibility in adolescent idiopathic scoliosis

Magnetic Resonance Imaging (MRI) offers a valuable research tool for the assessment of 3D spinal deformity in AIS, however the horizontal patient position imposed by conventional scanners removes the axial compressive loading on the spine. The objective of this study was to design, construct and test an MRI compatible compression device for research into the effect of axial loading on spinal deformity using supine MRI scans. The device was evaluated by performing unloaded and loaded supine MRI scans on a series of 10 AIS patients. The patient group had a mean initial (unloaded) major Cobb angle of 43±7º, which increased to 50±9º on application of the compressive load. The 7° increase in mean Cobb angle is consistent with that reported by a previous study comparing standing versus supine posture in scoliosis patients (Torell et al, 1985. Spine 10:425-7).