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.

Regulating IVF and pre-implantation tissue-typing for the creation of "saviour siblings" : a harm analysis

Scientific discoveries, developments in medicine and health issues are the constant focus of media attention and the principles surrounding the creation of so called ‘saviour siblings’ are of no exception. The development in the field of reproductive techniques has provided the ability to genetically analyse embryos created in the laboratory to enable parents to implant selected embryos to create a tissue-matched child who may be able to cure an existing sick child. The research undertaken in this thesis examines the regulatory frameworks overseeing the delivery of assisted reproductive technologies (ART) in Australia and the United Kingdom and considers how those frameworks impact on the accessibility of in vitro fertilisation (IVF) procedures for the creation of ‘saviour siblings’. In some jurisdictions, the accessibility of such techniques is limited by statutory requirements. The limitations and restrictions imposed by the state in relation to the technology are analysed in order to establish whether such restrictions are justified. The analysis is conducted on the basis of a harm framework. The framework seeks to establish whether those affected by the use of the technology (including the child who will be created) are harmed. In order to undertake such evaluation, the concept of harm is considered under the scope of John Stuart Mill’s liberal theory and the Harm Principle is used as a normative tool to judge whether the level of harm that may result, justifies state intervention or restriction with the reproductive decision-making of parents in this context. The harm analysis conducted in this thesis seeks to determine an appropriate regulatory response in relation to the use of pre-implantation tissue-typing for the creation of ‘saviour siblings’. The proposals outlined in the last part of this thesis seek to address the concern that harm may result from the practice of pre-implantation tissue-typing. The current regulatory frameworks in place are also analysed on the basis of the harm framework established in this thesis. The material referred to in this thesis reflects the law and policy in place in Australia and the UK at the time the thesis was submitted for examination (December 2009).

Announcement and legislation enactment improve children’s seating position in regional areas

New legislation requires all children 7 years and younger to use child-specific Australian Standards approved restraints suitable to their age and restricts seating young children in the front of cars. Observations of child seating position and restraint use were undertaken in Toowoomba and Rockhampton before the Queensland legislation was announced (T1), after the announcement but before it was enacted (T2) and after it came into force (T3). From T1 to T2, the percentage of children seated in the rear increased (69% to 75%), with a further increase from T2 to T3 (75% to 77%). This pattern was clear when there were one or two children in the car, but not when there were 3 or more. The effect on restraint use was more complex. After the announcement (T2) the percentage of children using adult seatbelts significantly increased regardless of the number of child passengers. However, once the legislation was enacted (T3) there was a significant increase in the percentage of children using child seats/boosters where there was one or two child passengers. Where there were three or more children in the vehicle there was little change in restraint choice between pre (T1) and post (T3) legislation.

Research-led educational renewal and economic-cultural convergence

The 'Queensland Model' grew out of three convergent agendas: educational renewal, urban redevelopment, and the Queensland state government's 'Smart State' strategy.

Adoption Law and Practice [2nd ed.]

This book addresses the modern law relating to adoption. It comes at a time of fundamental change in adoption practice as, increasingly, Irish couples look outside the jurisdiction for the child that will make their family complete.---------- * Examines and explains the new regulatory framework and the law now governing domestic and intercountry adoption.---------- * Provides a guide to the changes outlined in the Adoption Bill 2008 which also consolidates the provisions of seven previous statutes and incorporates the Hague Convention into Irish statute law.---------- * Considers the responsibilities of the new Adoption Authority, and the roles of other administrative and legal bodies.---------- * Sets out the adoption process, explaining the complexities of intercountry adoption, giving consideration to the interface between adoption and children in care and dealing with the rights of the parties involved.

Legal ethics in a post-Westphalian world : building the international rule of law and otehr tasks

The emergence of strong sovereign states after the Treaty of Westphalia turned two of the most cosmopolitan professions (law and arms) into two of the least cosmopolitan. Sovereign states determined the content of the law within their borders – including which, if any, ecclesiastical law was to be applied; what form of economic regulation was adopted; and what, if any, international law applied. Similarly, states sought to ensure that all military force was at their disposal in national armies. The erosion of sovereignty in a post-Westphalian world may significantly reverse these processes. The erosion of sovereignty is likely to have profound consequences for the legal profession and the ethics of how, and for what ends, it is practised. Lawyers have played a major role in the civilization of sovereign states through the articulation and institutionalisation of key governance values – starting with the rule of law. An increasingly global profession must take on similar tasks. The same could be said of the military. This essay will review the concept of an international rule of law and its relationship to domestic conceptions and outline the task of building the international rule of law and the role that lawyers can and should play in it.