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.

Health law in Australia

Health Law in Australia is the first book to deal with health law on a comprehensive national basis. In a field of law that is becoming increasingly important and where the demand for expertise is rapidly expanding, Health Law in Australia takes a logical, structured approach to an examination of the law in all Australian jurisdictions. By covering all the major areas in this diverse field of law, Health Law in Australia enhances the understanding of the discipline as a whole. Beginning with an exploration of the general principles of health law, including chapters on “Medical Negligence”, “Children and Consent”, and “Confidentiality, Privacy, and Access to Health Records”, the book goes on to consider beginning-of-life and end-of-life issues before concluding with chapters on emerging areas in health law, such as biotechnology and medical research. The contributing authors include national leaders in the field who are specialists in these areas of health law and who can therefore reveal to readers the results of their research. Health Law in Australia has been written for those with a legal background and is essential reading for undergraduate law students, postgraduate law students, researchers and scholars in the disciplines of law, health and medicine, as well as legal practitioners, government departments and bodies in the health area, and private health providers.

Health law : scope, sources and forces

• For the purposes of this chapter, “health law” encapsulates regulation of the medical and health professions, the administration of health services and the maintenance of public health to the extent that it is connected to the provision of health services. • There are diverging views as to whether health law can be regarded as a discrete “area of law”. • Health law draws on other areas of law such as tort law, criminal law and family law. It is also draws upon other disciplines, most notably medical and health ethics. • Social and economic forces have influenced the development and direction of health law, and these forces may become even more influential as the century develops. • The increasingly globalised world has implications for Australia’s health systems and raises questions and creates commitments in respect of the international community. • Technological developments, including in respect of treatment, diagnosis and information management, create ongoing challenges for health law. • Patient rights, human rights and consumerism are increasingly key drivers in the development of health law. • Health law is significant to contemporary Australian society because of the gravity of the topics that fall within its ambit, its social relevance to so many aspects of human existence and endeavour, the important role it plays in protecting the vulnerable, and the extent to which it engages with fundamental principles of justice.

Access to public sector information : law, technology and policy

On the back of the growing capacity of networked digital information technologies to process and visualise large amounts of information in a timely, efficient and user-driven manner we have seen an increasing demand for better access to and re-use of public sector information (PSI). The story is not a new one. Share knowledge and together we can do great things; limit access and we reduce the potential for opportunity. The two volumes of this book seek to explain and analyse this global shift in the way we manage public sector information. In doing so they collect and present papers, reports and submissions on the topic by leading authors and institutions from across the world. These in turn provide people tasked with mapping out and implementing information policy with reference material and practical guidance. Volume 1 draws together papers on the topic by policymakers, academics and practitioners while Volume 2 presents a selection of the key reports and submissions that have been published over the last few years.

Drugs, dental professionals and the law

Dentists have the privilege of possessing, administering and prescribing drugs, including highly addictive medications, to their patients. But because drugs are often vulnerable to being abused by all members of society, including dentists and their patients, and because drugs can be dangerous, they are tightly regulated in Canada by the federal and provincial/territorial governments. Regulatory and professional dental bodies also provide guidance for their members about how to best administer and prescribe drugs. This chapter outlines the regulation by federal and provincial/territorial governments in this area, examines the professional practice requirements set out by regulatory/professional bodies and the issue of drug abuse by dental professional and patients. It is important to note from the outset that governmental and professional regulations, policies and practices differ from province to province and territory to territory. This chapter aims to alert dentists to possible legal and professional issues surrounding the possession, administration and prescription of drugs. For detailed specific information about regulation, policies, ethical standards and professional practice standards in Canada or their province/ territory, dentists should contact their insurer or professional association.

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.

Digital constitutionalism and the role of the rule of law in the governance of virtual communities

This project proposes a new conceptual framework for the regulation of social networks and virtual communities. By applying a model based upon the rule of law, this thesis addresses the growing tensions that revolve around the public use of private networks. This research examines the shortcomings of traditional contractual governance models and cyberlaw theory and provides a reconstituted approach that will allow public constitutional-type interests to be recognised in the interpretation and enforcement of contractual doctrine.

The Conceptual Basis of Personal Information in Australian Privacy Law

Australian privacy law regulates how government agencies and private sector organisations collect, store and use personal information. A coherent conceptual basis of personal information is an integral requirement of information privacy law as it determines what information is regulated. A 2004 report conducted on behalf of the UK’s Information Commissioner (the 'Booth Report') concluded that there was no coherent definition of personal information currently in operation because different data protection authorities throughout the world conceived the concept of personal information in different ways. The authors adopt the models developed by the Booth Report to examine the conceptual basis of statutory definitions of personal information in Australian privacy laws. Research findings indicate that the definition of personal information is not construed uniformly in Australian privacy laws and that different definitions rely upon different classifications of personal information. A similar situation is evident in a review of relevant case law. Despite this, the authors conclude the article by asserting that a greater jurisprudential discourse is required based on a coherent conceptual framework to ensure the consistent development of Australian privacy law.

The role of the rule of law in virtual communities

There is a severe tendency in cyberlaw theory to delegitimize state intervention in the governance of virtual communities. Much of the existing theory makes one of two fundamental flawed assumptions: that communities will always be best governed without the intervention of the state; or that the territorial state can best encourage the development of communities by creating enforceable property rights and allowing the market to resolve any disputes. These assumptions do not ascribe sufficient weight to the value-laden support that the territorial state always provides to private governance regimes, the inefficiencies that will tend to limit the development utopian communities, and the continued role of the territorial state in limiting autonomy in accordance with communal values. In order to overcome these deterministic assumptions, this article provides a framework based upon the values of the rule of law through which to conceptualise the legitimacy of the private exercise of power in virtual communities. The rule of law provides a constitutional discourse that assists in considering appropriate limits on the exercise of private power. I argue that the private contractual framework that is used to govern relations in virtual communities ought to be informed by the values of the rule of law in order to more appropriately address the governance tensions that permeate these spaces. These values suggest three main limits to the exercise of private power: that governance is limited by community rules and that the scope of autonomy is limited by the substantive values of the territorial state; that private contractual rules should be general, equal, and certain; and that, most importantly, internal norms be predicated upon the consent of participants.