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.

Adults who lack capacity : substitute decision-making

• Mechanisms to facilitate consent to healthcare for adults who lack capacity are necessary to ensure that these adults can lawfully receive appropriate medical treatment when needed. • In Australia, the common law plays only a limited role in this context, through its recognition of advance directives and through the parens patriae jurisdiction of superior courts. • Substitute decision-making for adults who lack capacity is facilitated primarily by guardianship and other related legislation. This legislation, which has been enacted in all Australian States and Territories, permits a range of decision-makers to make different types of healthcare decisions. • Substitute decision-makers can be appointed by the adult or by a guardianship or other tribunal. Where there is no appointed decision-maker, legislation generally empowers those close to the adult to make the relevant decision. Most Australian jurisdictions have also provided for statutory advance directives. • For the most serious of decisions, such as non-therapeutic sterilisations, consent can only be provided by a Tribunal. Other decisions can generally be made by a range of substitute decision-makers. Some treatment, such as very minor treatment or that which is needed in an emergency, can be provided without consent. • Guardianship legislation generally establishes a set of principles and/or other criteria to guide healthcare decisions. Mechanisms to resolve disputes as to who is the appropriate decision-maker and how a decision should be made have also been established.

Withholding and withdrawing life-sustaining medical treatment

• At common law, a competent adult can refuse life-sustaining medical treatment, either contemporaneously or through an advance directive which will operate at a later time when the adult’s capacity is lost. • Legislation in most Australian jurisdictions also provides for a competent adult to complete an advance directive that refuses life-sustaining medical treatment. • At common law, a court exercising its parens patriae jurisdiction can consent to, or authorise, the withdrawal or withholding of life-sustaining medical treatment from an adult or child who lacks capacity if that is in the best interests of the person. A court may also declare that the withholding or withdrawal of treatment is lawful. • Guardianship legislation in most jurisdictions allows a substitute decision-maker, in an appropriate case, to refuse life-sustaining medical treatment for an adult who lacks capacity. • In terms of children, a parent may refuse life-sustaining medical treatment for his or her child if it is in the child’s best interests. • While a refusal of life-sustaining medical treatment by a competent child may be valid, this decision can be overturned by a court. • At common law and generally under guardianship statutes, demand for futile treatment need not be complied with by doctors.

The doctrine of double effect

• The doctrine of double effect is an exception to the general rule that taking active steps that end life is unlawful. • The essence of the doctrine at common law is intention. • Hastening a patient’s death through palliative care will be lawful provided the primary intention is to relieve pain, and not cause death, even if that death is foreseen. • Some States have enacted legislative excuses that deal with the provision of palliative care. • These statutory excuses tend to be stricter than the common law as they impose other requirements in addition to having an appropriate intent, such as adherence to some level of recognised medical practice.

Everyday life and the edges of existence : wrongs with no name or the wrong name?

Actions for wrongful life, as they have come unfortunately to be styled, encompass various types of claim. These include claims for alleged negligence after conception, those based on negligent advice or diagnosis prior to conception concerning possible effects of treatment given to the child's mother, contraception or sterilisation, or genetic disability. This distinguishes such claims from those for so called wrongful birth, which are claims by parents for the cost of raising either a healthy or a disabled child where the unplanned birth imposes costs on the parents as a result of clinical negligence. Two of the more controversial cases to have reached the High Court of Australia which are if interest to us here in the past decade are Cattanach v Melchior where the court, by a narrow majority (McHugh, Gummow, Kirby and Callinan JJ; Gleeson CJ, Hayne and Heydon dissenting) acknowledged recovery for wrongful birth. In the second joined appeals of Harriton v Stephens and Waller v James; Waller v Hoolahan the court overwhelmingly precluded a ‘wrongful life’ claim (Gleeson CJ, Gummow, Hayne, Callinan, Heydon and Crennan JJ; Kirby J dissenting). Both cases raised issues around the sanctity and value of life and the nature of harm and the assessment of damages, and this brief note affords us the opportunity to consider the way in which the ‘life as legal loss’ arguments were treated by the various judges in both cases.

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.

The legal framework of the Australian health system

This chapter is about the role of law in the creation and operation of Australian health systems. Accordingly, this chapter discusses how law regulates the way in which health services in Australia are funded, organised, regulated, managed, operated and governed. (The question of how health professionals are regulated is discussed in Chapter 15.) Although the focus of much of health law is on legal mechanisms for the resolution of disputes or disagreements between the state, health providers, professionals, patients and families and friends, and through dispute resolutions processes setting standards for practice, these are only some of the “jobs” that health law performs. In health systems where the state undertakes a significant role in regulating, funding, managing and providing health services, health law also performs an important constitutive function. Health law declares the values upon which the health system is based, shapes social processes to achieve public ends and provides a structure for the complex interactions that occur within a modern health system. Health law regulates decision-makers in health systems by establishing who has the power to participate in decisions and in what circumstances, establishing processes through which decisions are made and creating mechanisms for decision-makers to be held publicly accountable. It is this broader constitutive function of health law that is a primary focus of much of this chapter — how and why governments use their legislative powers to structure and shape the health system.

The regulation of health professionals

This chapter is about the role of law in the management of the health workforce in Australia. Health professionals play an important role in the health system as the providers of treatment and care — without health professionals health systems would not function. The relationship between health professionals and patients has always been complex and is often subject to some form of regulation by the state. The first surviving written reference to such legal regulation dates from 1795-1750 BCE when the Babylonian Code of Hammurabi stated: “If a physician make a large incision with the operating knife, and kill him, or open a tumor with the operating knife, and cut out the eye, his hands shall be cut off.” Alexander the Great recommended the crucifixion of health professionals who killed their patients. Fortunately, the law in Australia prescribes lesser penalties for erring health professionals, but at the heart of modern regulation are similar concerns to those that underpinned the ancient Babylonian Code — to create conditions to ensure the safety of patients and the provision of quality services by health professionals.

Play a role in the patent puzzle

Discusses the contentious issues surrounding computer software patents and patenting in connection with the Peer-to-Patent Australia project, a joint initiative of Queensland University of Technology (QUT) and New York Law School (NYLS) that operates with the support and endorsement of IP Australia, the government body housing Australia's patent office. Explains that the project is based on the successful Peer-to-Patent pilots run recently in the USA and Japan that are designed to improve the quality of issued patents and the patent examination process by facilitating community participation in that process. Describes how members of the public are allowed to put forward prior art references that will be considered by IP Australia's patent examiners when determining whether participating applications are novel and inventive, and therefore deserving of a patent. Concludes that, while Peer-to-Patent Australia is not a complete solution to the problems besetting patent law, the model has considerable advantages over the traditional model of patent examination

The patentability of non-physical inventions : lessons from the United States

Patent systems around the world are being pressed to recognise and protect challengingly new and exciting subject matter in order to keep pace with the rapid technological advancement of our age and the fact we are moving into the era of the ‘knowledge economy’. This rapid development and pressure to expand the bounds of what has traditionally been recognised as patentable subject matter has created uncertainty regarding what it is that the patent system is actually supposed to protect. Among other things, the patent system has had to contend with uncertainty surrounding claims to horticultural and agricultural methods, artificial living micro-organisms, methods of treating the human body, computer software and business methods. The contentious issue of the moment is one at whose heart lies the important distinction between what is a mere abstract idea and what is properly an invention deserving of the monopoly protection afforded by a patent. That question is whether purely intangible inventions, being methods that do not involve a physical aspect or effect or cause a physical transformation of matter, constitute patentable subject matter. This paper goes some way to addressing these uncertainties by considering how the Australian approach to the question can be informed by developments arising in the United States of America, and canvassing some of the possible lessons we in Australia might learn from the approaches taken thus far in the United States.