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 Guardian Reportage of the UK MP Expenses Scandal: a Case Study of Computational Journalism

The Guardian reportage of the United Kingdom Member of Parliament (MP) expenses scandal of 2009 used crowdsourcing and computational journalism techniques. Computational journalism can be broadly defined as the application of computer science techniques to the activities of journalism. Its foundation lies in computer assisted reporting techniques and its importance is increasing due to the: (a) increasing availability of large scale government datasets for scrutiny; (b) declining cost, increasing power and ease of use of data mining and filtering software; and Web 2.0; and (c) explosion of online public engagement and opinion.. This paper provides a case study of the Guardian MP expenses scandal reportage and reveals some key challenges and opportunities for digital journalism. It finds journalists may increasingly take an active role in understanding, interpreting, verifying and reporting clues or conclusions that arise from the interrogations of datasets (computational journalism). Secondly a distinction should be made between information reportage and computational journalism in the digital realm, just as a distinction might be made between citizen reporting and citizen journalism. Thirdly, an opportunity exists for online news providers to take a ‘curatorial’ role, selecting and making easily available the best data sources for readers to use (information reportage). These activities have always been fundamental to journalism, however the way in which they are undertaken may change. Findings from this paper may suggest opportunities and challenges for the implementation of computational journalism techniques in practice by digital Australian media providers, and further areas of research.

Automating computational proofs for public-key-based key exchange

We present an approach to automating computationally sound proofs of key exchange protocols based on public-key encryption. We show that satisfying the property called occultness in the Dolev-Yao model guarantees the security of a related key exchange protocol in a simple computational model. Security in this simpler model has been shown to imply security in a Bellare {Rogaway-like model. Furthermore, the occultness in the Dolev-Yao model can be searched automatically by a mechanisable procedure. Thus automated proofs for key exchange protocols in the computational model can be achieved. We illustrate the method using the well-known Lowe-Needham-Schroeder protocol.

The Guardian reportage of the UK MP expenses scandal: a case study of computational journalism

The Guardian reportage of the United Kingdom Member of Parliament (MP) expenses scandal of 2009 used crowdsourcing and computational journalism techniques. Computational journalism can be broadly defined as the application of computer science techniques to the activities of journalism. Its foundation lies in computer assisted reporting techniques and its importance is increasing due to the: (a) increasing availability of large scale government datasets for scrutiny; (b) declining cost, increasing power and ease of use of data mining and filtering software; and Web 2.0; and (c) explosion of online public engagement and opinion.. This paper provides a case study of the Guardian MP expenses scandal reportage and reveals some key challenges and opportunities for digital journalism. It finds journalists may increasingly take an active role in understanding, interpreting, verifying and reporting clues or conclusions that arise from the interrogations of datasets (computational journalism). Secondly a distinction should be made between information reportage and computational journalism in the digital realm, just as a distinction might be made between citizen reporting and citizen journalism. Thirdly, an opportunity exists for online news providers to take a ‘curatorial’ role, selecting and making easily available the best data sources for readers to use (information reportage). These activities have always been fundamental to journalism, however the way in which they are undertaken may change. Findings from this paper may suggest opportunities and challenges for the implementation of computational journalism techniques in practice by digital Australian media providers, and further areas of research.

The promise of computational journalism.

There are at least four key challenges in the online news environment that computational journalism may address. Firstly, news providers operate in a rapidly evolving environment and larger businesses are typically slower to adapt to market innovations. News consumption patterns have changed and news providers need to find new ways to capture and retain digital users. Meanwhile, declining financial performance has led to cost cuts in mass market newspapers. Finally investigative reporting is typically slow, high cost and may be tedious, and yet is valuable to the reputation of a news provider. Computational journalism involves the application of software and technologies to the activities of journalism, and it draws from the fields of computer science, social science and communications. New technologies may enhance the traditional aims of journalism, or may require “a new breed of people who are midway between technologists and journalists” (Irfan Essa in Mecklin 2009: 3). Historically referred to as ‘computer assisted reporting’, the use of software in online reportage is increasingly valuable due to three factors: larger datasets are becoming publicly available; software is becoming sophisticated and ubiquitous; and the developing Australian digital economy. This paper introduces key elements of computational journalism – it describes why it is needed; what it involves; benefits and challenges; and provides a case study and examples. Computational techniques can quickly provide a solid factual basis for original investigative journalism and may increase interaction with readers, when correctly used. It is a major opportunity to enhance the delivery of original investigative journalism, which ultimately may attract and retain readers online.

Numerical evaluation of pollutant dispersion at a toll plaza based on system dynamics and Computational Fluid Dynamics models

The health of tollbooth workers is seriously threatened by long-term exposure to polluted air from vehicle exhausts. Using traffic data collected at a toll plaza, vehicle movements were simulated by a system dynamics model with different traffic volumes and toll collection procedures. This allowed the average travel time of vehicles to be calculated. A three-dimension Computational Fluid Dynamics (CFD) model was used with a k–ε turbulence model to simulate pollutant dispersion at the toll plaza for different traffic volumes and toll collection procedures. It was shown that pollutant concentration around tollbooths increases as traffic volume increases. Whether traffic volume is low or high (1500 vehicles/h or 2500 vehicles/h), pollutant concentration decreases if electronic toll collection (ETC) is adopted. In addition, pollutant concentration around tollbooths decreases as the proportion of ETC-equipped vehicles increases. However, if the proportion of ETC-equipped vehicles is very low and the traffic volume is not heavy, then pollutant concentration increases as the number of ETC lanes increases.

An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields

In this paper, an enriched radial point interpolation method (e-RPIM) is developed the for the determination of crack tip fields. In e-RPIM, the conventional RBF interpolation is novelly augmented by the suitable trigonometric basis functions to reflect the properties of stresses for the crack tip fields. The performance of the enriched RBF meshfree shape functions is firstly investigated to fit different surfaces. The surface fitting results have proven that, comparing with the conventional RBF shape function, the enriched RBF shape function has: (1) a similar accuracy to fit a polynomial surface; (2) a much better accuracy to fit a trigonometric surface; and (3) a similar interpolation stability without increase of the condition number of the RBF interpolation matrix. Therefore, it has proven that the enriched RBF shape function will not only possess all advantages of the conventional RBF shape function, but also can accurately reflect the properties of stresses for the crack tip fields. The system of equations for the crack analysis is then derived based on the enriched RBF meshfree shape function and the meshfree weak-form. Several problems of linear fracture mechanics are simulated using this newlydeveloped e-RPIM method. It has demonstrated that the present e-RPIM is very accurate and stable, and it has a good potential to develop a practical simulation tool for fracture mechanics problems.

An advanced implicit meshless approach for fractional partial differential equation in computational mechanics

Recently, the numerical modelling and simulation for fractional partial differential equations (FPDE), which have been found with widely applications in modern engineering and sciences, are attracting increased attentions. The current dominant numerical method for modelling of FPDE is the explicit Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of time fractional diffusion equations. The discrete system of equations is obtained by using the RBF meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modelling and simulation for FPDE.

Sequential development of algebra knowledge : a cognitive analysis

Learning to operate algebraically is a complex process that is dependent upon extending arithmetic knowledge to the more complex concepts of algebra. Current research has shown a gap between arithmetic and algebraic knowledge and suggests a pre-algebraic level as a step between the two knowledge types. This paper examines arithmetic and algebraic knowledge from a cognitive perspective in an effort to determine what constitutes a pre-algebraic level of understanding. Results of a longitudinal study designed to investigate students' readiness for algebra are presented. Thirty-three students in Grades 7, 8, and 9 participated. A model for the transition from arithmetic to pre-algebra to algebra is proposed and students' understanding of relevant knowledge is discussed.