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.

Incorporating uncertainty in environmental models informed by imagery

In this thesis, the issue of incorporating uncertainty for environmental modelling informed by imagery is explored by considering uncertainty in deterministic modelling, measurement uncertainty and uncertainty in image composition. Incorporating uncertainty in deterministic modelling is extended for use with imagery using the Bayesian melding approach. In the application presented, slope steepness is shown to be the main contributor to total uncertainty in the Revised Universal Soil Loss Equation. A spatial sampling procedure is also proposed to assist in implementing Bayesian melding given the increased data size with models informed by imagery. Measurement error models are another approach to incorporating uncertainty when data is informed by imagery. These models for measurement uncertainty, considered in a Bayesian conditional independence framework, are applied to ecological data generated from imagery. The models are shown to be appropriate and useful in certain situations. Measurement uncertainty is also considered in the context of change detection when two images are not co-registered. An approach for detecting change in two successive images is proposed that is not affected by registration. The procedure uses the Kolmogorov-Smirnov test on homogeneous segments of an image to detect change, with the homogeneous segments determined using a Bayesian mixture model of pixel values. Using the mixture model to segment an image also allows for uncertainty in the composition of an image. This thesis concludes by comparing several different Bayesian image segmentation approaches that allow for uncertainty regarding the allocation of pixels to different ground components. Each segmentation approach is applied to a data set of chlorophyll values and shown to have different benefits and drawbacks depending on the aims of the analysis.

Improved correlation-based modal strain energy method for global damage detection of truss bridge structures

The previous investigations have shown that the modal strain energy correlation method, MSEC, could successfully identify the damage of truss bridge structures. However, it has to incorporate the sensitivity matrix to estimate damage and is not reliable in certain damage detection cases. This paper presents an improved MSEC method where the prediction of modal strain energy change vector is differently obtained by running the eigensolutions on-line in optimisation iterations. The particular trail damage treatment group maximising the fitness function close to unity is identified as the detected damage location. This improvement is then compared with the original MSEC method along with other typical correlation-based methods on the finite element model of a simple truss bridge. The contributions to damage detection accuracy of each considered mode is also weighed and discussed. The iterative searching process is operated by using genetic algorithm. The results demonstrate that the improved MSEC method suffices the demand in detecting the damage of truss bridge structures, even when noised measurement is considered.

Creation of a validated 3D finite element model of an ovine tibia

Introduction Ovine models are widely used in orthopaedic research. To better understand the impact of orthopaedic procedures computer simulations are necessary. 3D finite element (FE) models of bones allow implant designs to be investigated mechanically, thereby reducing mechanical testing. Hypothesis We present the development and validation of an ovine tibia FE model for use in the analysis of tibia fracture fixation plates. Material & Methods Mechanical testing of the tibia consisted of an offset 3-pt bend test with three repetitions of loading to 350N and return to 50N. Tri-axial stacked strain gauges were applied to the anterior and posterior surfaces of the bone and two rigid bodies – consisting of eight infrared active markers, were attached to the ends of the tibia. Positional measurements were taken with a FARO arm 3D digitiser. The FE model was constructed with both geometry and material properties derived from CT images of the bone. The elasticity-density relationship used for material property determination was validated separately using mechanical testing. This model was then transformed to the same coordinate system as the in vitro mechanical test and loads applied. Results Comparison between the mechanical testing and the FE model showed good correlation in surface strains (difference: anterior 2.3%, posterior 3.2%). Discussion & Conclusion This method of model creation provides a simple method for generating subject specific FE models from CT scans. The use of the CT data set for both the geometry and the material properties ensures a more accurate representation of the specific bone. This is reflected in the similarity of the surface strain results.

An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation

Recently, the numerical modelling and simulation for anomalous subdiffusion equation (ASDE), which is a type of fractional partial differential equation( FPDE) and has been found with widely applications in modern engineering and sciences, are attracting more and more attentions. The current dominant numerical method for modelling ASDE is 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 the non-linear ASDE. The discrete system of equations is obtained by using the 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 formulations 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 modeling and simulation of the ASDE. Therefore, the meshless technique should have good potential in development of a robust simulation tool for problems in engineering and science which are governed by the various types of fractional differential equations.

A point interpolation method with locally smoothed strain field (PIM-LS2) for mechanics problems using triangular mesh

A point interpolation method with locally smoothed strain field (PIM-LS2) is developed for mechanics problems using a triangular background mesh. In the PIM-LS2, the strain within each sub-cell of a nodal domain is assumed to be the average strain over the adjacent sub-cells of the neighboring element sharing the same field node. We prove theoretically that the energy norm of the smoothed strain field in PIM-LS2 is equivalent to that of the compatible strain field, and then prove that the solution of the PIM- LS2 converges to the exact solution of the original strong form. Furthermore, the softening effects of PIM-LS2 to system and the effects of the number of sub-cells that participated in the smoothing operation on the convergence of PIM-LS2 are investigated. Intensive numerical studies verify the convergence, softening effects and bound properties of the PIM-LS2, and show that the very ‘‘tight’’ lower and upper bound solutions can be obtained using PIM-LS2.

Elastic shear buckling characteristics of LiteSteel Beams

LiteSteel beam (LSB) is a new cold-formed steel hollow flange channel beam. The unique LSB section is produced by a patented manufacturing process involving simultaneous cold-forming and dual electric resistance welding. To date, limited research has been undertaken on the shear buckling behaviour of LSBs with torsionally rigid, rectangular hollow flanges. For the shear design of LSB web panels, their elastic shear buckling strength must be determined accurately including the potential post-buckling strength. Currently the elastic shear buckling coefficients of web panels are determined by assuming conservatively that the web panels are simply supported at the junction between the flange and web elements. Therefore finite element analyses were carried out to investigate the elastic shear buckling behaviour of LSB sections including the effect of true support conditions at the junction between their flange and web elements. An improved equation for the higher elastic shear buckling coefficient of LSBs was developed and included in the shear capacity equations of Australian cold-formed steel codes. Predicted ultimate shear capacity results were compared with available experimental results, both of which showed considerable improvement to the shear capacities of LSBs. A study on the shear flow distribution of LSBs was also undertaken prior to the elastic buckling analysis study. This paper presents the details of this investigation and the results including the shear flow distribution of LSBs. Keywords: LiteSteel beam, Elastic shear buckling, Shear flow, Cold-formed steel structures, Slender web, Hollow flanges.