Graph coloring applied to secure computation in non-Abelian groups

We study the natural problem of secure n-party computation (in the computationally unbounded attack model) of circuits over an arbitrary finite non-Abelian group (G,⋅), which we call G-circuits. Besides its intrinsic interest, this problem is also motivating by a completeness result of Barrington, stating that such protocols can be applied for general secure computation of arbitrary functions. For flexibility, we are interested in protocols which only require black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our investigations focus on the passive adversarial model, where up to t of the n participating parties are corrupted.

On secure multi-party computation in black-box groups

We study the natural problem of secure n-party computation (in the passive, computationally unbounded attack model) of the n-product function f G (x 1,...,x n ) = x 1 ·x 2 ⋯ x n in an arbitrary finite group (G,·), where the input of party P i is x i  ∈ G for i = 1,...,n. For flexibility, we are interested in protocols for f G which require only black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our results are as follows. First, on the negative side, we show that if (G,·) is non-abelian and n ≥ 4, then no ⌈n/2⌉-private protocol for computing f G exists. Second, on the positive side, we initiate an approach for construction of black-box protocols for f G based on k-of-k threshold secret sharing schemes, which are efficiently implementable over any black-box group G. We reduce the problem of constructing such protocols to a combinatorial colouring problem in planar graphs. We then give two constructions for such graph colourings. Our first colouring construction gives a protocol with optimal collusion resistance t < n/2, but has exponential communication complexity O(n*2t+1^2/t) group elements (this construction easily extends to general adversary structures). Our second probabilistic colouring construction gives a protocol with (close to optimal) collusion resistance t < n/μ for a graph-related constant μ ≤ 2.948, and has efficient communication complexity O(n*t^2) group elements. Furthermore, we believe that our results can be improved by further study of the associated combinatorial problems.

Finite element analysis of vulnerable atherosclerotic plaques: A comparison of mechanical stresses within carotid plaques of acute and recently symptomatic patients with carotid artery disease

Objectives: There is considerable evidence that patients with carotid artery stenosis treated immediately after the ischaemic cerebrovascular event have a better clinical outcome than those who have delayed treatment. Biomechanical assessment of carotid plaques using high-resolution MRI can help examine the relationship between the timing of carotid plaque symptomology and maximum simulated plaque stress concentration. Methods: Fifty patients underwent high-resolution multisequence in vivo MRI of their carotid arteries. Patients with acute symptoms (n=25) underwent MRI within 72 h of the onset of ischaemic cerebrovascular symptoms, whereas recently symptomatic patients (n=25) underwent MRI from 2 to 6 weeks after the onset of symptoms. Stress analysis was performed based on the geometry derived from in vivo MRI of the symptomatic carotid artery at the point of maximum stenosis. The peak stresses within the plaques of the two groups were compared. Results: Patient demographics were comparable for both groups. All the patients in the recently symptomatic group had severe carotid stenosis in contrast to patients with acute symptoms who had predominantly mild to moderate carotid stenosis. The simulated maximum stresses in patients with acute symptoms was significantly higher than in recently symptomatic patients (median (IQR): 313310 4 dynes/cm 2 (295 to 382) vs 2523104 dynes/cm 2 (236 to 311), p=0.02). Conclusions: Patients have extremely unstable, high-risk plaques, with high stresses, immediately after an acute cerebrovascular event, even at lower degrees of carotid stenoses. Biomechanical stress analysis may help us refine our risk-stratification criteria for the management of patients with carotid artery disease in future.

On the zeros of the Abelian integrals for a class of Liénard systems

A planar polynomial differential system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for the general nonlinear dynamical systems. In this paper, we investigated a class of Liénard systems of the form x'=y, y'=f(x)+y g(x) with deg f=5 and deg g=4. We proved that the related elliptic integrals of the Liénard systems have at most three zeros including multiple zeros, which implies that the number of limit cycles bifurcated from the periodic orbits of the unperturbed system is less than or equal to 3.

An experimental and finite element investigation of the biomechanics of vertebral compression fractures

Osteoporosis is a disease characterized by low bone mass and micro-architectural deterioration of bone tissue, with a consequent increase in bone fragility and susceptibility to fracture. Osteoporosis affects over 200 million people worldwide, with an estimated 1.5 million fractures annually in the United States alone, and with attendant costs exceeding $10 billion dollars per annum. Osteoporosis reduces bone density through a series of structural changes to the honeycomb-like trabecular bone structure (micro-structure). The reduced bone density, coupled with the microstructural changes, results in significant loss of bone strength and increased fracture risk. Vertebral compression fractures are the most common type of osteoporotic fracture and are associated with pain, increased thoracic curvature, reduced mobility, and difficulty with self care. Surgical interventions, such as kyphoplasty or vertebroplasty, are used to treat osteoporotic vertebral fractures by restoring vertebral stability and alleviating pain. These minimally invasive procedures involve injecting bone cement into the fractured vertebrae. The techniques are still relatively new and while initial results are promising, with the procedures relieving pain in 70-95% of cases, medium-term investigations are now indicating an increased risk of adjacent level fracture following the procedure. With the aging population, understanding and treatment of osteoporosis is an increasingly important public health issue in developed Western countries. The aim of this study was to investigate the biomechanics of spinal osteoporosis and osteoporotic vertebral compression fractures by developing multi-scale computational, Finite Element (FE) models of both healthy and osteoporotic vertebral bodies. The multi-scale approach included the overall vertebral body anatomy, as well as a detailed representation of the internal trabecular microstructure. This novel, multi-scale approach overcame limitations of previous investigations by allowing simultaneous investigation of the mechanics of the trabecular micro-structure as well as overall vertebral body mechanics. The models were used to simulate the progression of osteoporosis, the effect of different loading conditions on vertebral strength and stiffness, and the effects of vertebroplasty on vertebral and trabecular mechanics. The model development process began with the development of an individual trabecular strut model using 3D beam elements, which was used as the building block for lattice-type, structural trabecular bone models, which were in turn incorporated into the vertebral body models. At each stage of model development, model predictions were compared to analytical solutions and in-vitro data from existing literature. The incremental process provided confidence in the predictions of each model before incorporation into the overall vertebral body model. The trabecular bone model, vertebral body model and vertebroplasty models were validated against in-vitro data from a series of compression tests performed using human cadaveric vertebral bodies. Firstly, trabecular bone samples were acquired and morphological parameters for each sample were measured using high resolution micro-computed tomography (CT). Apparent mechanical properties for each sample were then determined using uni-axial compression tests. Bone tissue properties were inversely determined using voxel-based FE models based on the micro-CT data. Specimen specific trabecular bone models were developed and the predicted apparent stiffness and strength were compared to the experimentally measured apparent stiffness and strength of the corresponding specimen. Following the trabecular specimen tests, a series of 12 whole cadaveric vertebrae were then divided into treated and non-treated groups and vertebroplasty performed on the specimens of the treated group. The vertebrae in both groups underwent clinical-CT scanning and destructive uniaxial compression testing. Specimen specific FE vertebral body models were developed and the predicted mechanical response compared to the experimentally measured responses. The validation process demonstrated that the multi-scale FE models comprising a lattice network of beam elements were able to accurately capture the failure mechanics of trabecular bone; and a trabecular core represented with beam elements enclosed in a layer of shell elements to represent the cortical shell was able to adequately represent the failure mechanics of intact vertebral bodies with varying degrees of osteoporosis. Following model development and validation, the models were used to investigate the effects of progressive osteoporosis on vertebral body mechanics and trabecular bone mechanics. These simulations showed that overall failure of the osteoporotic vertebral body is initiated by failure of the trabecular core, and the failure mechanism of the trabeculae varies with the progression of osteoporosis; from tissue yield in healthy trabecular bone, to failure due to instability (buckling) in osteoporotic bone with its thinner trabecular struts. The mechanical response of the vertebral body under load is highly dependent on the ability of the endplates to deform to transmit the load to the underlying trabecular bone. The ability of the endplate to evenly transfer the load through the core diminishes with osteoporosis. Investigation into the effect of different loading conditions on the vertebral body found that, because the trabecular bone structural changes which occur in osteoporosis result in a structure that is highly aligned with the loading direction, the vertebral body is consequently less able to withstand non-uniform loading states such as occurs in forward flexion. Changes in vertebral body loading due to disc degeneration were simulated, but proved to have little effect on osteoporotic vertebra mechanics. Conversely, differences in vertebral body loading between simulated invivo (uniform endplate pressure) and in-vitro conditions (where the vertebral endplates are rigidly cemented) had a dramatic effect on the predicted vertebral mechanics. This investigation suggested that in-vitro loading using bone cement potting of both endplates has major limitations in its ability to represent vertebral body mechanics in-vivo. And lastly, FE investigation into the biomechanical effect of vertebroplasty was performed. The results of this investigation demonstrated that the effect of vertebroplasty on overall vertebra mechanics is strongly governed by the cement distribution achieved within the trabecular core. In agreement with a recent study, the models predicted that vertebroplasty cement distributions which do not form one continuous mass which contacts both endplates have little effect on vertebral body stiffness or strength. In summary, this work presents the development of a novel, multi-scale Finite Element model of the osteoporotic vertebral body, which provides a powerful new tool for investigating the mechanics of osteoporotic vertebral compression fractures at the trabecular bone micro-structural level, and at the vertebral body level.

Mechanisms of change in negative thinking and urinary monoamines in depressed patients during acute treatment with group cognitive behavior therapy and antidepressant medication

This naturalistic study investigated the mechanisms of change in measures of negative thinking and in 24-h urinary metabolites of noradrenaline (norepinephrine), dopamine and serotonin in a sample of 43 depressed hospital patients attending an eight-session group cognitive behavior therapy program. Most participants (91%) were taking antidepressant medication throughout the therapy period according to their treating Psychiatrists' prescriptions. The sample was divided into outcome categories (19 Responders and 24 Non-responders) on the basis of a clinically reliable change index [Jacobson, N.S., & Truax, P., 1991. Clinical significance: a statistical approach to defining meaningful change in psychotherapy research. Journal of Consulting and Clinical Psychology, 59, 12–19.] applied to the Beck Depression Inventory scores at the end of the therapy. Results of repeated measures analysis of variance [ANOVA] analyses of variance indicated that all measures of negative thinking improved significantly during therapy, and significantly more so in the Responders as expected. The treatment had a significant impact on urinary adrenaline and metadrenaline excretion however, these changes occurred in both Responders and Non-responders. Acute treatment did not significantly influence the six other monoamine metabolites. In summary, changes in urinary monoamine levels during combined treatment for depression were not associated with self-reported changes in mood symptoms.

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.

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.

Effect of poly(acrylic acid) end-group functionality on inhibition of calcium oxalate crystal growth

A number of series of poly(acrylic acids) (PAA) of differing end-groups and molecular weights prepared using atom transfer radical polymerization were used as inhibitors for the crystallization of calcium oxalate at 23 and 80°C. As measured by turbidimetry and conductivity and as expected from previous reports, all PAA series were most effective for inhibition of crystallization at molecular weights of 1500–4000. However, the extent of inhibition was in general strongly dependent on the hydrophobicity and molecular weight of the end-group. These results may be explicable in terms of adsorption/desorption of PAA to growth sites on crystallites. The overall effectiveness of the series didn't follow a simple trend with end-group hydrophobicity, suggesting self-assembly behavior or a balance between adsorption and desorption rates to crystallite surfaces may be critical in the mechanism of inhibition of calcium oxalate crystallization.

Effect of poly(acrylic acid) molecular mass and end-group functionality on calcium oxalate crystal morphology and growth

A number of series of poly(acrylic acids) (PAA) of differing end-groups and molecular mass were used to study the inhibition of calcium oxalate crystallization. The effects of the end-group on crystal speciation and morphology were significant and dramatic, with hexyl-isobutyrate end groups giving preferential formation of calcium oxalate dihydrate (COD) rather than the more stable calcium oxalate monohydrate (COM), while both more hydrophobic end-groups and less-hydrophobic end groups led predominantly to formation of the least thermodynamically stable form of calcium oxalate, calcium oxalate trihydrate. Conversely, molecular mass had little impact on calcium oxalate speciation or crystal morphology. It is probable that the observed effects are related to the rate of desorption of the PAA moiety from the crystal (lite) surfaces and that the results point to a major role for end-group as well as molecular mass in controlling desorption rate.

Optimism about safety and group-serving interpretations of safety among pedestrians and cyclists in relation to road use in general and under low light conditions

Drivers are known to be optimistic about their risk of crash involvement, believing that they are less likely to be involved in a crash than other drivers. However, little comparative research has been conducted among other road users. In addition, optimism about crash risk is conceptualised as applying only to an individual’s assessment of his or her personal risk of crash involvement. The possibility that the self-serving nature of optimism about safety might be generalised to the group-level as a cyclist or a pedestrian, i.e., becoming group-serving rather than self-serving, has been overlooked in relation to road safety. This study analysed a subset of data collected as part of a larger research project on the visibility of pedestrians, cyclists and road workers, focusing on a set of questionnaire items administered to 406 pedestrians, 838 cyclists and 622 drivers. The items related to safety in various scenarios involving drivers, pedestrians and cyclists, allowing predictions to be derived about group differences in agreement with items based on the assumption that the results would exhibit group-serving bias. Analysis of the responses indicated that specific hypotheses about group-serving interpretations of safety and responsibility were supported in 22 of the 26 comparisons. When the nine comparisons relevant to low lighting conditions were considered separately, seven were found to be supported. The findings of the research have implications for public education and for the likely acceptance of messages which are inconsistent with current assumptions and expectations of pedestrians and cyclists. They also suggest that research into group-serving interpretations of safety, even for temporary roles rather than enduring groups, could be fruitful. Further, there is an implication that gains in safety can be made by better educating road users about the limitations of their visibility and the ramifications of this for their own road safety, particularly in low light.