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<p>The capability of encrypting top secret information remains as a major research problem in the GGH cryptosystem, which depends on various attacking methods. The early approaches to attacking the GGH cryptosystem mainly relied on special properties of the lattice generated by the vectors of the private key. Consequently, those attacks are not appropriate for general cases.</p> <p>This thesis presents a GGH attacking method for general cases. A lattice basis reduction algorithm is applied to the public key to get a better basis, which is used to decrypt the ciphertext. In the proposed approach, we concentrate on three lattice reduction algorithms: the LLL algorithm, the approximate optimally-reduced algorithm, and the optimally-reduced algorithm. We have implemented a package in MATLAB for the GGH cryptosystem and the three algorithms. We experimented with two groups of experiments and obtained promising results for lattices of low dimensions.</p> / Thesis / Master of Science (MSc)

This dissertation presents the development of numerical models based on lattice Boltzmann (LB) and cellular automaton (CA) methods for solving phase change and microstructural evolution problems. First, a new variation of the LB method is discussed for solving the heat conduction problem with phase change. In contrast to previous explicit algorithms, the latent heat source term is treated implicitly in the energy equation, avoiding iteration steps and improving the formulation stability and efficiency. The results showed that the model can deal with phase change problems more accurately and efficiently than explicit LB models. Furthermore, a new numerical technique is introduced for simulating dendrite growth in three dimensions. The LB method is used to calculate the transport phenomena and the CA is employed to capture the solid/liquid interface. It is assumed that the dendritic growth is driven by the difference between the local actual and local equilibrium composition of the liquid in the interface. The evolution of a threedimensional (3D) dendrite is discussed. In addition, the effect of undercooling and degree of anisotropy on the kinetics of dendrite growth is studied. Moreover, effect of melt convection on dendritic solidification is investigated using 3D simulations. It is shown that convection can change the kinetics of growth by affecting the solute distribution around the dendrite. The growth features of twodimensional (2D) and 3D dendrites are compared. Furthermore, the change in growth kinetics and morphology of Al-Cu dendrites is studied by altering melt undercooling, alloy composition and inlet flow velocity. The local-type nature of LB and CA methods enables efficient scaling of the model in petaflops supercomputers, allowing the simulation of large domains in 3D. The model capabilities with large scale simulations of dendritic solidification are discussed and the parallel performance of the algorithm is assessed. Excellent strong scaling up to thousands of computing cores is obtained across the nodes of a computer cluster, along with near-perfect weak scaling. Considering the advantages offered by the presented model, it can be used as a new tool for simulating 3D dendritic solidification under convection.

<p> We investigated the local electronic properties of the triangular-lattice materials NaxCO2 (x = 0.3, 0.72) and the superconductor Na0.3Co02-1.3H20 by 59Co and 170 Nuclear Magnetic Resonance(NMR). For Na0.72Co02 , 59Co NMR line shape shows clearly that there are two types of Co sites - Co(A) site and Co(B) site. The electronic character of Co(A) site is close to that of the less magnetic Co+3-like ion with spin rv 0, while the electronic character of Co(B) site is close to that of the strongly magnetic Co+4-like ions with spin ~ 1/2. The temperature dependence of the Knight shifts suggests that the Co(A) and Co(B) sites are electronically coupled, which is not consistent with simple phase separation. The local Co electronic environments propagate to the adjacent 0 layers through p-d hybridization. Therefore, there are two types of oxygen sites, O(A) site and O(B) site. We introduced a different route to do K vs x plot analysis for the Co sites and determined that for the Co sites, Karbitat(A) is 1.816 % and Kspin(A) is about rv 0.2 %; Karbitat(B) is 4.0255 % and Kspin(B) is at least 1.5 %. For Na0 .3Co02 , 59Co NMR line shape shows that the Co valence is averaged out in this material. There are two types of oxygen sites, 0( C) site and O(D) site, presumably because of the nearest neighbor Na+ sites. The constant behavior of Knight shifts below 100 K for both the Co and 0 sites suggests the emergence of a low temperature canonical Fermi-liquid behavior. For the superconductor Na0.3CoO2-1.3H20, both 59Co and 170 NMR line shapes show that there is only one type of Co site and oxygen site. The Knight shifts of 59Co and 170 are temperature independent below 100 K down to Tc. Combined with our spin-lattice relaxation 1/T 1 T measurements, we can rule out the possibility of ferromagnetic scenario of spin excitations above Tc. </p> / Thesis / Master of Science (MSc)

<p> Entropy of mixing for random mass defects is examined and equations for the phonon contribution are developed. The Green's function method used gives the entropy change due to the phonons at all temperatures, but for experimental comparison the high temperature region is used. There simple formulae obtain, but the mass defect is not sufficient to account for the observed entropy changes. This suggests that further work is necessary. </p> / Thesis / Master of Science (MSc)

The procedure introduced by Cornish⁴ for estimating a missing value in a lattice design, has been extended here and modified to include not only intra-block information, but also inter-block information as well. The analysis of the lattice design has been presented with some simplification applicable to this type of lattice; thus, allowing a more uniform development than has appeared before. The implications in the limit were examined for the use of the modification of intra-block estimation formula, and it was shown that the form used was an appropriate practical tool. An actual experiment was analyzed, where certain plots were considered missing, and it was shown that the additional use of the inter-block information, as evidenced by the modified formula, would give a much better estimate of a missing value than had been possible previously in the presence of significant inter-block variation. The lattice design has thus been made stable in the event of loss of and plot value. This design has been proven useful in industrial analysis, and its utility may now be protected by a more complete recovery of missing information. / M.S.

Ultra-High Performance Concrete (UHPC) is an evolving structural material that has attracted interest in the civil engineering industry recently. Currently, it is being used mainly for highway infrastructure in the US and also being explored for various other applications. The existing design guides on UHPC in countries like Japan, Australia and France are not as detailed as the concrete or steel guides. Most of the sections made of UHPC are slender due to its superior mechanical properties which are expected to simplify construction. Being an expensive material to use, making slender sections also helps to minimize the overall cost of the structure and makes it competitive with that of high strength steel and prestressed concrete. It has also been demonstrated to have very high compressive strength and considerable tensile strength. To begin with, an introduction on UHPC and its current applications around the world is presented, followed by a review of the existing design guides on UHPC. The importance as well as the methodology to measure fracture energy of concrete with factors to be considered for fiber reinforced concrete is discussed in detail. The main motive of this research is to introduce a creative modeling concept which served as the theoretical basis for the development of a computer program called Lattice 3D. The program is a modeling tool for engineers studying the behavior of UHPC, and in the future will be developed into a finite element protocol for analyzing complex structures made of UHPC. Parametric studies on lattice models of thin simply supported plates in compression and three-point bending of beams have been demonstrated in this research. Experimental tests conducted on briquette specimens under uniaxial tension are also discussed. / Master of Science

Lattice structures are widely used in nowadays industries in combination with additive manufacturing technology to obtain components with a limited weight and tuneable mechanical properties. However, industries still find challenging a complete implementation of these metamaterials in the product development due to the complexity given by an accurate prediction of the mechanical and fatigue properties. To overcome this limitation, analytical and numerical techniques are developed, to help designers to achieve the desired performances. Finite Element simulations are a common tool utilized in this sense, where solid models can provide accurate results. Nevertheless, the implementation of this technique requires high computational costs, often not compatible with an iterative design process where versions of the component are constantly updated considering the feedback provided by actors having different backgrounds and product interactions. Accurate and computationally efficient simulations strategies are thus required. The proposed thesis investigates three possible simulations ideas able to describe the mechanical properties of the lattice-based components. Two main properties are studied: the lattice structure elastic behaviour, which is important to determine the in-service behaviour of the designed component and the fatigue resistance, which defines the component service duration. Homogenization technique is the first numerical method analysed and it is pivoted on the idea of substituting the intricate lattice geometries with a solid fictitious material displaying the same elastic properties. In this framework, a case study is analysed, where the design process of a total hip replacement prosthetic device is developed. The workflow starts with a preliminary experimental campaign on lattice specimens with the aim of determining the printing quality, the mechanical properties, and the biological characteristics. In this phase, a verification of the homogenization predictions is performed. On this base, the best specimens’ configurations are selected to design and manufacture the prosthetic device. The second simulation technique leverages on the observation of the onedimensional nature of the strut-based lattice structures. Lattice structures’ behaviour can thus be simulated through the usage of truss and beam elements, depending on the stretching or bending dominated nature of the lattice topologies. Based on this observation, two different paths are followed, the first one aiming to improve the fatigue life of lattice components by acting of their orientation in the printing chamber. It is known that printing orientation influences the surface quality of the components and, in lattice struts this effect can be directly linked to a variation in the fatigue life. An optimization algorithm is thus developed, aiming to optimize the fatigue resistance of the manufactured components. Following this idea, a control and an optimized lattice batch are printed and an improvement in the fatigue resistance is found, even if not as large as expected by the simulations. Improvements in the predictions can be observed if the as-build geometry of the struts is considered. The second path is devoted to the computation of the corrective coefficients able to properly describe the elastic properties of bending dominated lattice structures. One-dimensional simulations are normally too severe for bending dominated lattice topologies, and a compensation has to be provided to match the elastic properties calculated trough computational efficient beam models and lattice ones. To address this problem, an optimization routine is developed, where the compensation factors are computed comparing the elastic properties of the beam models and a homogenised solid model taken as reference. A benchmark testing between the beam model, - built with the so computed compensation coefficients - a homogenised, and a solid model is developed. Compensated beam models are found to be able to improve the predictions of lattice structures elastic properties if compared to the homogenization techniques, showing a comparable computational time. Nevertheless, a reduced accuracy is found in presence of dense lattice structures, where the hypothesis of one-dimensional is weaker. The third analysed simulation method aims to obtain a precise fatigue life estimation at the expense of computational time. Starting from an as-build geometry reconstructed trough CT-scan analysis, a finite element simulation built with solid elements is performed. To reduce the computational cost, an innovative finite element theory is adopted, the Finite Cell Method. A two-step simulation is performed, and thanks to the usage of the average strain energy density method, the fatigue life estimation can be obtained. An excellent agreement is found; however, a complete validation is required for this method before its safe implementation in the design process.

Three different definitions for the determinant of a matrix over arbitrary lattices have been developed to determine which properties and relations were reminiscent of the determinant or permanent of elementary algebra. In each determinant there are properties concerning: the elements of the matrix in the expansion of its determinant; the determinant of a matrix and its transpose; a principle of duality for rows and columns; the interchange of rows and columns; the determinant of a matrix formed from another by a row or column meet of certain elements; and evaluations of certain special matrices. An expansion by row or column is given for one determinant and a lemma on inverses is proven in light of another. A preliminary section on Lattice Theory is also included. / Master of Science