In the thesis we study the elliptic curves and its use in cryptography. Elliptic curvesencompasses a vast area of mathematics. Elliptic curves have basics in group theory andnumber theory. The points on elliptic curve forms a group under the operation of addition.We study the structure of this group. We describe Hasse’s theorem to estimate the numberof points on the curve. We also discuss that the elliptic curve group may or may not becyclic over finite fields. Elliptic curves have applications in cryptography, we describe theapplication of elliptic curves for discrete logarithm problem and ElGamal cryptosystem.

This work is composed of two independent parts, both addressing problems related to algebraic curves over finite fields. In the first part, we characterize all irreducible plane curves defined over Fq which are Frobenius non-classical for different powers of q. Such characterization gives rise to many previously unknown curves which turn out to have some interesting properties. For instance, for n [greater-than or equal to] 3 a curve which is both q- and qn-Frobenius non-classical will have its number of Fqn-rational points attaining the Stöhr-Voloch bound. In the second part, we study the arc property of several plane curves and present new complete (N, d)-arcs in PG(2, q). Some of these arcs (viewed as linear (N, 3,N - d)-codes) are just a small constant away from the Griesmer bound and for some small values of q the bound is achieved. In addition, this part also answers a question of Voloch about the arc property of a certain family of curves with many rational points, and another question of Giulietti et al about the arc property of q-Frobenius non-classical plane curves. / text

An important part of mathematics is the construction of good definitions. Some things, like planar graphs, are trivial to define, and other concepts, like compact sets, arise from putting a name on often used requirements (although the notion of compactness has changed over time to be more general). In other cases, such as in set theory, the natural definitions may yield undesired and even contradictory results, and it can be necessary to use a more complicated formalization.    The notion of a curve falls in the latter category. While it is intuitively clear what a curve is – line segments, empty geometric shapes, and squiggles like this: – it is not immediately clear how to make a general definition of curves. Their most obvious characteristic is that they have no width, so one idea may be to view curves as what can be drawn with a thin pen. This definition, however, has the weakness that even such a line has the ability to completely fill a square, making it a bad definition of curves. Today curves are generally defined by the condition of having no width, that is, being one-dimensional, together with the conditions of being compact and connected, to avoid strange cases.    In this thesis we investigate this definition and a few examples of curves.

Compliant suspensions are often used in micromechanical devices and precision mechanisms as substitutes for kinematic joints. While their small-displacement behavior is easily captured in simple formulae, large-displacement behavior requires nonlinear finite element analysis. In this work, we present a method that helps capture the geometrically nonlinear behavior of compliant suspensions using parameterized non-dimensional maps. The maps are created by performing one nonlinear finite element analysis for any one loading condition for one instance of a suspension of a given topology and fixed proportions. These maps help retrieve behavioral information for any other instance of the same suspension with changed size, cross-section dimensions, material, and loading. Such quantities as multi-axial stiffness, maximum stress, natural frequency, etc. ,can be quickly and accurately estimated from the maps. These quantities are non-dimensionalized using suitable factors that include loading, size, cross-section, and material properties. The maps are useful in not only understanding the limits of performance of the topology of a given suspension with fixed proportions but also in design. We have created the maps for 20 different suspensions. Case studies are included to illustrate the effectiveness of the method in microsystem design as well as in precision mechanisms. In particular, the method and 2D plots of non-dimensional kinetoelastic maps provide a comprehensive view of sensitivity, cross-axis sensitivity, linearity, maximum stress, and bandwidth for microsensors and microactuators.

We investigate in this thesis the generic properties of curves in the Minkowski plane R2 1 and of smooth Lorentzian surfaces. The generic properties of curves in R2 1 are obtained by studying the contacts of curves in R2 1 with lines and pseudo-circles. These contacts are captured by the singularities of the families of height and distancesquared functions on the curves. On the other hand, the generic properties of smooth Lorentzian surfaces are obtained by studying certain Binary Differential Equations defined on the surfaces.

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This thesis investigates two problems that are discrete analogues of two harmonic analytic problems which lie in the heart of research in the field. More specifically, we consider discrete analogues of the maximal Kakeya operator conjecture and of the recently solved endpoint multilinear Kakeya problem, by effectively shrinking the tubes involved in these problems to lines, thus giving rise to the problems of counting joints and multijoints with multiplicities. In fact, we effectively show that, in R3, what we expect to hold due to the maximal Kakeya operator conjecture, as well as what we know in the continuous case due to the endpoint multilinear Kakeya theorem by Guth, still hold in the discrete case. In particular, let L be a collection of L lines in R3 and J the set of joints formed by L, that is, the set of points each of which lies in at least three non-coplanar lines of L. It is known that |J| = O(L3/2) ( first proved by Guth and Katz). For each joint x ∈ J, let the multiplicity N(x) of x be the number of triples of non-coplanar lines through x. We prove here that X x2J N(x)1=2 = O(L3=2); while we also extend this result to real algebraic curves in R3 of uniformly bounded degree, as well as to curves in R3 parametrized by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, involving three finite collections of lines in R3; a multijoint formed by them is a point that lies in (at least) three non-coplanar lines, one from each collection. We finally present some results regarding the joints problem in different field settings and higher dimensions.