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Elliptic curves have a rich mathematical history dating back to Diophantus (c. 250 C.E.), who used a form of these cubic equations to find right triangles of integer area with rational sides. In more recent times the deep mathematics of elliptic curves was used by Andrew Wiles et. al., to construct a proof of Fermat's last theorem, a problem which challenged mathematicians for more than 300 years. In addition, elliptic curves over finite fields find practical application in the areas of cryptography and coding theory. For such problems, knowing the order of the group of points satisfying the elliptic curve equation is important to the security of these applications. In 1985 René Schoof published a paper [5] describing a polynomial time algorithm for solving this problem. In this thesis we explain some of the key mathematical principles that provide the basis for Schoof's method. We also present an implementation of Schoof's algorithm as a collection of Mathematica functions. The operation of each algorithm is illustrated by way of numerical examples. / Master of Science

Elliptic curves are the basis for a relative new class of public-key schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. Efficient algorithms for elliptic curves can be classified into low-level algorithms, which deal with arithmetic in the underlying finite field and high-level algorithms, which operate with the group operation. This thesis describes three new algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm describes the application of the Karatsuba-Ofman Algorithm to multiplication in composite fields GF((2n)m). The second algorithm deals with efficient inversion in composite Galois fields of the form GF((2n)m). The third algorithm is an entirely new approach which accelerates the multiplication of points which is the core operation in elliptic curve public-key systems. The algorithm explores computational advantages by computing repeated point doublings directly through closed formulae rather than from individual point doublings. Finally we apply all three algorithms to an implementation of an elliptic curve system over GF((216)11). We provide ablolute performance measures for the field operations and for an entire point multiplication. We also show the improvements gained by the new point multiplication algorithm in conjunction with the k-ary and improved k-ary methods for exponentiation.

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.

This thesis aims to prove the following statement, where the Weierstrass p-function has algebraic invariants and complex multiplication by Q(alpha): "If beta_1,..., beta_n are algebraic numbers which are linearly independent over Q(alpha), then p(beta_1),...,p(beta_n) are algebraically independent over Q." This was proven by Philippon in 1983, and the proof in this thesis follows his ideas. The difference lies in the strength of the tools used, allowing certain arguments to be simplified. This thesis shows that the above result is equivalent to imposing the restriction (beta_1,...,beta_n)=(1,beta,...,beta^{n-1}), where n=[Q(alpha,beta):Q(alpha)]. The core of the proof consists of developing height estimates, constructing representations for morphisms between products of elliptic curves, and finding height and degree estimates on large families of polynomials which are small at a point in Q(alpha,beta,g_2,g_3)(p(1),p'(1),...,p(beta^{n-1}),p'(beta^{n-1})). An application of Philippon's zero estimate (1986) and his criterion of algebraic independence (1984) is then used to obtain the main result.

<pre>This dissertation concerns two questions involving the injectivity of specialization homomorphisms for elliptic surfaces. We primarily focus on elliptic surfaces over the projective line defined over the rational numbers. The specialization theorem of Silverman proven in 1983 says that, for a fixed surface, all but finitely many specialization homomorphisms are injective. Given a subgroup of the group of rational sections with explicit generators, we thus ask the following.</pre><pre>Given some rational number, how can we effectively determine whether or not the associated specialization map is injective?</pre><pre>What is the set of rational numbers such that the corresponding specialization maps are injective?</pre><pre>The classical specialization theorem of Neron proves that there is a set S which differs from a Hilbert subset of the rational numbers by finitely many elements such that for each number in S the associated specialization map is injective. We expand this into an effective procedure that determines if some rational number is in S, yielding a partial answer to question 1. Computing the Hilbert set provides a partial answer to question 2, and we carry this out for some examples. We additionally expand an effective criterion of Gusic and Tadic to include elliptic surfaces with a rational 2-torsion curve.<br></pre>

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We provide an explicit algorithm for computing the trace of an endomorphism of an elliptic curve which is given by a chain of small-degree isogenies. We analyze its complexity, determining that if the length of the chain, the degree of the isogenies, and the log of the field-size are all O(n), the trace of the endomorphism can be computed in O(n⁶) bit operations. This makes explicit a theorem of Kohel which states that such a polynomial time algorithm exists. The given procedure is based on Schoof's point-counting algorithm. / Master of Science / The developing technology of quantum computers threatens to render current cryptographic systems (that is, systems for protecting stored or transmitted digital information from unauthorized third parties) ineffective. Among the systems proposed to ensure information security against attacks by quantum computers is a cryptographic scheme known as SIKE. In this thesis, we provide and analyze an algorithm that comprises one piece of a potential attack against SIKE by a classical computer. The given algorithm is also useful more generally in the field of arithmetic geometry.

Integral solutions to y² = X³ + k, where either the x's or the y's, or both, are in arithmetic progression are studied. When both the x's and the y's are in arithmetic progression, then this situation is completely solved. One set of solutions where the y's formed an arithmetic progression of length 4 have already been constructed. In this dissertation, we construct infinitely many set of solutions where there are 4 x's in arithmetic progression and we also disprove Mohanty's Conjecture[8] by constructing infinitely many set of solutions where there are 4, 5 and 6 y's in arithmetic progression.

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