Thesis advisor: Benjamin Howard / In the last twenty years, Elliptic Curve Cryptography has become a standard for the transmission of secure data. The purpose of my thesis is to develop the necessary theory for the implementation of elliptic curve cryptosystems, using elementary number theory, abstract algebra, and geometry. This theory is based on developing formulas for adding rational points on an elliptic curve. The set of rational points on an elliptic curve form a group over the addition law as it is defined. Using the group law, my study continues into computing the torsion subgroup of an elliptic curve and considering elliptic curves over finite fields. With a brief introduction to cryptography and the theory developed in the early chapters, my thesis culminates in the explanation and implementation of three elliptic curve cryptosystems in the Java programming language. / Thesis (BA) — Boston College, 2007. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Mathematics. / Discipline: College Honors Program.

The purpose of this study was to determine how the study of number theory might affect high school students' metacognitive functioning, mathematical curiosity, and/or attitudes towards mathematics. The study utilized questionnaire and/or interview responses of seven high school students from New York City and 33 high school students from Dalian, China. The questionnaire components served to measure and compare the students' metacognitive functioning, mathematical curiosity, and mathematics attitudes before and after they worked on a number theory problem set included with the questionnaire. Interviews with 13 of these students also helped to reveal any changes in their metacognitive tendencies and/or mathematics attitudes or curiosity levels after the students had worked on said number theory problems. The investigator sought to involve very motivated as well as less motivated mathematics students in the study. The participation of a large group of Chinese students enabled the investigator to obtain a diverse set of data elements, and also added an international flavor to the research. All but one of the 40 participating students described or presented some evidence of metacognitive enhancement, greater mathematical curiosity, and/or improved attitudes towards mathematics after the students had worked on the assigned number theory problems. The results of the study thus have important implications for the value of number theory coursework by high school students, with respect to the students' metacognitive processes as well as their feelings about mathematics as an academic discipline.

This paper is based on Landau's book "Foundations of Analysis" which constitutes a development of the number system founded on the Peano axioms for natural numbers. In order to show mastery of the subject matter this paper gives a somewhat different organization of material and modified or more detailed proofs of theorems. In situations where proofs become rather routine re pet it ions of previously noted techniques the proofs are omitted. The following symbols and notation are used. Natural numbers are denoted by lower case letters such as a,b,c, ... x,y,z. Sets are denoted by upper case letters such as M, N, ... X, Y, Z. If a is an element of M, this will be written atM, The denial of this is written at M. The symbol 3 /x is read "There exists an unique x". If x and y are names for the same number we write x=y. It is assumed that the relation= is an equivalence relation; i.e., (1) x=x, (2) if x=y, then y=x, (3) u x=y and y=z, then x=z. Throughout this paper there will be no special attempt to distinguish between the name of a number and the number itself. For example, the phrase" if xis a number" will be used in place of "if x is the name of a number."

In the classical case we make use of Pells equation to compute units in the ring OF. Consider the parallel to the classical case and the quadratic field extension that creates the ring OK. We use the generalized Pell's equation to find the units in this ring since they are solutions. Through the use of continued fractions we may further characterize this ring and compute its units.

This project investigates the development of four different proofs of the law of quadratic reciprocity, in order to study the critical reasoning process that drives discovery in mathematics. We begin with an examination of the first proof of this law given by Gauss. We then describe Gauss’ fourth proof of this law based on Gauss sums, followed by a look at Eisenstein’s geometric simplification of Gauss’ third proof. Finally, we finish with an examination of one of the modern proofs of this theorem published in 1991 by Rousseau. Through this investigation we aim to analyze the different strategies used in the development of each of these proofs, and in the process gain a better understanding of this theorem.

The main objective of this project is to find the original symmetric presentations of some very important finite groups and to give our constructions of some of these groups. We have found the Mathieu sporadic group M11, HS × D5, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L2(149) as homomorphic images of the monomial progenitors 11*4 :m (5 :4), 5*6 :m S5 and 149*2 :m D37. We have also discovered 24 : S3 × C2, 24 : A5, (25 : S4), 25 : S3 × S3, 33 : S4 × C2, S6, 29: PGL(2,7), 22 • (S6 : S6), PGL(2,19), ((A5 : A5 × A5) : D6), 6 • (U4(3): 2), 2 • PGL(2,13), S7, PGL (2,8), PSL(2,19), 2 × PGL(2,81), 25 : (S6 × A5), 26 : S4 × D3, U(4,3), 34 : S4, 32 :D6, 2 • (PGL(2,7) :PSL(2,7), 22 : (S5 : S5) and 23 : (PSL3(4) : 2) as homomorphic images of the permutation progenitors 2*8 : (2 × 4 : 2), 2*16: (2 × 4 :C2 × C2), 2*9: (S3 × S3), 2*9: (S3 × A3), 2*9: (32 × 23) and 2*9: (33 × A3). We have also constructed 24: S3 × C2, 24 : A5, (25: S4), 25 : S3 × S3,: 33: S4 × C2, S6, M11 and U (3,5) by using the technique of double coset enumeration. We have determined the isomorphism types of the most of the images mentioned in this thesis. We demonstrate our work for the following examples: 34 : (32 * 23) × 2, 29 : PGL(2,7), 2•S6, (54 : (D4 × S3)), and 3: •PSL(2,19) ×2.

Doctor of Philosophy (PhD) / In this work, we investigate methods for computing explicitly with homomorphisms (and particularly endomorphisms) of Jacobian varieties of algebraic curves. Our principal tool is the theory of correspondences, in which homomorphisms of Jacobians are represented by divisors on products of curves. We give families of hyperelliptic curves of genus three, five, six, seven, ten and fifteen whose Jacobians have explicit isogenies (given in terms of correspondences) to other hyperelliptic Jacobians. We describe several families of hyperelliptic curves whose Jacobians have complex or real multiplication; we use correspondences to make the complex and real multiplication explicit, in the form of efficiently computable maps on ideal class representatives. These explicit endomorphisms may be used for efficient integer multiplication on hyperelliptic Jacobians, extending Gallant--Lambert--Vanstone fast multiplication techniques from elliptic curves to higher dimensional Jacobians. We then describe Richelot isogenies for curves of genus two; in contrast to classical treatments of these isogenies, we consider all the Richelot isogenies from a given Jacobian simultaneously. The inter-relationship of Richelot isogenies may be used to deduce information about the endomorphism ring structure of Jacobian surfaces; we conclude with a brief exploration of these techniques.

In 1980, L. Adleman, C. Pomerance, and R. Rumely invented the first cyclotomicprimality test, and shortly after, in 1981, a simplified and more efficient versionwas presented by H.W. Lenstra for the Bourbaki Seminar. Later, in 2008, ReneSchoof presented an updated version of Lenstra's primality test. This thesis presents adetailed description of the cyclotomic primality test as described by Schoof, along withsuggestions for implementation. The cornerstone of the test is a prime congruencerelation similar to Fermat's little theorem" that involves Gauss or Jacobi sumscalculated over cyclotomic fields. The algorithm runs in very nearly polynomial time.This primality test is currently one of the most computationally efficient tests and isused by default for primality proving by the open source mathematics systems Sageand PARI/GP. It can quickly test numbers with thousands of decimal digits.

Soit un $f(X)$ un polynôme à coefficients rationnels, $S$ un ensemble infini du nombres rationnels. Soit $f(S)$ l' ensemble image de $f(X)$ sur $S$. Si $g(X)$ est un polynôme telle que $f(S)=g(S)$ on dit que $g$ parametrise l'ensemble $f(S)$. En plus de la solution $g=f$ on peut imposer autre conditions sur le polynôme $g$; par example, si $f(S)\subset\Z$, on peut se demander si il y a un polynôme à coefficients entiers que parametrise l'ensemble $f(S)$. De plus, si l'image $f(S)$ est parametrisé par un polynôme $g$, on peut demander si il y a de relations entre les polynômes $f$ et $g$. Par example, si $h$ est un polynôme linéaire et on pose $g=f\circ h$, évidemment le polynôme $g$ parametrise l'ensemble $f(\Q)$. Réciproquement, si nous avons que $f(\Q)=g(\Q)$ (ou aussi $f(\Z)=g(\Z)$) alors par le théorème d'irréductibilité de Hilbert il y a un polynôme linéaire $h$ telle que $g=f\circ h$. Donc, si $g$ est un polynôme que parametrise l'ensemble $f(S)$, pour un ensemble infinie de nombres rationnels, nous nous demandons si il y a un polynôme $h$ telle que $f=g\circ h$. Il y a de théorèmes par Kubota que donnons de réponses positif sous certain conditions. Le but de ce thèse est l'étude de certain aspects de cet deux problèmes lié à la parametrisation de les ensembles image de polynômes.

The problem of finding the crossing number of an arbitrary knot or link is a hard problem in general. Only for very special classes of knots and links can we solve this problem. Often we can only hope to find a lower bound on the crossing number Cr(K) of a knot or a link K by computing the Jones polynomial of K, V(K). The crossing number Cr(K) is bounded from below by the difference between the greatest degree and the smallest degree of the polynomial V(K). However the computation of the Jones polynomial of an arbitrary knot or link is also difficult in general. The goal of this thesis is to find closed formulas for the smallest and largest exponents of the Jones polynomial for certain classes of knots and links. This allows us to find a lower bound on the crossing number for these knots and links very quickly. These formulas for the smallest and largest exponents of the Jones polynomial are constructed from special rational tangles expansions and using these formulas, we can extend these results to for [sic] special cases of Montesinos knots and links.