Number Theory Commons^™

Explicit Composition Identities For Higher Composition Laws In The Quadratic Case, Ajith A. Nair

Dissertations, Theses, and Capstone Projects

The theory of Gauss composition of integer binary quadratic forms provides a very useful way to compute the structure of ideal class groups in quadratic number fields. In addition to that, Gauss composition is also important in the problem of representations of integers by binary quadratic forms. In 2001, Bhargava discovered a new approach to Gauss composition which uses 2x2x2 integer cubes, and he proved a composition law for such cubes. Furthermore, from the higher composition law on cubes, he derived four new higher composition laws on the following spaces - 1) binary cubic forms, 2) pairs of binary quadratic …

Boolean Group Structure In Class Groups Of Positive Definite Quadratic Forms Of Primitive Discriminant, Christopher Albert Hudert Jr.

Student Research Submissions

It is possible to completely describe the representation of any integer by binary quadratic forms of a given discriminant when the discriminant’s class group is a Boolean group (also known as an elementary abelian 2-group). For other discriminants, we can partially describe the representation using the structure of the class group. The goal of the present project is to find whether any class group with 32 elements and a primitive positive definite discriminant is a Boolean group. We find that no such class group is Boolean.

Rsa Algorithm, Evalisbeth Garcia Diazbarriga

ATU Research Symposium

I will be presenting about the RSA method in cryptology which is the coding and decoding of messages. My research will focus on proving that the method works and how it is used to communicate secretly.

Pairs Of Quadratic Forms Over P-Adic Fields, John Hall

Theses and Dissertations--Mathematics

Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$.

The Vulnerabilities To The Rsa Algorithm And Future Alternative Algorithms To Improve Security, James Johnson

Cybersecurity Undergraduate Research Showcase

The RSA encryption algorithm has secured many large systems, including bank systems, data encryption in emails, several online transactions, etc. Benefiting from the use of asymmetric cryptography and properties of number theory, RSA was widely regarded as one of most difficult algorithms to decrypt without a key, especially since by brute force, breaking the algorithm would take thousands of years. However, in recent times, research has shown that RSA is getting closer to being efficiently decrypted classically, using algebraic methods, (fully cracked through limited bits) in which elliptic-curve cryptography has been thought of as the alternative that is stronger than …

Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring, Joseph Swanson

All Dissertations

Let R be a domain. We look at the algebraic and integral closure of a polynomial ring, R[x], in its power series ring, R[[x]]. A power series α(x) ∈ R[[x]] is said to be an algebraic power series if there exists F (x, y) ∈ R[x][y] such that F (x, α(x)) = 0, where F (x, y) ̸ = 0. If F (x, y) is monic, then α(x) is said to be an integral power series. We characterize the units of algebraic and integral power series. We show that the only algebraic power series with infinite radii of convergence are …

(R2050) Dual Quaternion Matrices And Matlab Applications, Kemal Gökhan Nalbant, Salim Yüce

Applications and Applied Mathematics: An International Journal (AAM)

There are many studies in the literature on real quaternions and real quaternion matrices. There are few studies in the literature on dual quaternions. Definitions of the matrices of dual quaternions used in this study will be given. The originality of our research, the set of dual quaternion matrix we studied, will be defined for the first time in this study, and its properties will be given. Moreover, this study is critical because it is an applied study related to dual quaternion matrices. It will be easier to solve examples with large matrix sizes with MATLAB. People who use different …

Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein

Doctoral Dissertations

Given an ordinary elliptic curve E over a field 𝕜 of characteristic p, there is an elliptic curve E over the Witt vectors W(𝕜) for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over 𝔽_p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on …

Explicit Constructions Of Canonical And Absolute Minimal Degree Lifts Of Twisted Edwards Curves, William Coleman Bitting Iv

Doctoral Dissertations

Twisted Edwards Curves are a representation of Elliptic Curves given by the solutions of bx^2 + y^2 = 1 + ax^2y^2. Due to their simple and unified formulas for adding distinct points and doubling, Twisted Edwards Curves have found extensive applications in fields such as cryptography. In this thesis, we study the Canonical Liftings of Twisted Edwards Curves and the associated lift of points Elliptic Teichmu ̈ller Lift. The coordinate functions of the latter are proved to be polynomials, and their degrees and derivatives are computed. Moreover, an algorithm is described for explicit computations, and some properties of the general …

Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof, Zachary Warren

Senior Honors Theses

It is a well-known property of the integers, that given any nonzero a ∈ Z, where a is not a unit, we are able to write a as a unique product of prime numbers. This is because the Fundamental Theorem of Arithmetic (FTA) holds in the integers and guarantees (1) that such a factorization exists, and (2) that it is unique. As we look at other domains, however, specifically those of the form O(√D) = {a + b√D | a, b ∈ Z, D a negative, squarefree integer}, we find that …

A Visual Tour Of Dynamical Systems On Color Space, Jonathan Maltsman

HMC Senior Theses

We can think of a pixel as a particle in three dimensional space, where its x, y and z coordinates correspond to its level of red, green, and blue, respectively. Just as a particle’s motion is guided by physical rules like gravity, we can construct rules to guide a pixel’s motion through color space. We can develop striking visuals by applying these rules, called dynamical systems, onto images using animation engines. This project explores a number of these systems while exposing the underlying algebraic structure of color space. We also build and demonstrate a Visual DJ circuit board for …

Mathematical Structure Of Musical Tuning Systems, Shay Joel Francis Spitzer

Senior Projects Spring 2023

Over the course of history, western music has created a unique mathematical problem for itself. From acoustics, we know that two notes sound good together when they are related by simple ratios consisting of low primes. The problem arises when we try to build a finite set of pitches, like the 12 notes on a piano, that are all related by such ratios. We approach the problem by laying out definitions and axioms that seek to identify and generalize desirable properties. We can then apply these ideas to a broadened algebraic framework. Rings in which low prime integers can be …

Elliptic Curves Over Finite Fields, Christopher S. Calger

Honors Theses

The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of F_q-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use …

(R1979) Permanent Of Toeplitz-Hessenberg Matrices With Generalized Fibonacci And Lucas Entries, Hacène Belbachir, Amine Belkhir, Ihab-Eddine Djellas

Applications and Applied Mathematics: An International Journal (AAM)

In the present paper, we evaluate the permanent and determinant of some Toeplitz-Hessenberg matrices with generalized Fibonacci and generalized Lucas numbers as entries.We develop identities involving sums of products of generalized Fibonacci numbers and generalized Lucas numbers with multinomial coefficients using the matrix structure, and then we present an application of the determinant of such matrices.

Reduction Of L-Functions Of Elliptic Curves Modulo Integers, Félix Baril Boudreau

Electronic Thesis and Dissertation Repository

Let $\mathbb{F}_q$ be a finite field of size $q$, where $q$ is a power of a prime $p \geq 5$. Let $C$ be a smooth, proper, and geometrically connected curve over $\mathbb{F}_q$. Consider an elliptic curve $E$ over the function field $K$ of $C$ with nonconstant $j$-invariant. One can attach to $E$ its $L$-function $L(T,E/K)$, which is a generating function that contains information about the reduction types of $E$ at the different places of $K$. The $L$-function of $E/K$ was proven to be a polynomial in $\mathbb{Z}[T]$.

In 1985, Schoof devised an algorithm to compute the zeta function of an …

Efficiency Of Homomorphic Encryption Schemes, Kyle Yates

All Theses

In 2009, Craig Gentry introduced the first fully homomorphic encryption scheme using bootstrapping. In the 13 years since, a large amount of research has gone into improving efficiency of homomorphic encryption schemes. This includes implementing leveled homomorphic encryption schemes for practical use, which are schemes that allow for some predetermined amount of additions and multiplications that can be performed on ciphertexts. These leveled schemes have been found to be very efficient in practice. In this thesis, we will discuss the efficiency of various homomorphic encryption schemes. In particular, we will see how to improve sizes of parameter choices in homomorphic …

The Hfd Property In Orders Of A Number Field, Grant Moles

All Theses

We will examine orders R in a number field K. In particular, we will look at how the generalized class number of R relates to the class number of its integral closure R. We will then apply this to the case when K is a quadratic field to produce a more specific relation. After this, we will focus on orders R which are half-factorial domains (HFDs), in which the irreducible factorization of any element α∈R has fixed length. We will determine two cases in which R is an HFD if and only if its ring of …

Unomaha Problem Of The Week (2021-2022 Edition), Brad Horner, Jordan M. Sahs

UNO Student Research and Creative Activity Fair

The University of Omaha math department's Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester's end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes.

Now there are three difficulty tiers to POW problems, roughly corresponding to …

On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi

Rose-Hulman Undergraduate Mathematics Journal

It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$-invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of either 0 …

An Overview Of Monstrous Moonshine, Catherine E. Riley

Channels: Where Disciplines Meet

The Conway-Norton monstrous moonshine conjecture set off a quest to discover the connection between the Monster and the J-function. The goal of this paper is to give an overview of the components of the conjecture, the conjecture itself, and some of the ideas that led to its solution. Special focus is given to Klein's J-function.

John Horton Conway: The Man And His Knot Theory, Dillon Ketron

Electronic Theses and Dissertations

John Horton Conway was a British mathematician in the twentieth century. He made notable achievements in fields such as algebra, number theory, and knot theory. He was a renowned professor at Cambridge University and later Princeton. His contributions to algebra include his discovery of the Conway group, a group in twenty-four dimensions, and the Conway Constellation. He contributed to number theory with his development of the surreal numbers. His Game of Life earned him long-lasting fame. He contributed to knot theory with his developments of the Conway polynomial, Conway sphere, and Conway notation.

The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

Electronic Theses, Projects, and Dissertations

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …

Cryptography Through The Lens Of Group Theory, Dawson M. Shores

Electronic Theses and Dissertations

Cryptography has been around for many years, and mathematics has been around even longer. When the two subjects were combined, however, both the improvements and attacks on cryptography were prevalent. This paper introduces and performs a comparative analysis of two versions of the ElGamal cryptosystem, both of which use the specific field of mathematics known as group theory.

A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, Caroline Nunn

Rose-Hulman Undergraduate Mathematics Journal

Niven’s theorem states that the sine, cosine, and tangent functions are rational for only a few rational multiples of π. Specifically, for angles θ that are rational multiples of π, the only rational values of sin(θ) and cos(θ) are 0, ±½, and ±1. For tangent, the only rational values are 0 and ±1. We present a proof of this fact, along with a generalization, using the structure of ideals in imaginary quadratic rings. We first show that the theorem holds for the tangent function using elementary properties of Gaussian integers, before extending the approach to other imaginary quadratic rings. We …

A Study In Applications Of Continued Fractions, Karen Lynn Parrish

Electronic Theses, Projects, and Dissertations

This is an expository study of continued fractions collecting ideas from several different sources including textbooks and journal articles. This study focuses on several applications of continued fractions from a variety of levels and fields of mathematics. Studies begin with looking at a number of properties that pertain to continued fractions and then move on to show how applications of continued fractions is relevant to high school level mathematics including approximating irrational numbers and developing new ideas for understanding and solving quadratics equations. Focus then continues to more advanced applications such as those used in the studies of number theory …

Algorithms Related To Triangle Groups, Bao The Pham

LSU Doctoral Dissertations

Given a finite index subgroup of $\PSL_2(\Z)$, one can talk about the different properties of this subgroup. These properties have been studied extensively in an attempt to classify these subgroups. Tim Hsu created an algorithm to determine whether a subgroup is a congruence subgroup by using permutations \cite{hsu}. Lang, Lim, and Tan also created an algorithm to determine if a subgroup is a congruence subgroup by using Farey Symbols \cite{llt}. Sebbar classified torsion-free congruence subgroups of genus 0 \cite{sebbar}. Pauli and Cummins computed and tabulated all congruence subgroups of genus less than 24 \cite{ps}. However, there are still some problems …

Irreducibility And Galois Groups Of Random Polynomials, Hanson Hao, Eli Navarro, Henri Stern

Rose-Hulman Undergraduate Mathematics Journal

In 2015, I. Rivin introduced an effective method to bound the number of irreducible integral polynomials with fixed degree d and height at most N. In this paper, we give a brief summary of this result and discuss the precision of Rivin's arguments for special classes of polynomials. We also give elementary proofs of classic results on Galois groups of cubic trinomials.

Disjointness Of Linear Fractional Actions On Serre Trees, Henry W. Talbott

Rose-Hulman Undergraduate Mathematics Journal

Serre showed that, for a discrete valuation field, the group of linear fractional transformations acts on an infinite regular tree with vertex degree determined by the residue degree of the field. Since the p-adics and the polynomials over the finite field of order p act on isomorphic trees, we may ask whether pairs of actions from these two groups are ever conjugate as tree automorphisms. We analyze permutations induced on finite vertex sets, and show a permutation classification result for actions by these linear fractional transformation groups. We prove that actions by specific subgroups of these groups are conjugate only …

Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), Marcus L. Shell

Theses

This paper explores and elaborates on a method of solving Pell’s equation as introduced by Norman Wildberger. In the first chapters of the paper, foundational topics are introduced in expository style including an explanation of Pell’s equation. An explanation of continued fractions and their ability to express quadratic irrationals is provided as well as a connection to the Stern-Brocot tree and a convenient means of representation for each in terms of 2×2 matrices with integer elements. This representation will provide a useful way of navigating the Stern-Brocot tree computationally and permit us a means of computing continued fractions without the …

On Elliptic Curves, Montana S. Miller

MSU Graduate Theses

An elliptic curve over the rational numbers is given by the equation y² = x³+Ax+B. In our thesis, we study elliptic curves. It is known that the set of rational points on the elliptic curve form a finitely generated abelian group induced by the secant-tangent addition law. We present an elementary proof of associativity using Maple. We also present a relatively concise proof of the Mordell-Weil Theorem.