Matching Items (15)

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p-Adic Numbers with an Emphasis on q-Volkenborn Integration

Description

Similar to the real numbers, the p-adic fields are completions of the rational numbers. However, distance in this space is determined based on divisibility by a prime number, p, rather

Similar to the real numbers, the p-adic fields are completions of the rational numbers. However, distance in this space is determined based on divisibility by a prime number, p, rather than by the traditional absolute value. This gives rise to a peculiar topology which offers significant simplifications for p-adic continuous functions and p-adic integration than is present in the real numbers. These simplifications may present significant advantages to modern physics – specifically in harmonic analysis, quantum mechanics, and string theory. This project discusses the construction of the p-adic numbers, elementary p-adic topology, p-adic continuous functions, introductory p-adic measure theory, the q-Volkenborn distribution, and applications of p-adic numbers to physics. We define q-Volkenborn integration and its connection to Bernoulli numbers.

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Created

Date Created
  • 2020-05

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A Statistic on a Super Catalan Structure

Description

The Super Catalan numbers are a known set of numbers which have so far eluded a combinatorial interpretation. Several weighted interpretations have appeared since their discovery, one of which was

The Super Catalan numbers are a known set of numbers which have so far eluded a combinatorial interpretation. Several weighted interpretations have appeared since their discovery, one of which was discovered by William Kuszmaul in 2017. In this paper, we connect the weighted Super Catalan structure created previously by Kuszmaul and a natural $q$-analogue of the Super Catalan numbers. We do this by creating a statistic $\sigma$ for which the $q$ Super Catalan numbers, $S_q(m,n)=\sum_X (-1)^{\mu(X)} q^{\sigma(X)}$. In doing so, we take a step towards finding a strict combinatorial interpretation for the Super Catalan numbers.

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Agent

Created

Date Created
  • 2018-05

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The Number Field Sieve

Description

This thesis project is focused on studying the number field sieve. The number field sieve is a factoring algorithm which uses algebraic number theory and is one of the fastest

This thesis project is focused on studying the number field sieve. The number field sieve is a factoring algorithm which uses algebraic number theory and is one of the fastest known factoring algorithms today. Factoring large integers into prime factors is an extremely difficult problem, yet also extremely important in cryptography. The security of the cryptosystem RSA is entirely based on the difficulty of factoring certain large integers into a product of two distinct large primes. While the number field sieve is one of the fastest factoring algorithms known, it is still not efficient enough to factor cryptographic sized integers.

In this thesis we will examine the algorithm of the number field sieve and discuss some important advancements. In particular, we will focus on the advancements that have been done in the polynomial selection step, the first main step of the number field sieve. The polynomial selected determines the number field by which computations are carried out in the remainder of the algorithm. Selection of a good polynomial allows for better time efficiency and a higher probability that the algorithm will be successful in factoring.

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Created

Date Created
  • 2020-05

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An Investigation of Supersingular Elliptic Curves in Quantum-Resistant Cryptography

Description

Many current cryptographic algorithms will eventually become easily broken by Shor's Algorithm once quantum computers become more powerful. A number of new algorithms have been proposed which are not compromised

Many current cryptographic algorithms will eventually become easily broken by Shor's Algorithm once quantum computers become more powerful. A number of new algorithms have been proposed which are not compromised by quantum computers, one of which is the Supersingular Isogeny Diffie-Hellman Key Exchange Protocol (SIDH). SIDH works by having both parties perform random walks between supersingular elliptic curves on isogeny graphs of prime degree and eventually end at the same location, a shared secret.<br/><br/>This thesis seeks to explore some of the theory and concepts underlying the security of SIDH, especially as it relates to finding supersingular elliptic curves, generating isogeny graphs, and implementing SIDH. As elliptic curves and SIDH may be an unfamiliar topic to many readers, the paper begins by providing a brief introduction to elliptic curves, isogenies, and the SIDH Protocol. Next, the paper investigates more efficient methods of generating supersingular elliptic curves, which are important for visualizing the isogeny graphs in the algorithm and the setup of the protocol. Afterwards, the paper focuses on isogeny maps of various degrees, attempting to visualize isogeny maps similar to those used in SIDH. Finally, the paper looks at an implementation of SIDH in PARI/GP and work is done to see the effects of using isogenies of degree greater than 2 and 3 on the security, runtime, and practicality of the algorithm.

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Created

Date Created
  • 2021-05

Explaining Electronic Voting Protocols to Non-Technical Audiences

Description

As computers become a more embedded aspect of daily life, the importance of communicating ideas in computing and technology to the general public has become increasingly apparent. One such growing

As computers become a more embedded aspect of daily life, the importance of communicating ideas in computing and technology to the general public has become increasingly apparent. One such growing technology is electronic voting. The feasibility of explaining electronic voting protocols was directly investigated through the generation of a presentation based on journal articles and papers identified by the investigator. Extensive use of analogy and visual aids were used to explain various cryptographic concepts. The presentation was then given to a classroom of ASU freshmen, followed by a feedback survey. A self-evaluation on the presentation methods is conducted, and a procedure for explaining subjects in computer science is proposed based on the researcher's personal process.

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Created

Date Created
  • 2018-05

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Almost-Primes Near Factorials

Description

In this paper, we study the prime factorizations of numbers slightly larger than the factorial function. While these are closely related to the factorial prime, they have more inherent structure,

In this paper, we study the prime factorizations of numbers slightly larger than the factorial function. While these are closely related to the factorial prime, they have more inherent structure, which allows for explicit results as of yet not established on factorial prime. Case in point, the main result of this paper is that these numbers, which are described in concrete terms below, cannot be prime powers outside of a handful of small cases; this is a generalization of a classical result stating they cannot be primes. Minor explicit results and heuristic analysis are then given to further characterize the set.

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Created

Date Created
  • 2019-12

Battleship: An Application of Multiparty Computation Encryption

Description

Cheating in Battleship is effortless. Battleship is a popular two-player board game where each player strategically places five ships on his or her concealed board. During this game, one can

Cheating in Battleship is effortless. Battleship is a popular two-player board game where each player strategically places five ships on his or her concealed board. During this game, one can easily move their ships during a play, falsify an attack, or not even place their ships. A solution to this concern is implementing multiparty computation (MPC) encryption to ensure that the location of both players’ ships and the result of attacking a ship is true. This document details the creation and security of a Battleship program that implements an MPC encryption method known as Poker Over the Telephone.

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Created

Date Created
  • 2021-05

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Some diophantine problems

Description

Diophantine arithmetic is one of the oldest branches of mathematics, the search

for integer or rational solutions of algebraic equations. Pythagorean triangles are

an early instance. Diophantus of Alexandria wrote the

Diophantine arithmetic is one of the oldest branches of mathematics, the search

for integer or rational solutions of algebraic equations. Pythagorean triangles are

an early instance. Diophantus of Alexandria wrote the first related treatise in the

fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat.

The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals.

The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves.

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Agent

Created

Date Created
  • 2019

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Classifying lambda-modules up to isomorphism and applications to Iwasawa theory

Description

In Iwasawa theory, one studies how an arithmetic or geometric object grows as its field of definition varies over certain sequences of number fields. For example, let $F/\mathbb{Q}$ be a

In Iwasawa theory, one studies how an arithmetic or geometric object grows as its field of definition varies over certain sequences of number fields. For example, let $F/\mathbb{Q}$ be a finite extension of fields, and let $E:y^2 = x^3 + Ax + B$ with $A,B \in F$ be an elliptic curve. If $F = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots F_\infty = \bigcup_{i=0}^\infty F_i$, one may be interested in properties like the ranks and torsion subgroups of the increasing family of curves $E(F_0) \subseteq E(F_1) \subseteq \cdots \subseteq E(F_\infty)$. The main technique for studying this sequence of curves when $\Gal(F_\infty/F)$ has a $p$-adic analytic structure is to use the action of $\Gal(F_n/F)$ on $E(F_n)$ and the Galois cohomology groups attached to $E$, i.e. the Selmer and Tate-Shafarevich groups. As $n$ varies, these Galois actions fit into a coherent family, and taking a direct limit one obtains a short exact sequence of modules $$0 \longrightarrow E(F_\infty) \otimes(\mathbb{Q}_p/\mathbb{Z}_p) \longrightarrow \Sel_E(F_\infty)_p \longrightarrow \Sha_E(F_\infty)_p \longrightarrow 0 $$ over the profinite group algebra $\mathbb{Z}_p[[\Gal(F_\infty/F)]]$. When $\Gal(F_\infty/F) \cong \mathbb{Z}_p$, this ring is isomorphic to $\Lambda = \mathbb{Z}_p[[T]]$, and the $\Lambda$-module structure of $\Sel_E(F_\infty)_p$ and $\Sha_E(F_\infty)_p$ encode all the information about the curves $E(F_n)$ as $n$ varies. In this dissertation, it will be shown how one can classify certain finitely generated $\Lambda$-modules with fixed characteristic polynomial $f(T) \in \mathbb{Z}_p[T]$ up to isomorphism. The results yield explicit generators for each module up to isomorphism. As an application, it is shown how to identify the isomorphism class of $\Sel_E(\mathbb{Q_\infty})_p$ in this explicit form, where $\mathbb{Q}_\infty$ is the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, and $E$ is an elliptic curve over $\mathbb{Q}$ with good ordinary reduction at $p$, and possessing the property that $E(\mathbb{Q})$ has no $p$-torsion.

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Agent

Created

Date Created
  • 2011

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One- and two-variable p-adic measures in Iwasawa theory

Description

In 1984, Sinnott used $p$-adic measures on $\mathbb{Z}_p$ to give a new proof of the Ferrero-Washington Theorem for abelian number fields by realizing $p$-adic $L$-functions as (essentially) the $Gamma$-transform of

In 1984, Sinnott used $p$-adic measures on $\mathbb{Z}_p$ to give a new proof of the Ferrero-Washington Theorem for abelian number fields by realizing $p$-adic $L$-functions as (essentially) the $Gamma$-transform of certain $p$-adic rational function measures. Shortly afterward, Gillard and Schneps independently adapted Sinnott's techniques to the case of $p$-adic $L$-functions associated to elliptic curves with complex multiplication (CM) by realizing these $p$-adic $L$-functions as $Gamma$-transforms of certain $p$-adic rational function measures. The results in the CM case give the vanishing of the Iwasawa $mu$-invariant for certain $mathbb{Z}_p$-extensions of imaginary quadratic fields constructed from torsion points of CM elliptic curves.

In this thesis, I develop the theory of $p$-adic measures on $mathbb{Z}_p^d$, with particular interest given to the case of $d>1$. Although I introduce these measures within the context of $p$-adic integration, this study includes a strong emphasis on the interpretation of $p$-adic measures as $p$-adic power series. With this dual perspective, I describe $p$-adic analytic operations as maps on power series; the most important of these operations is the multivariate $Gamma$-transform on $p$-adic measures.

This thesis gives new significance to product measures, and in particular to the use of product measures to construct measures on $mathbb{Z}_p^2$ from measures on $mathbb{Z}_p$. I introduce a subring of pseudo-polynomial measures on $mathbb{Z}_p^2$ which is closed under the standard operations on measures, including the $Gamma$-transform. I obtain results on the Iwasawa-invariants of such pseudo-polynomial measures, and use these results to deduce certain continuity results for the $Gamma$-transform. As an application, I establish the vanishing of the Iwasawa $mu$-invariant of Yager's two-variable $p$-adic $L$-function from measure theoretic considerations.

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Agent

Created

Date Created
  • 2015