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EWU Masters Thesis Collection
Elliptic curves and their cryptographic applications.
Samuel L. Wenberg , Eastern Washington University
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Master of Science (MS) in Mathematics
Mathematics
"This thesis is a basic overview of elliptic curves and their applications to Cryptography. We begin with basic definitions and a demonstration that, given an elliptic curve addition, the points of an elliptic curve form a mathematical group. We then proceed to delve further into the mathematics, discussing torsion points on the group of elliptic curves before investigating the behavior of elliptic curves over finite fields wherein is given a proof of Hasse's Theorem on elliptic curves. With these tools, we discuss the discrete log problem, and the connection between elliptic curves and the field of cryptography. Finally, we look at elliptic curves over C and establish a trapdoor isomorphism between elliptic curves, and a topological torus"--Document.
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Wenberg, Samuel L., "Elliptic curves and their cryptographic applications" (2013). EWU Masters Thesis Collection . 160. https://dc.ewu.edu/theses/160
Since August 12, 2014
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Home > Graduate Research > Master's Theses > 3794
Master's Theses
Elliptic curves and cryptography.
Senorina Ramos Vazquez , San Jose State University Follow
Publication Date
Spring 2010
Degree Type
Degree name.
Master of Science (MS)
Mathematics
Timothy Hsu
Cryptography, Elliptic Curves, Group of an Elliptic Curve
Subject Areas
In this expository thesis we study elliptic curves and their role in cryptography. In doing so we examine an intersection of linear algebra, abstract algebra, number theory, and algebraic geometry, all of which combined provide the necessary background. First we present background information on rings, fields, groups, group actions, and linear algebra. Then we delve into the structure and classification of finite fields as well as construction of finite fields and computation in finite fields. We next explore logarithms in finite fields and introduce the Diffie-Hellman key exchange system. Subsequently, we take a look at the projective and affine planes and we examine the action of the general linear group of degree 3 (over K) on the points of the projective plane P2(K). We then explore the geometry of the projective plane with Desargues Theorem. Next, we study conics, quadratic forms, and methods of counting intersection of curves. Finally, we study forms of degree 3 and we are able to explore cubics and the group law on an elliptic curve which leads us to our ultimate goal of examining the role of elliptic curves in cryptography.
Recommended Citation
Vazquez, Senorina Ramos, "Elliptic Curves and Cryptography" (2010). Master's Theses . 3794. DOI: https://doi.org/10.31979/etd.6fat-tnvm https://scholarworks.sjsu.edu/etd_theses/3794
Since December 15, 2010
https://doi.org/10.31979/etd.6fat-tnvm
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What a lovely hat
Is it made out of tin foil , paper 2008/100, accelerating the scalar multiplication on elliptic curve cryptosystems over prime fields.
Patrick Longa
Elliptic curve cryptography (ECC), independently introduced by Koblitz and Miller in the 80's, has attracted increasing attention in recent years due to its shorter key length requirement in comparison with other public-key cryptosystems such as RSA. Shorter key length means reduced power consumption and computing effort, and less storage requirement, factors that are fundamental in ubiquitous portable devices such as PDAs, cellphones, smartcards, and many others. To that end, a lot of research has been carried out to speed-up and improve ECC implementations, mainly focusing on the most important and time-consuming ECC operation: scalar multiplication. In this thesis, we focus in optimizing such ECC operation at the point and scalar arithmetic levels, specifically targeting standard curves over prime fields. At the point arithmetic level, we introduce two innovative methodologies to accelerate ECC formulae: the use of new composite operations, which are built on top of basic point doubling and addition operations; and the substitution of field multiplications by squarings and other cheaper operations. These techniques are efficiently exploited, individually or jointly, in several contexts: to accelerate computation of scalar multiplications, and the computation of pre-computed points for window-based scalar multiplications (up to 30% improvement in comparison with previous best method); to speed-up computations of simple side-channel attack (SSCA)-protected implementations using innovative atomic structures (up to 22% improvement in comparison with scalar multiplication using original atomic structures); and to develop parallel formulae for SIMD-based applications, which are able to execute three and four operations simultaneously (up to 72% of improvement in comparison with a sequential scalar multiplication). At the scalar arithmetic level, we develop new sublinear (in terms of Hamming weight) multibase scalar multiplications based on NAF-like conversion algorithms that are shown to be faster than any previous scalar multiplication method. For instance, proposed multibase scalar multiplications reduce computing times in 10.9% and 25.3% in comparison with traditional NAF for unprotected and SSCA-protected scenarios, respectively. Moreover, our conversion algorithms overcome the problem of converting any integer to multibase representation, solving an open problem that was defined as hard. Thus, our algorithms make the use of multiple bases practical for applications as ECC scalar multiplication for first time.
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Elliptic Curve Cryptography - Implementation and Performance Testing of Curve Representations
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Elliptic Curve Cryptography Implementation and Performance Testing of Curve ... Master's Thesis Spring 2017. Elliptic Curve Cryptography Implementation and Performance Testing of Curve Representations Olav Wegner Eide 30th April 2017. ii. Abstract Public-key cryptography makes it possible to create digital signatures and do key
Abstract. In the recent years, the need of information security has rapidly increased due to an enormous growth of data transmission. In this thesis, we study the uses of elliptic curves in the cryptography. We discuss the elliptic curves over finite fields, attempts to attack; discrete logarithm, Pollard's rho algorithm, baby-step giant-step ...
Thesis Prepared for the Degree of MASTER OF SCIENCE UNIVERSITY OF NORTH TEXAS August 2007 . Huang, Jian. FPGA Implementations of Elliptic Curve Cryptography and Tate Pairing over Binary Field. Master of Science (Computer Engineering), August 2007, 70 pp., 12 tables, ... 21 figures, references, 52 titles. Elliptic curve cryptography (ECC) is an ...
under which the points on the elliptic curves form an abelian group. Then moving to a brief introduction to both public, and private key cryptography. This will lead into an explanation of the discrete logarithm problem along with an implementation using the elliptic curve group over F p. This thesis will conclude with an exploration
ELLIPTIC CURVES AND CRYPTOGRAPHY by Senorina R. V¶azquez In this expository thesis we study elliptic curves and their role in cryptography. In doing so we examine an intersection of linear algebra, abstract algebra, number theory, and algebraic geometry, all of which combined provide the necessary background.
life. In this master thesis we present a lightweight BSD-based implementation of the Elliptic Curve Cryptography (ECC) for the Contiki OS and its evaluation. We show the feasibility of the implementation and use of this cryptography in the IoT by a thorough evaluation of the solution by analyzing the performance using
thesis, are brought together to realize four high-speed implementations on x86-64 processors at the 128-bit security level. Presented results set new speed records for elliptic curve scalar multiplication and introduce up to 34% of cost reduction in comparison with the best previous results in the literature.
"This thesis is a basic overview of elliptic curves and their applications to Cryptography. We begin with basic definitions and a demonstration that, given an elliptic curve addition, the points of an elliptic curve form a mathematical group. We then proceed to delve further into the mathematics, discussing torsion points on the group of elliptic curves before investigating the behavior of ...
Emphasis is given to elliptic curve cryptography methods which make use of more advanced mathematical concepts. Contents 1. Introduction 1 2. Public-key Cryptography Systems Overview 2 2.1. Preliminaries 2 2.2. Discrete Logarithm Problem 3 2.3. Di e-Hellman Key Exchange 3 2.4. Other Public Cryptosystems 4 3. Elliptic Curve Cryptography 5 3.1.
A thesis submitted for the degree of Master of Science of Fiji National University October 2020. ii We certify that, as the assessors, we have read this thesis and that in our ... [25], elliptic curve cryptography (ECC) began to be employed for commercial applications. As a result, a significant amount of research has been devoted to analyzing the
A mathematical object called an elliptic curve can be used in the construction of public key cryptosystems. This thesis focuses on speeding up elliptic curve cryptography which is an attractive alternative to traditional public key cryptosystems such as RSA. Speeding up elliptic curve cryptography can be
We have in this thesis laid the initial foundation for post-quantum cryptography based on supersingular elliptic curve isogenies. This area has great potential and we hope that the wider cryptographic community will express an interest and perform more research in this direction. 80. Bibliography.
In this expository thesis we study elliptic curves and their role in cryptography. In doing so we examine an intersection of linear algebra, abstract algebra, number theory, and algebraic geometry, all of which combined provide the necessary background. First we present background information on rings, fields, groups, group actions, and linear algebra. Then we delve into the structure and ...
A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Masters of Applied Science in the Department of Electrical and Computer Engineering O Majid Khabbazian, 2004 University of Victoria ... Elliptic curve cryptography is being considered by standard organizations such as the National
Public-key cryptography Publication info Published elsewhere. Master's Thesis, University of Ottawa, June 2007 Keywords Elliptic curve cryptography multibase NAF Contact author(s) plonga @ uwaterloo ca History 2008-03-10: received Short URL https://ia.cr/2008/100 License CC BY
Elliptic curve cryptography (ECC) is a preferred method to implement these services, due to its low computational complexity compared to other modular arithmetic systems such as the Rivest-Shamir-Adleman algorithm (RSA). This thesis investigates how the use of different curve representations impact the performance of the Elliptic Curve Discrete ...
CERTIFICATE This is to certify that the M.Phil dissertation entitled "A Brief study of Elliptic Curve Cryptography" submitted to Madurai Kamraj University, Madurai, for the award of Master of Philosophy in Computer Science, is a bonafied record of research work and investigations done by Sri. E. Kesavulu Reddy (A5A6686903), under my supervision as a Research Scholar (Part - Time) of the ...
Elliptic Curve (EC) is the most recent and advanced technique of Elliptic Curve Cryptography (ECC). EC is often used to improve the security of open communication networks and to let specific persons with confirmed identities into the Modern Digital Era (MDE). Users of MDE make use of many technologies, such as social media, the cloud, and the ...
The world of cryptography is always expanding and growing. In this paper, we set out to explore the use of elliptic curves in the cryptography of today, as well as the cryptography of the future.We also offer our own original cryptosystem, CSDH. This system on its ownoffers some moderate level of security. It shares many similarities to the ...
Elliptic-Curve Cryptography (ECC) belongs to public-key cryptography which is widely used for secure communications on the Internet of Things (IoT) applications. ... (AXI) link, is a system-memory mapped transaction bus with master-slave functionality. The 32-bit MicroBlaze Core Processor is closely related to the timer, security engine ...
into consideration is that of nding an explicit isogeny between two given elliptic curves E 1 and E 2 over a nite eld F q. Remember that an elliptic curve E over F q is an equation of the form y2 = x3 + Ax + B, where A;B 2F q are eld elements that satisfy 4A3 + 27B2 6= 0. When equipped with the point at in nity 1:= (0 : 1 : 0) its set of ...
A beamer presentation on elliptic curves. Contribute to ctesta01/thesis-presentation development by creating an account on GitHub. ... Elliptic Curves Thesis Defense.tex. ... 1985 & Elliptic Curve Cryptography is born \\ 1987 & Lenstra's Integer Factorization Algorithm \\ 1995 & Wiles' Modularity Theorem \\ 2006 & Elkies' Discovery of a Rank ...
Writing a master's thesis on elliptic curve cryptography can be an overwhelming challenge that requires extensive research and a deep understanding of complex topics. Seeking assistance from professional writers who specialize in elliptic curve cryptography can help students save time, ensure high quality work, meet deadlines, and receive personalized support throughout the thesis writing ...
This document discusses how a website called HelpWriting.net can assist students struggling with writing a master's thesis on elliptic curve cryptography. It offers customized solutions, thorough research, clear writing, timely delivery, and unlimited revisions. Their team of experts specializes in cryptography and related fields to ensure high-quality support. Students can have their thesis ...