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Dissertations / Theses on the topic 'Elliptic curve points group'

Author: Grafiati

Published: 30 May 2022

Last updated: 31 July 2025

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1

McGee, John J. "René Schoof's Algorithm for Determining the Order of the Group of Points on an Elliptic Curve over a Finite Field." Thesis, Virginia Tech, 2006. http://hdl.handle.net/10919/31911.

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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 t

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2

Hişil, Hüseyin. "Elliptic curves, group law, and efficient computation." Thesis, Queensland University of Technology, 2010. https://eprints.qut.edu.au/33233/1/H%C3%BCseyin_Hi%C5%9Fil_Thesis.pdf.

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This thesis is about the derivation of the addition law on an arbitrary elliptic curve and efficiently adding points on this elliptic curve using the derived addition law. The outcomes of this research guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology. Mastered by the 19th century mathematicians, the study of the theory of elliptic curves has been active for decades. Elliptic curves over finite fields made their way into public key cryptography in late 1980’s with independent pro

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3

Li, Chao. "2-Selmer groups and Heegner points on elliptic curves." Thesis, Harvard University, 2015. http://nrs.harvard.edu/urn-3:HUL.InstRepos:17464036.

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This thesis studies several aspects of the arithmetic of elliptic curves. In particular, we explore the prediction of the Birch and Swinnerton-Dyer conjecture when the 2-Selmer group has rank one. For certain elliptic curves $E/\mathbb{Q}: y^2=F(x)$ with additive reduction at 2, we determine their 2-Selmer ranks in terms of the 2-rank of the class group of the cubic field $L=\mathbb{Q}[x]/F(x)$. We then interpret this result as a mod 2 congruence between the Hasse-Weil $L$-function of $E$ and a degree two Artin $L$-function associated to the cubic field $L$. When the class number of $L$ is o

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4

Kouchaki, Barzi Behnaz. "Points of High Order on Elliptic Curves : ECDSA." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-58449.

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This master thesis is about Elliptic Curve Digital Signature Algorithm or ECDSA and two of the known attacks on this security system. The purpose of this thesis is to find points that are likely to be points of high order on an elliptic curve. If we have a point P of high order and if Q = mP, then we have a large set of possible values of m. Therefore it is hard to solve the Elliptic Curve Discrete Logarithm Problem or ECDLP. We have investigated on the time of finding the solution of ECDLP for a certain amount of elliptic curves based on the order of the point which is used to create the digi

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5

DeLorme, Cheryl Lynn 1969. "On the Shafarevich-Tate group of an elliptic curve." Diss., The University of Arizona, 1997. http://hdl.handle.net/10150/288706.

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This dissertation work concentrates on finding non-trivial elements in the Shafarevich-Tate group of an elliptic curve. The set of K-rational points on an elliptic curve, E, are known to form a finitely generated abelian group. My results are of interest when trying to find the rank of this group, which in general is a hard problem. The Selmer group of E,S(E/K), can be used to give a bound on this rank, and the obstruction to using this to find the exact rank is the Shafarevich-Tate group, scIII(E/K). There is a pairing on scIII(E/K), called the Cassels-Tate pairing, which is non-degenerate mo

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6

Ito, Koki. "Elliptic hypergeometric functions associated to the configuration space of three-points on an affine elliptic curve." 京都大学 (Kyoto University), 2006. http://hdl.handle.net/2433/144133.

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Kyoto University (京都大学)<br>0048<br>新制・課程博士<br>博士(理学)<br>甲第12064号<br>理博第2958号<br>新制||理||1443(附属図書館)<br>23900<br>UT51-2006-J59<br>京都大学大学院理学研究科数学・数理解析専攻<br>(主査)教授深谷賢治, 教授河野明, 助教授梅田亨<br>学位規則第4条第1項該当

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7

Lester, Jeremy W. "The Elliptic Curve Group Over Finite Fields: Applications in Cryptography." Youngstown State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1348847698.

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8

Weimerskirch, Andre. "The Application of the Mordell-Weil Group to Cryptographic Systems." Digital WPI, 2001. https://digitalcommons.wpi.edu/etd-theses/321.

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This thesis examines the Mordell-Weil group for application in cryptography. This approach has recently been proposed by Gerhard Frey. The use of the Mordell-Weil group for discrete logarithm schemes is a variant of elliptic curve cryptosystems. We extended the original idea by Frey with the goal of a performance improvement. The arithmetic complexity using the Mordell-Weil group will be compared to ordinary elliptic curve cryptosystems. The main goals of this thesis are (1) to investigate the algorithmic complexity of Mordell-Weil cryptosystems relative to elliptic curve cryptosystems; (2) th

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9

Wade, Darryl Gene. "The tropical Jacobian of an elliptic curve is the group S¹(Q) /." Diss., CLICK HERE for online access, 2008. http://contentdm.lib.byu.edu/ETD/image/etd2521.pdf.

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10

Wilcox, Nicholas. "A Computational Introduction to Elliptic and Hyperelliptic Curve Cryptography." Oberlin College Honors Theses / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=oberlin1528649455201473.

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11

HIRATA, Tomio, and Daisuke ADACHI. "Refined Computations for Points of the Form 2kP Based on Montgomery Trick." Institute of Electronics, Information and Communication Engineers, 2006. http://hdl.handle.net/2237/15065.

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12

Wade, Darryl Gene. "The Tropical Jacobian of a Tropical Elliptic Curve Is S^1(Q)." BYU ScholarsArchive, 2008. https://scholarsarchive.byu.edu/etd/1868.

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We establish consistent deﬁnitions for divisors, principal divisors, and Jacobians of a tropical elliptic curve and show that for a tropical elliptic cubic C , the associated Jacobian (or zero divisor class group) is the group S^1(Q).

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13

Pocienė, Jurgita. "Elipsinių kreivių taškų skaičiavimo algoritmai ir jų taikymai." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2006. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2006~D_20060608_201450-20142.

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Pociene, Jurgita. Informatics Master’s Final Thesis. Elliptic Curve Points Calculation Algorithms and their Application. Work leader dr. R. Steuding. Siauliai University. Siauliai, 2006. 35 pages In the work I analyse calculation algorithms of the points on elliptic curves above a body Fp (the body is above primary numbers’ field) and their application opportunities. Basic aims, set for the master’s thesis (to analyse elliptic curve points calculation algorithms and to compare them, to review application of elliptic curve points calculation algorithms, to realize Schoof elliptic curve points c

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14

Охріменко, Андрій Олександрович, та Andriy Okhrimenko. "Методи арифметичних перетворень в полях і кільцях для криптографічних застосувань". Thesis, Національний авіаційний університет, 2020. https://er.nau.edu.ua/handle/NAU/44626.

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Дисертаційна робота присвячена розв’язанню актуальної науково-практичної задачі дослідження і розробки нових методів арифметичних перетворень над великими цілими числами з відкладеним переносом для підвищення швидкодії реалізації криптографічних перетворень, що мають місце в інформаційно-телекомунікаційних системах центрів сертифікації ключів національної інфраструктури відкритих ключів України. В роботі запропоновано метод представлення цілих чисел з відкладеним переносом, який за рахунок можливості відкласти операцію переносу зі старших розрядів в молодші та операцію займу з молодших р

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15

Lam, Matthew. "A Combinatorial Exploration of Elliptic Curves." Scholarship @ Claremont, 2015. http://scholarship.claremont.edu/hmc_theses/91.

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At the intersection of algebraic geometry, number theory, and combinatorics, an interesting problem is counting points on an algebraic curve over a finite field. When specialized to the case of elliptic curves, this question leads to a surprising connection with a particular family of graphs. In this document, we present some of the underlying theory and then summarize recent results concerning the aforementioned relationship between elliptic curves and graphs. A few results are additionally further elucidated by theory that was omitted in their original presentation.

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16

Figueiredo, Rob. "Securing group communication in dynamic, disadvantaged networks : implementation of an elliptic-curve pairing-based cryptography library." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/41602.

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Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.<br>Includes bibliographical references (p. 155-158).<br>This thesis considers the problem of securing communication among dynamic groups of participants without relying on an online group keying service. As a solution, we offer the design and implementation of the Public Key Group Encryption (PKGE) service. It is a cryptography library, written in C, and designed to be shared among all communications applications on any particular system. PKGE imposes low communication overhead

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17

Kirlar, Baris Bulent. "Isomorphism Classes Of Elliptic Curves Over Finite Fields Of Characteristic Two." Master's thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/2/12606489/index.pdf.

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In this thesis, the work of Menezes on the isomorphism classes of elliptic curves over finite fields of characteristic two is studied. Basic definitions and some facts of the elliptic curves required in this context are reviewed and group structure of elliptic curves are constructed. A fairly detailed investigation is made for the isomorphism classes of elliptic curves due to Menezes and Schoof. This work plays an important role in Elliptic Curve Digital Signature Algorithm. In this context, those isomorphism classes of elliptic curves recommended by National Institute of Standards and Tech

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18

Gebremichael, Teklay. "Lightweight Cryptographic Group Key Management Protocols for the Internet of Things." Licentiate thesis, Mittuniversitetet, Institutionen för informationssystem och –teknologi, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:miun:diva-35607.

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The Internet of Things (IoT) is increasingly becoming an integral component of many applications in consumer, industrial and other areas. Notions such as smart industry, smart transport, and smart world are, in large part, enabled by IoT. At its core, the IoT is underpinned by a group of devices, such as sensors and actuators, working collaboratively to provide a required service. One of the important requirements most IoT applications are expected to satisfy is ensuring the security and privacy of users. Security is an umbrella term that encompasses notions such as confidentiality, integrity

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19

Hertgen, Alan. "Modèles de Néron et groupes formels." Thesis, Bordeaux, 2016. http://www.theses.fr/2016BORD0028/document.

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Dans cette thèse, on aborde plusieurs questions autour des modèles de Néron de variétés abéliennes sur un corps de valuation discrète. On dit qu’une variété abélienne a réduction scindée si la suite exacte définissant le groupe des composantes de la fibre spéciale est scindée. On donne un exemple de variété abélienne modérément ramifiée qui n’a pas réduction scindée. Pour les variétés jacobiennes,on montre que l’on obtient réduction scindée après toute extension modérément ramifiée de degré plus grand qu’une constante ne dépendant que de la dimension.On considère aussi le lien avec le conducte

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20

MERCURI, PIETRO. "Rational Points on Modular Curves." Doctoral thesis, 2014. http://hdl.handle.net/11573/918265.

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I explain a way to compute Fourier coefficients of modular forms associated to normalizer of non-split Cartan subgroups of GL(2,Z/pZ) and how, using these coefficients, one can compute explicit equations of modular curves associated to same subgroup. I attached some tables containing some examples of results of this method.

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21

Ambrose, Christopher Daniel. "On Artin's primitive root conjecture." Doctoral thesis, 2014. http://hdl.handle.net/11858/00-1735-0000-0022-5F1A-F.

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Artins Vermutung über Primitivwurzeln besagt, dass es zu jeder ganzen Zahl a, die weder 0, ±1 noch eine Quadratzahl ist, unendlich viele Primzahlen p gibt, sodass a eine Primitivwurzel modulo p ist, d.h. a erzeugt eine multiplikative Untergruppe von Q*, dessen Reduktion modulo p Index 1 in (Z/pZ)* hat. Dies wirft die Frage nach Verteilung von Index und Ordnung dieser Reduktion in (Z/pZ)* auf, wenn man p variiert. Diese Arbeit widmet sich verallgemeinerten Fragestellungen in Zahlkörpern: Ist K ein Zahlkörper und Gamma eine endlich erzeugte unendliche Untergruppe von K*, so werden Momente von In

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22

Wu, Shyi-Tsong, and 吳錫聰. "Authentication and Group Secure Communications Using Elliptic Curve Cryptography." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/28590399095058355278.

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博士<br>國立臺灣科技大學<br>電子工程系<br>93<br>With the explosion of the Internet as well as the wireless and mobile communications, it faces a growing need for security. Both for secure web transaction and for secure messaging, an efficient public key system is required. The Elliptic Curve Cryptography delivers the highest security strength per bit of key in any known public key system. It well suits to the applications such as the smart card systems and the wireless/mobile communications. In this thesis, we apply the ECC and the bilinear pairings on elliptic curve to the authentication and the group secur

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23

Yu, Li-An, and 游立安. "Group Communication Security System Based on Elliptic Curve Cryptography." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/79485013915734344657.

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博士<br>國立交通大學<br>網路工程研究所<br>102<br>Many emerging applications, such as Instant Communication in Vehicular Networks and Video Conferencing, adopt group communication model. In this thesis, we use Elliptic Curve Cryptography (ECC) as the basis to develop a Group Communication Security System (GCSS). The GCSS system provides security mechanisms for “Group Member Join”, “One-to-Many Data Transmission”, “One-to-One Data Transmission” and “Group Member Leave”. In “Group Member Join”, GCSS integrates Pre-Shared Key (PSK) authentication with Elliptic Curve Diffie-Hellman (ECDH) key exchange protocol to

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24

Huang, Wei-Lun, and 黃韋綸. "Multiple Group Key Distribution Protocols Based on Elliptic Curve Cryptography." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/68950415588453489087.

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碩士<br>南台科技大學<br>資訊管理系<br>98<br>The purpose of this study is to develop two multiple group key distribution protocols based on elliptic curve cryptography. They will allow all members in a group to share multiple group keys after executing the proposed protocol. Comparing with the existed schemes, our protocols are more efficient and more suitable for many applications. Besides, the security of the proposed protocols is the same with breaking elliptic curve cryptosystem. When the members join or leave, our protocols can renew group keys efficiently. And we prove that our protocols can protect t

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25

Thompson, Benjamin L. "Poncelet-type theorems and points of finite order on a curve in its Jacobian." Thesis, 2021. https://hdl.handle.net/2144/42677.

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For nearly three centuries mathematicians have been interested in polygons which simultaneously circumscribe and inscribe quadrics. They have shown in many contexts (real, complex, non-euclidean, higher dimensional, etc.) that such polygons may be ``rotated'' while maintaining their circum-inscribed quality. Of particular interest has been conditions on the quadrics which guarantee the existence of such polygons. In 1854 Arthur Cayley provided conditions for closure general to polygons of any size in the complex projective plane. We show that under suitable circumstances the curve, defi

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26

Wu, Chung-Yi, and 吳中一. "A Group-Oriented Thresholded Multisignature Scheme Based On the Elliptic Curve Cryptosystem." Thesis, 1999. http://ndltd.ncl.edu.tw/handle/83076472624811260747.

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碩士<br>國防管理學院<br>國防資訊研究所<br>87<br>Due to the popularity of hackers in Internet, secrecy preservation is one of the most acceptable in the related works in interconnections of Internets. While secrecy is required, a lot of many cryptosystems have been allocated to hold the function of secrecy. Some examples like RSA or ElGamal cryptosystems have been used to keep secrecy from illegal displacement . Since 1995, Elliptic Curve cryptosystem(ECC) has been studied to reveal the applicability in signatures, it may have been found to find signatures with new scheme. ECC is a new scheme for signatures w

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27

Wang, Yen-Sheng, and 王彥勝. "A Survey on Q-torsion group of elliptic curve and Mazur''s Theorem." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/32999567369275210930.

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碩士<br>國立臺灣大學<br>數學研究所<br>101<br>Let K be a number eld and E=K be an elliptic curve, that is, a smooth projective curve of genus 1 with an distinguished K-rational point chosen. By the Mordell-Weil Theorem, the group of points E(K) is a nitely generated abelian group. Its structure is of the form: E(K) = Etors(K) Zr According to this theorem, we know that Etors(K) is a nite group. In 1977, Mazur [Maz] proved a beautiful theorem for K = Q. It determines all the possible torsion structures of Etors(Q). In this thesis, we try to survey on the proof of this tremendous theorem as well as that o

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28

Cowan, Alexander. "Fourier expansions for Eisenstein series twisted by modular symbols and the distribution of multiples of real points on an elliptic curve." Thesis, 2019. https://doi.org/10.7916/d8-76ah-m845.

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This thesis consists of two unrelated parts. In the first part of this thesis, we give explicit expressions for the Fourier coefficients of Eisenstein series E∗(z, s, χ) twisted by modular symbols ⟨γ, f⟩ in the case where the level of f is prime and equal to the conductor of the Dirichlet character χ. We obtain these expressions by computing the spectral decomposition of an automorphic function closely related to E∗(z, s, χ). We then give applications of these expressions. In particular, we evaluate sums such as Σχ(γ)⟨γ, f⟩, where the sum is over γ ∈ Γ∞\Γ0(N) with c^2 + d^2 < X, with c a

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