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Elliptic Curve Cryptography Overview
 
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John Wagnon discusses the basics and benefits of Elliptic Curve Cryptography (ECC) in this episode of Lightboard Lessons. Check out this article on DevCentral that explains ECC encryption in more detail: https://devcentral.f5.com/articles/real-cryptography-has-curves-making-the-case-for-ecc-20832
Views: 158187 F5 DevCentral
An Introduction to Elliptic Curve Cryptography
 
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Cryptography and Network Security by Prof. D. Mukhopadhyay, Department of Computer Science and Engineering, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in
Views: 29845 nptelhrd
CloudFlare meet-up: Michael Hamburg talks elliptic curves
 
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CloudFlare hosts regular meetups in its San Francisco office. In the latest meetup, we invited people from academia and industry to talk about the interesting cryptographic algorithms or protocols they are working on. From hyperelliptic curves, lattice-based cryptography, new block chain modes, fully homomorphic cryptography, memory-hard hashing algorithms, to more obscure and promising ideas, this is the place to geek out. Michael is a cryptographer and software engineer. He did a PhD under Dan Boneh, and is working on a Cryptography Research. Elliptic curves have been the "next big thing" in cryptography for many years now, but they turn out to be very tricky to implement securely. Montgomery curves and Edwards curves give faster and simpler implementations. Dan Bernstein's Curve25519 and Ed25519 have caught on. Now there is now interest in a stronger curve, without sacrificing too much of this speed or simplicity. I'll discuss some alternatives in this space, in particular the curve "Ed448-Goldilocks."
Views: 1096 Cloudflare
NETWORK SECURITY - DIFFIE HELLMAN KEY EXCHANGE ALGORITHM
 
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This Algorithm is used to exchange the secret /symmetric key between sender and receiver. This exchange of key can be done with the help of public key and private key step 1 Assume prime number p step 2 Select a such that a is primitive root of p and a less than p step 3 Assume XA private key of user A step 4 Calculate YA public key of user A with the help of formula step 5 Assume XB private key of user B step 6 Calculate YB public key of user B with the help of formula step 7 Generate K secret Key using YB and XA with the help of formula at Sender side. step 8 Generate K secret Key using YA and XB with the help of formula at Receiver side.
Image Encryption using Elliptic Curve Cryptography in MATLAB
 
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This video demonstrate the process of image encryption using elliptical curve cryptography. The complete code for it is available at https://free-thesis.com/product/image-encryption-decryption-using-ecc/. This is the code which simulates the encryption and decryption of an image using random and private keys in MATLAB. The elliptic curve cryptography is applied to achieve the security of any image before transmitting it to some one so that no other can see the data hidden in the image. At the receiver end the destined user will already have the decryption key used for this. If key is altered, image will not be decrypted.
Most Hyperelliptic Curves Over Q Have No Rational Points - Manjul  Bhargava
 
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Manjul Bhargava Princeton University April 18, 2013 For more videos, visit http://video.ias.edu
encryption - decryption text by the elliptic curve cryptography depending on AlGammal   system
 
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Android application for encryption - decryption text by the elliptic curve cryptography depending on AlGammal system
qDSA Small and Secure Digital Signatures with Curve based Diffie Hellman Key Pairs
 
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Paper by Joost Renes and Benjamin Smith, presented at Asiacrypt 2017. See https://www.iacr.org/cryptodb/data/paper.php?pubkey=28278
Views: 92 TheIACR
Signcryption
 
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Signcryption In cryptography, signcryption is a public-key primitive that simultaneously performs the functions of both digital signature and encryption.Encryption and digital signature are two fundamental cryptographic tools that can guarantee the confidentiality, integrity, and non-repudiation. -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=bjYvFKDDpLE
Views: 304 WikiAudio
Elliptic Curve Cryptography Video 2
 
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Advance Cyber Security. Finding the coordinates of P_1+P_2 Point addition. Based on a Cubic curve with one real component
Views: 11638 Israel Reyes
Igor Shparlinski: Group structures of elliptic curves #2
 
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Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like. This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying the group structure, arithmetic structure of the number of points (primality, smoothness, etc.) and certain divisibility conditions. These questions are related to such celebrated problems as Lang-Trotter and Sato-Tate conjectures. More recently the interest to these questions was re-fueled by the needs of pairing based cryptography. In a series of talks we will describe the state of art in some of these directions, demonstrate the richness of underlying mathematics and pose some open questions. Recording during the thematic meeting: "Frobenius distribution on curves" the February 19, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France)
Chantal David: Distributions of Frobenius of elliptic curves #2
 
01:00:47
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area In all the following, let an elliptic curve E defined over Q without complex multiplication. For every prime ℓ, let E[ℓ]=E[ℓ](Q) be the group of ℓ-torsion points of E, and let Kℓ be the field extension obtained from Q by adding the coordinates of the ℓ-torsion points of E. This is a Galois extension of Q , and Gal(Kℓ/Q)⊆GL2(Z/ℓZ). [...] Recording during the thematic meeting: "Frobenius distributions on curves" the February 18, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent
CloudFlare meet-up: Brian Warner from Mozilla talks cryptography
 
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CloudFlare hosts regular meetups in its San Francisco office. In the latest meetup, we invited people from academia and industry to talk about the interesting cryptographic algorithms or protocols they are working on. From hyperelliptic curves, lattice-based cryptography, new block chain modes, fully homomorphic cryptography, memory-hard hashing algorithms, to more obscure and promising ideas, this is the place to geek out. Brian Warner is a security engineer with the Mozilla Cloud Services group, working on Firefox Sync and Firefox Accounts. In this CloudFlare meetup session, Brian talks about the cryptography he uses in Firefox Sync, describing what they've changed in the last couple of years, the 3 different protocols they've used, problems encountered and compromises made.
Views: 413 Cloudflare
Elliptic Curve Cryptography Demo on Android
 
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Elliptic Curve Cryptography Demo on Android Emulator.
Views: 784 Pival Infotech
A Secure Key Predistribution Scheme for WSN Using ECC and HCC
 
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Security in wireless sensor networks (WSNs) is an upcoming research field which is quite different from traditional network security mechanisms.Many applications are dependent on the secure operation of a WSN,and have serious effects if the network is disrupted. Therefore,it is necessary to protect communication between sensor nodes.Key management plays an essential role in achieving security in WSNs.To achieve security, various key predistribution schemes have been proposed in the literature. A secure key management technique in WSN is a real challenging task.In this project, a novel approach to the above problem by making use of Elliptic Curve Cryptography (ECC) and Hyperelliptic Curve Cryptosystem(HECC) is presented.In the proposed scheme, a seed key, which is a distinct point in an elliptic curve, is assigned to each sensor node prior to its deployment. The private key ring for each sensor node is generated using the point doubling mathematical operation over the seed key. When two nodes share a common private key, then a link is established between these two nodes. By suitably choosing the value of the prime field and key ring size, the probability of two nodes sharing the same private key could be increased. The performance is evaluated in terms of connectivity and resilience against node capture. The results show that the performance comaprsion for the proposed scheme ECC and HECC with polynomial genus 2.
Views: 190 VERILOG COURSE TEAM
Elliptic curves to the rescue: tackling availability and attack potential in DNSSEC
 
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Speaker: Roland van Rijswijk-Deij, SURFnet Over the past decade, we have seen the gradual rollout of DNSSEC across the name space, with adoption growing slowly but steadily. While DNSSEC was introduced to solve security problems in the DNS, it is not without its own problems. In particular, it suffers from two big problems: 1) Use of DNSSEC can lead to fragmentation of DNS responses, which impacts the availability of signed domains due to resolvers being unable to receive fragmented responses and 2) DNSSEC can be abused to create potent denial-of-service attacks based on amplification. Arguably, the choice of the RSA cryptosystem as default algorithm for DNSSEC is the root cause of these problems. RSA signatures need to be large to be cryptographically strong. Given that DNS responses can contain multiple signatures, this has a major impact on the size of these responses. Using elliptic curve cryptography, we can solve both problems with DNSSEC, because ECC offers much better cryptographic strength with far smaller keys and signatures. But using ECC will introduce one new problem: signature validation - the most commonly performed operation in DNSSEC - can be up to two orders of magnitude slower than with RSA. Thus, we run the risk of pushing workload to the edges of the network by introducing ECC in DNSSEC. This talk discusses solid research results that show 1) the benefits of using ECC in terms of solving open issues in DNSSEC, and 2) that the potential new problem of CPU use for signature validation on resolvers is not prohibitive, to such an extent that even if DNSSEC becomes universally deployed, the signature validations a resolver would need to perform can easily be handled on a single modern CPU core. Based on these results, we call for an overhaul of DNSSEC where operators move away from using RSA to using elliptic curve-based signature schemes.
Views: 357 TeamNANOG
Special vs Random Curves: Could the Conventional Wisdom Be Wrong?
 
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The conventional wisdom in cryptography is that for greatest security one should choose parameters as randomly as possible. In particular, in elliptic and hyperelliptic curve cryptography this means making random choices of the coefficients of the defining equation. One can often achieve greater efficiency by working with special curves, but that should be done only if one is willing to risk a possible lowering of security. Namely, the extra structure that allows for greater efficiency could also some day lead to specialized attacks that would not apply to random curves. This way of thinking is reasonable, and it is uncontroversial. However, some recent work opens up the possibility that it might sometimes be wrong. This talk is based on a joint paper with Alfred Menezes and Ann Hibner Koblitz.
Views: 172 Microsoft Research
Chantal David: Distributions of Frobenius of elliptic curves #1
 
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Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area In all the following, let an elliptic curve E defined over Q without complex multiplication. For every prime ℓ, let E[ℓ]=E[ℓ](Q) be the group of ℓ-torsion points of E, and let Kℓ be the field extension obtained from Q by adding the coordinates of the ℓ-torsion points of E. This is a Galois extension of Q , and Gal(Kℓ/Q)⊆GL2(Z/ℓZ). [...] Recording during the thematic meeting: "Frobenius distribution on curves" the February 17, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent
Faster Pairing Computations on Curves with High-Degree Twists.
 
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Talk at pkc 2010. Authors: Craig Costello, Tanja Lange, Michael Naehrig. See http://www.iacr.org/cryptodb/data/paper.php?pubkey=23413
Views: 235 TheIACR
CloudFlare meet-up: Trevor Perrin on end-to-end secure messages
 
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CloudFlare hosts regular meetups in its San Francisco office. In the latest meetup, we invited people from academia and industry to talk about the interesting cryptographic algorithms or protocols they are working on. From hyperelliptic curves, lattice-based cryptography, new block chain modes, fully homomorphic cryptography, memory-hard hashing algorithms, to more obscure and promising ideas, this is the place to geek out. Trevor Perrin is an independent consultant who designs and reviews cryptographic systems. There's been a recent surge of interest in end-to-end security for applications like chat, text messaging, and email. Besides deployment of existing protocols like OTR, PGP, and S/MIME, a number of projects are working on "next-generation" protocols to improve usability and security, protect new forms of communication. Trevor discusses a few such protocol designs, focusing on TextSecure and Pond as examples.
Views: 1157 Cloudflare
Cover and Decomposition Index Calculus on Elliptic Curve ...
 
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Talk at eurocrypt 2012. Authors: Antoine Joux, Vanessa Vitse. See http://www.iacr.org/cryptodb/data/paper.php?pubkey=24240
Views: 1178 TheIACR
ANT X talk on Elliptic Curves over Q(sqrt(5))
 
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See http://wstein.org/talks/2012-07-10-sqrt5/sqrt5.pdf
Views: 152 William Stein
Final Year Projects 2015 | Use of elliptic curve cryptography for multimedia encryption
 
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Including Packages ======================= * Base Paper * Complete Source Code * Complete Documentation * Complete Presentation Slides * Flow Diagram * Database File * Screenshots * Execution Procedure * Readme File * Addons * Video Tutorials * Supporting Softwares Specialization ======================= * 24/7 Support * Ticketing System * Voice Conference * Video On Demand * * Remote Connectivity * * Code Customization ** * Document Customization ** * Live Chat Support * Toll Free Support * Call Us:+91 967-778-1155 Visit Our Channel: http://www.youtube.com/clickmyproject Mail Us: [email protected]
Views: 1199 myproject bazaar
3rd BIU Winter School on Cryptography: The basics of elliptic curves - Nigel Smart
 
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The 3rd Bar-Ilan Winter School on Cryptography: Bilinear Pairings in Cryptography, which was held between February 4th - 7th, 2013. The event's program: http://crypto.biu.ac.il/winterschool2013/schedule2013.pdf For All 2013 Winter school Lectures: http://www.youtube.com/playlist?list=PLXF_IJaFk-9C4p3b2tK7H9a9axOm3EtjA&feature=mh_lolz Dept. of Computer Science: http://www.cs.biu.ac.il/ Bar-Ilan University: http://www1.biu.ac.il/indexE.php
Views: 5261 barilanuniversity
Composition Laws
 
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The parametrization of ideal classes of quadratic rings by binary quadratic forms has been an important tool for computing class numbers of quadratic fields. We will discuss how in this classical theorem, the integers can be replaced by the projective line, quadratic rings are then replaced by hyperelliptic curves, and ideal classes are replaced by line bundles on those curves. This gives a very explicit parametrization of line bundles on hyperelliptic curves by certain forms that are 'binary quadratic forms over the projective line.
Views: 32 Microsoft Research
Cryptographic Problems in Algebraic Geometry Lecture
 
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AGNES is a series of weekend workshops in algebraic geometry. One of our goals is to introduce graduate students to a broad spectrum of current research in algebraic geometry. AGNES is held twice a year at participating universities in the Northeast. Lecture presented by Kristin Lauter.
Views: 1602 Brown University
Introduction to Elliptic Curves - Part 1 of 8
 
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“Introduction to Elliptic Curves,” by Álvaro Lozano-Robledo. This is an overview of the theory of elliptic curves, discussing the Mordell-Weil theorem, how to compute the torsion subgroup of an elliptic curve, the 2-descent algorithm, and what is currently known about rank and torsion subgroups of elliptic curves. This is a video from CTNT, the Connecticut Summer School in Number Theory that took place at UConn during August 8th - 14th, 2016, organized by Keith Conrad, Amanda Folsom, Alvaro Lozano-Robledo, and Liang Xiao. For more information, see http://ctnt-summer.math.uconn.edu/
Views: 2979 UConn Mathematics
Elliptic Curve Digital Signature Algorithm (ECDSA) in NS2
 
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Elliptic Curve Digital Signature Algorithm (ECDSA) in ns2: To get this project in ONLINE or through TRAINING Sessions, Contact: JP INFOTECH, Old No.31, New No.86, 1st Floor, 1st Avenue, Ashok Pillar, Chennai -83.Landmark: Next to Kotak Mahendra Bank. Pondicherry Office: JP INFOTECH, #45, Kamaraj Salai,Thattanchavady, Puducherry -9.Landmark: Next to VVP Nagar Arch. Mobile: (0) 9952649690, Email: [email protected], web: www.jpinfotech.org, Blog: www.jpinfotech.blogspot.com Hardware implementation of Elliptic Curve Digital Signature Algorithm (ECDSA) on Koblitz Curves This paper presents Elliptic Curve Digital Signature Algorithm (ECDSA) hardware implementation over Koblitz subfield curves with 163-bit key length. We designed ECDSA with the purpose to improve performance and security respectively by using elliptic curve point multiplication on Koblitz curves to compute the public key and a key stream generator “W7” to generate private key. Different blocs of ECDSA are implemented on a reconfigurable hardware platform (Xilinx xc6vlx760-2ff1760). We used the hardware description language VHDL (VHSIC Hardware Description Language) for compartmental validation. The design requires 0.2 ms, 0.8 ms and 0.4 ms with 7 %, 13 % and 5 % of the device resources on Slice LUT for respectively key generation, signature generation and signature verification. The proposed ECDSA implementation is suitable to the applications that need: low-bandwidth communication, low-storage and low-computation environments. In particular our implementation is suitable to smart cards and wireless devices.
Views: 4528 jpinfotechprojects
Elliptic Curves in Sage
 
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Sage (http://sagemath.org) is the most feature rich general purpose free open source software for computing with elliptic curves. In this talk, I'll describe what Sage can compute about elliptic curves and how it does some of these computation, then discuss what Sage currently can't compute but should be able to (e.g., because Magma can).
Views: 795 Microsoft Research
Scalable Zero Knowledge via Cycles of Elliptic Curves
 
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Speaker: Alessandro Chiesa, ETH Zurich 'The First Greater Tel Aviv Area Symposium' School of Computer Science Tel-Aviv University, 13.11.14
Views: 1134 TAUVOD
Amicable Pairs and Aliquot Cycles for Elliptic Curves
 
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An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good reduction for E satisfying #E(Fp) = q and #E(Fq) = p. Aliquot cycles are analogously defined longer cycles. Although rare for non-CM curves, amicable pairs are -- surprisingly -- relatively abundant in the CM case. We present heuristics and conjectures for the frequency of amicable pairs and aliquot cycles, and some results for the CM case (including the especially intricate j=0 case). We present some open problems and computational challenges arising from this work. This is joint work with Joseph H. Silverman.
Views: 68 Microsoft Research
Igor Shparlinski: Group structures of elliptic curves #3
 
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Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like. This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying the group structure, arithmetic structure of the number of points (primality, smoothness, etc.) and certain divisibility conditions. These questions are related to such celebrated problems as Lang-Trotter and Sato-Tate conjectures. More recently the interest to these questions was re-fueled by the needs of pairing based cryptography. In a series of talks we will describe the state of art in some of these directions, demonstrate the richness of underlying mathematics and pose some open questions. Recording during the thematic meeting: "Frobenius distribution on curves" the February 21, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France)
Advances in the CM method for elliptic curves
 
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The complex multiplication method (CM method) builds an algebraic curve over a given finite field GF(q) and having an easily computable cardinality. Used at first for elliptic curves, this method is one of the building blocks of the ECPP algorithm that proves the primality of large integers, and it appeared interesting for other applications, the most recent of which being the construction of pairing friendly curves. The aim of the talk is to recall the method, give some applications, and survey recent advances on several parts of the method, due to various authors, concentrating on elliptic curves. This includes class invariant computations, and the potential use of the Montgomery/Edwards parametrization of elliptic curves.
Views: 86 Microsoft Research
Expander Graphs Based on GRH with an Application to Elliptic Curve Cryptography
 
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Authors: David Jao, University of Waterloo, Ontario CA Stephen D. Miller, Rutgers Universtiy, New Brunswick NJ, Ramarathnam Venkatesan, Microsoft Research Manusript #:JNT-D-08-00174
Views: 1143 JournalNumberTheory
MMUSSL - Elliptic Curves
 
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The Michigan Math Undergraduate Summer Seminar Lunch (MMUSSL) is a mini course series organized by, given by, and aimed at undergraduate math concentrators at the University of Michigan, with the goal of giving students a chance to share their knowledge of mathematics that interest them. All of the speakers are currently or recently graduated students at the University of Michigan. Sorry for the poor video quality. -------------------- Title: Elliptic Curves (1/1) Speaker: Gwyn Moreland Date: 6/11/14 Description: Elliptic curves arise in many problems in mathematics as a useful tool. This is much in part due to their structure and the multitude of theorems about them, especially their torsion groups. Not only that, they also generate some fun math on their own, such as the open problem of finding elliptic curves of arbitrarily high rank. The first talk will serve as an abridged introduction to elliptic curves. We will discuss their origin (parametrizations of integrands) and give a definition of an elliptic curve. We will also introduce some of the important theorems surrounding them (Nagell-Lutz, Mordell-Weil, Mazur) and then lastly look at some of their applications and where they appear in math today (BSD, cryptography).
Views: 898 Juliette Bruce
Interview Igor Shparlinski : Jean Morlet Chair (First Semester 2014)
 
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Jean-Morlet Chair on 'Number Theory and its Applications to Cryptography' Beneficiaries : Jean-Morlet Chair : Igor SHPARLINSKI School of Mathematics and Statistics University of New South Wales Sydney, Australia [email protected] Local project leader : David KOHEL I2M - Institut de Mathématiques de Marseille Aix-Marseille Université [email protected] General themes This chair was linked in parts to the thematic month on 'Arithmetics' which took part in February 2014 at CIRM. Igor Shparlinski has a career in Number theory and its applications to cryptography, with significant overlap with the research interests of the groups Dynamique Arithmétique, Combinatoire (DAC) and Arithmétique et Théorie de l'Information (ATI) in Marseille. The idea was to start the month with a week on 'Unlikely Intersections' followed by a workshop organized by members of the DAC research group. Weeks 3 and 4 were on 'Frobenius distributions' and were co-organized with the ATI group. The focus was to introduce and explore new directions of research around the proof of the Sato-Tate conjecture, its generalizations, and the related Lang-Trotter conjecture. Continuing the progression to the interactions of arithmetics with geometry, the thematic month closed with a week on the topic 'On the Conjectures of Lang and Volta'. The project was concentrated around several areas of number theory and its applications to quasi-Monte Carlo methods and cryptography. For both applications, the notion of pseudorandomness plays a very crucial role and thus they both require high quality pseudorandom number generators and randomness extractors. In turn, these applications lead to several subtle questions of analytic and combinatorial number theory, which are of intrinsic mathematical interest and involve the study of distribution of integers with prescribed arithmetic or combinatorial structure (e.g primes or smooth numbers and numbers with prescribed digit expansions). One of the new directions envisaged was to obtain polynomial analogues of several important results and conjectures which are known in the number case. Furthermore, driven by applications to elliptic curve cryptography, the project also addressed several theoritic and algorithmic questions related to elliptic and higher genus curves. The above applications were used on a combination of advanced number theory methods such as a) bounds of exponential and character sums; b) sieve methods and c) Subspace theorem and other Diophantine methods, which are developed by the members of DAC as well as the methods of algebraic geometry and commutative algebra such as d) effective forms of Hilbert's Nullstellensatz; e) Newton polytopes and f) Hilbert's Irreducibility theorem, which are developed by the members of ATI. The potential applications to pseudorandomness are of main interest to the members of DAC, while the applications to elliptic curve cryptography are one of the main directions of ATI. More specifically, the project consisted of the following closely related and cross-fertilising areas: 1. Pseudorandom number generators 2. Integers of cryptographic interest 3. Distribution of points in small boxes on curves over finite fields 4. Arithmetic and group theoretic properties of elliptic curves over finite fields. Interview : July 2014 By Stéphanie Vareilles
Modular curves over Z ☆ Mathematics Lecture
 
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lll➤ Gratis Crypto-Coins: https://crypto-airdrops.de ) More about the coarse moduli space, compactifying modular curves via generalized elliptic curves, and defining modular curves over all over Z. That´s what you will learn in this lesson. Also have a look at the other parts of the course, and thanks for watching. This video was made by another YouTube user and made available for the use under the Creative Commons licence "CC-BY". His channel can be found here: https://www.youtube.com/channel/UC5f0ii9uewnsgu0WuyNkfLQ
Computing Elliptic Curves over Q(√5)
 
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I will discuss creating (conjectural) tables of elliptic curves over Q(√5) ordered by conductor up to the first curve of rank 2. We computed these curves by first computing weight (2,2) Hilbert modular forms over Q(√5) using an algorithm of Lassina Dembélé. Using various methods we constructed the (conjecturally) corresponding elliptic curves. I will also discuss newer work towards partially extending these results to the first curve of rank 3. This is joint work with Jonathan Bober, Joanna Gaski, Ariah Klages-Mundt, Benjamin LeVeque, R. Andrew Ohana, Sebastian Pancratz, Ashwath Rabindranath, Paul Sharaba, Ari Shnidman, William Stein, and Christelle Vincent.
Views: 232 Microsoft Research
Faster Computation of the Tate Pairing
 
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Christophe Arene, *Tanja Lange, Michael Naehrig, Christophe Ritzenthaler *Department of Mathematics and Computer Science Technische Universiteit Eindhoven P.O. Box 513, 5600 MB Eindhoven Netherlands Email: [email protected] Manuscript number: JNT-D-09-00332R1
Views: 1839 JournalNumberTheory
Elliptic Curves
 
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This is an animation I did with Sage Math (http://www.sagemath.org/) for a presentation about Lenstra's Factoring Algorithm. The code for it is really simple: x, y = var('x y') T = (-7.5, 7.5) P = [] for A in [-10,..,10]:  for B in [-10,..,10]:    P.append(implicit_plot(y^2==x^3+A*x+B, T, T)) animate(P) It contains (or at least should) 21^2 or 441 frames, all of them a different elliptic curve.
Solving Diophantine equations using elliptic curves + Introduction to SAGE by Chandrakant Aribam
 
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12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution. An elliptic curve, say E, can be represented by points on a cubic equation as below with certain A, B ? Q: y2 = x3 + Ax +B A Theorem of Mordell says that that E(Q), the set of rational points of E, is a finitely generated abelian group, and thus, E(Q) = Zr ? T, for some non-negative integer r and a finite group T. Here, r is called the algebraic rank of E. The Birch and Swinnerton-Dyer conjecture relates the algebraic rank of E to the value of the L-function, L(E, s), attached to E at s = 1. Further theoretical understanding, corroborated by computations lead to a stronger version of the BSD conjecture. This refined version of the BSD conjecture provides a very precise formula for the leading term of L(E, s) at s = 1, the coefficient of (s - 1)r, in terms of various arithmetical data attached to E. Thus, the computational side of the BSD conjecture goes hand in hand with the advanced concepts in the theory of Elliptic curves. In this program, the computational aspects of the BSD conjecture with various illustrative examples, as well as p-adic L-functions, which are the p-adic analogues of the L-functions and other theoretical aspects which are important for the BSD conjecture will be discussed. CONTACT US: [email protected] PROGRAM LINK: https://www.icts.res.in/program/bsdtc2016
Seminario de Cómputo Científico - 03nov2016
 
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The end of paring based cryptography using small characteristic finite fields. Gora Adj Abstract A necessary condition for the security of a cryptosystem based on bilinear pairings over elliptic or hyperelliptic curves is that the discrete logarithm problem in the subjacent curve subgroups and the finite field subgroup must be hard. In recent years, there have been several dramatic improvements in algorithms for computing discrete logarithms in small characteristic finite fields, that consequently placed the security of the small-characteristic pairing-based cryptography in a state of uncertainty. In this talk, we will discuss these new algorithms and tell how they drastically impact the security of cryptosystems based on pairings that utilize finite fields of small characteristic. Plática dictada por el profesor Gora Adj del Departamento de Ciencias de la Computación, Cinvestav-IPN, dentro del Seminario de Cómputo Científico el día 3 de noviembre de 2016. Para más información visite la página del Laboratorio de Cómputo Científico: http://tikhonov.fciencias.unam.mx/
Sato-Tate Distributions in Genus 2 - Andrew Sutherland
 
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Andrew Sutherland Massachusetts Institute of Technology For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. Under the generalized Sato-Tate conjecture, this is equal to the distribution of characteristic polynomials of random matrices in a closed subgroup ST(A) of USp(4). The Sato-Tate group ST(A) may be defined in terms of the Galois action on any Tate module of A, and must satisfy a certain set of constraints (the Sato-Tate axioms). Up to conjugacy, we find that there are exactly 55 subgroups of USp(4) that satisfy these axioms. By analyzing the possible Galois-module structures on the R-algebra generated by the endomorphisms of A (the Galois type), we are able to establish a matching with Sato-Tate groups, proving that at most 52 of the 55 subgroups of USp(4) that satisfy the Sato-Tate axioms can actually arise for some A and k, of which at most 34 can occur when k = Q. After a large-scale numerical search, we are able to exhibit explicit examples, as Jacobians of hyperelliptic curves, that realize all 52 of the possible Sato-Tate groups of an abelian surface. I will give an overview of these results, including graphic animations of several examples. Time permitting, I will also discuss a recent computational breakthrough by David Harvey that may greatly facilitate extensions of this work to genus 3. This is joint work with Francesc Fite, Victor Rotger, and Kiran Kedlaya, and also with David Harvey. For more videos, visit http://video.ias.edu
An Elliptic Curve Package for Mathematica
 
23:45
For the latest information, please visit: http://www.wolfram.com Speaker: John McGee Wolfram developers and colleagues discussed the latest in innovative technologies for cloud computing, interactive deployment, mobile devices, and more.
Views: 156 Wolfram
Genus-2 curves with a given number of points
 
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This is a report on joint work with Kristin Lauter and Peter Stevenhagen. Broker and Stevenhagen have shown that in practice it is not hard to produce an elliptic curve (over some finite field) with a given number N of points, provided that the factorization of N is known. In his talk this week, Stevenhagen will show that the natural generalization of this method to produce genus-2 curves with a given number of points on their Jacobian is an exponential algorithm. I will consider the related problem of constructing a genus-2 curve over some finite field such that the curve itself has a given number N of points. The idea of explicit
Views: 94 Microsoft Research

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