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Implementation of 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: 12438 nptelhrd
Elliptic Curve Cryptography & Diffie-Hellman
 
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Today we're going over Elliptic Curve Cryptography, particularly as it pertains to the Diffie-Hellman protocol. The ECC Digital Signing Algorithm was also discussed in a separate video concerning Bitcoin's cryptography.
Views: 50055 CSBreakdown
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: 29485 nptelhrd
Elliptic Curve Cryptography, A very brief and superficial introduction
 
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by Ron Garret Bay Area Lisp and Scheme Meetup http://balisp.org/ Sat 30 Apr 2016 Hacker Dojo Mountain View, CA Abstract This will be a beginner’s introduction to elliptic curve cryptography using Lisp as a pedagogical tool. Cryptography generally relies heavily on modular arithmetic. Lisp’s ability to change the language syntax and define generic functions provides opportunities to implement modular arithmetic operations much more cleanly than other languages. Video notes The audio for the introduction and for the questions from the audience is hard to hear. I will try to improve on that in the next batch of talks. — Arthur
Views: 3177 Arthur Gleckler
Application of Elliptic Curves to 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: 10148 nptelhrd
Prng Implementation - Applied Cryptography
 
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This video is part of an online course, Applied Cryptography. Check out the course here: https://www.udacity.com/course/cs387.
Views: 3049 Udacity
Elliptic Curve Diffie Hellman
 
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A short video I put together that describes the basics of the Elliptic Curve Diffie-Hellman protocol for key exchanges.
Views: 108136 Robert Pierce
Bitcoin 101 - Elliptic Curve Cryptography - Part 4 - Generating the Public Key (in Python)
 
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Welcome to part four in our series on Elliptic Curve Cryptography. I this episode we dive into the development of the public key. In just 44 lines of code, with no special functions or imports, we produce the elliptic curve public key for use in Bitcoin. Better still, we walk you through it line by line, constant by constant. Nothing makes the process clearer and easier to understand than seeing it in straight forward code. If you've been wondering about the secp256k1 (arguably the most important piece of code in Bitcoin), well then this is the video for you. This is part 4 of our upcoming series on Elliptic Curves. Because of such strong requests, even though this is part 4, it is the first one we are releasing. In the next few weeks we will release the rest of the series. Enjoy. Here's the link to our Python code (Python 2.7.6): https://github.com/wobine/blackboard101/blob/master/EllipticCurvesPart4-PrivateKeyToPublicKey.py Here's the private key and the link to the public address that we use. Do you know why it is famous? Private Key : A0DC65FFCA799873CBEA0AC274015B9526505DAAAED385155425F7337704883E Public Address on Blockchain.info https://blockchain.info/address/1JryTePceSiWVpoNBU8SbwiT7J4ghzijzW Here's the private key we use at the end: 42F615A574E9CEB29E1D5BD0FDE55553775A6AF0663D569D0A2E45902E4339DB Public Address on Blockchain.info https://blockchain.info/address/16iTdS1yJhQ6NNQRJqsW9BF5UfgWwUsbF Welcome to WBN's Bitcoin 101 Blackboard Series -- a full beginner to expert course in bitcoin. Please like, subscribe, comment or even drop a little jangly in our bitcoin tip jar 1javsf8GNsudLaDue3dXkKzjtGM8NagQe. Thanks, WBN
Views: 20976 CRI
Elliptic curve cryptography
 
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Elliptic curve cryptography is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. One of the main benefits in comparison with non-ECC cryptography is the same level of security provided by keys of smaller size. Elliptic curves are applicable for encryption, digital signatures, pseudo-random generators and other tasks. They are also used in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic curve factorization. This video is targeted to blind users. Attribution: Article text available under CC-BY-SA Creative Commons image source in video
Views: 2702 Audiopedia
Lecture 17: Elliptic Curve Cryptography (ECC) by Christof Paar
 
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For slides, a problem set and more on learning cryptography, visit www.crypto-textbook.com
Elliptic Curve Cryptography
 
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Adding two rational points will create a third rational point
Views: 34377 Israel Reyes
Breaking ECDSA (Elliptic Curve Cryptography) - rhme2 Secure Filesystem v1.92r1 (crypto 150)
 
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We are going to recover a ECDSA private key from bad signatures. Same issue the Playstation 3 had that allowed it to be hacked. -------------------------------------- Twitter: https://twitter.com/LiveOverflow Website: http://liveoverflow.com/ Subreddit: https://www.reddit.com/r/LiveOverflow/
Views: 26617 LiveOverflow
Elliptic Curve Cryptography Authentication by NXP Semiconductors
 
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NXP Semiconductors introduces A1006 Secure Authenticator, using ECC.
Views: 1091 Interface Chips
Elliptic Curve Cryptography
 
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Views: 2207 @Scale
The Twofish Encryption Algorithm
 
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Twofish is a block cipher by Counterpane Labs, published in 1998. It was one of the five Advanced Encryption Standard (AES) finalists, and was not selected as AES. Twofish has a 128-bit block size, a key size ranging from 128 to 256 bits, and is optimized for 32-bit CPUs. Currently there is no successful cryptanalysis of Twofish. https://www.schneier.com/academic/twofish/ This animation is designed by Abdullah AlQahtani [email protected]
Views: 9635 Hemaya Group
NETWORK SECURITY - DES (DATA ENCRYPTION STANDARD) ALGORITHM
 
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DES algorithm follows the Feistel Structure Most of the Block cipher algorithms follows Feistel Structure BLOCK SIZE - 64 bits Plain Text No. of Rounds - 16 Rounds Key Size - 64 bits Sub Key Size - 48 bits No. of Sub Keys - 16 Sub Keys Cipher Text - 64 bits
Views: 116465 Sundeep Saradhi Kanthety
Elliptic Curve ElGamal Cryptosystem
 
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In this video I primarily do through the Elliptic Curve ElGamal crytposystem (Bob's variables/computations, Alice's variables/computations, what is sent, and how it is decrypted by Bob). In addition, I go over the basics of elliptic curves such as their advantages and how they are written. Digital Signatures - ElGamal: https://www.youtube.com/watch?v=Jo3wHnIH4y832,rpd=4,rpg=7,rpgr=0,rpm=t,rpr=d,rps=7 Reference: Trappe, W., & Washington, L. (2006). Introduction to cryptography: With coding theory (2nd ed.). Upper Saddle River, N.J.: Pearson Prentice Hall.
Views: 9113 Theoretically
Applied Cryptography: The Discrete Log Problem - Part 3
 
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This video covers the definition of discrete logarithm and the discrete logarithm problem. We also give several examples.
Views: 1881 Leandro Junes
Elliptic Curve Cayley Diagram in 3D (Ubigraph)
 
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Used SAGE and Ubigraph. This is a group isomorphic to Z/26 + Z/26 of points on Elliptic Curve defined by y^2 = x^3 + 673*x over Finite Field of size 677 Ubigraph's layout can't seem to sort out a perfect torus in this diagram, but the group structure says that's what it should be. There are 676 points total, including two small torsion E[r], r=13 suitable for Weil pairing which hopefully will be in a future video.
Views: 7710 Andrew L
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.
3rd BIU Winter School on Cryptography: Applications of Elliptic Curves to Cryptography - 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: 1394 barilanuniversity
21. Cryptography: Hash Functions
 
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MIT 6.046J Design and Analysis of Algorithms, Spring 2015 View the complete course: http://ocw.mit.edu/6-046JS15 Instructor: Srinivas Devadas In this lecture, Professor Devadas covers the basics of cryptography, including desirable properties of cryptographic functions, and their applications to security. License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
Views: 66975 MIT OpenCourseWare
NETWORK SECURITY - DIGITAL SIGNATURE ALGORITHM (DSA)
 
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Digital Signature : If the Sender Private key is used at encryption then it is called digital signature. This digital Signature is implemented two approaches 1) RSA Approach 2) DSS Approach.
Secret Key Exchange (Diffie-Hellman) - Computerphile
 
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How do we exchange a secret key in the clear? Spoiler: We don't - Dr Mike Pound shows us exactly what happens. Mathematics bit: https://youtu.be/Yjrfm_oRO0w Computing Limit: https://youtu.be/jv2H9fp9dT8 https://www.facebook.com/computerphile https://twitter.com/computer_phile This video was filmed and edited by Sean Riley. Computer Science at the University of Nottingham: https://bit.ly/nottscomputer Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
Views: 181764 Computerphile
Dual EC or the NSA's Backdoor: Explanations
 
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This video is an explanation following the paper Dual EC: A Standardized Backdoor by Daniel J. Bernstein, Tanja Lange and Ruben Niederhagen I have a blog here: www.cryptologie.net And you should follow me on twitter here: https://twitter.com/lyon01_david
Views: 4642 David Wong
How to Reveal the Secrets of an Obscure White-Box Implementation | Junwei Wang | RWC 2018
 
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Technical talks from the Real World Crypto conference series.
Views: 559 Real World Crypto
Asymmetric encryption - Simply explained
 
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How does public-key cryptography work? What is a private key and a public key? Why is asymmetric encryption different from symmetric encryption? I'll explain all of these in plain English! 🐦 Follow me on Twitter: https://twitter.com/savjee ✏️ Check out my blog: https://www.savjee.be 👍🏻 Like my Facebook page: https://www.facebook.com/savjee
CTNT 2018 - "Elliptic curves over finite fields" (Lecture 1) by Erik Wallace
 
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This is lecture 1 of a mini-course on "Elliptic curves over finite fields", taught by Erik Wallace, during CTNT 2018, the Connecticut Summer School in Number Theory. For more information about CTNT and other resources and notes, see https://ctnt-summer.math.uconn.edu/
Views: 306 UConn Mathematics
Endomorphisms, isogeny graphs, and moduli
 
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I will present a retrospective of aspects of my thesis, in light of applications in the last 14 years since its birth. In particular, I will focus on explicit isogenies, moduli of elliptic curves and CM structure, the 'local' Galois module structures of l-torsion and l-isogeny graphs, and 'global' structure of action visa class groups and isogenies. The focus will be directed principally towards ordinary elliptic curves over finite fields, but I will discuss briefly the supersingular case and generalizations to higher dimension.
Views: 487 Microsoft Research
Post-Quantum Key Exchange for the TLS Protocol from the Ring Learning with Errors Problem
 
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Post-Quantum Key Exchange for the TLS Protocol from the Ring Learning with Errors Problem Douglas Stebila Presented at the 2015 IEEE Symposium on Security & Privacy May 18--20, 2015 San Jose, CA http://www.ieee-security.org/TC/SP2015/ ABSTRACT Lattice-based cryptographic primitives are believed to offer resilience against attacks by quantum computers. We demonstrate the practicality of post-quantum key exchange by constructing cipher suites for the Transport Layer Security (TLS) protocol that provide key exchange based on the ring learning with errors (R-LWE) problem, we accompany these cipher suites with a rigorous proof of security. Our approach ties lattice-based key exchange together with traditional authentication using RSA or elliptic curve digital signatures: the post-quantum key exchange provides forward secrecy against future quantum attackers, while authentication can be provided using RSA keys that are issued by today's commercial certificate authorities, smoothing the path to adoption. Our cryptographically secure implementation, aimed at the 128-bit security level, reveals that the performance price when switching from non-quantum-safe key exchange is not too high. With our R-LWE cipher suites integrated into the Open SSL library and using the Apache web server on a 2-core desktop computer, we could serve 506 RLWE-ECDSA-AES128-GCM-SHA256 HTTPS connections per second for a 10 KiB payload. Compared to elliptic curve Diffie-Hellman, this means an 8 KiB increased handshake size and a reduction in throughput of only 21%. This demonstrates that provably secure post-quantum key-exchange can already be considered practical.
Daniel J. Bernstein - How to manipulate standards - project bullrun
 
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Daniel J. Bernstein - How to manipulate standards - project bullrun Daniel Julius Bernstein (sometimes known simply as djb; born October 29, 1971) is a German-American[2] mathematician, cryptologist, programmer, and professor of mathematics and computer science at the Eindhoven University of Technology and research professor at the University of Illinois at Chicago. His computer software programs qmail, publicfile, and djbdns were released as license-free software. This was used by some of the people that were offended by his criticism to stop the distribution of his software, so that Linux distributions such as Debian which used qmail internally did not distribute qmail. OpenBSD a security focused operating system had the majority of its security exploits as a result of its decision to stay with Sendmail and BIND and removed qmail and djbdns from its ports as part of the license dispute. This issue was resolved when Bernstein released the source code of his projects into public domain software in 2007. Bernstein designed his Salsa20 stream cipher in 2005 and submitted to eSTREAM for review, another variant, ChaCha20, is published by him in 2008. He also designed Curve25519, a public key cryptography scheme based on elliptic curve in 2005, and worked as the lead researcher on its Ed25519 implementation of EdDSA. Without any adoptions at first, after nearly a decade later, Edward Snowden's disclosure about the mass surveillance by the National Security Agency, especially a backdoor inside Dual_EC_DRBG, suspicions of the NIST's P curve constants[3] led to concerns[4] that the NSA had chosen values that gave them an advantage in factoring[5] public keys.[6] Since then Curve25519 and EdDSA has attracted much attention and became the de facto replacement of NIST P curve. Google has also selected ChaCha20 along with Bernstein's Poly1305 message authentication code as a replacement for RC4 in TLS, which is used for Internet security.[7] Many protocols based on his works have now standardized and used in a variety of applications, such as Apple iOS,[8] Linux kernel,[9] OpenSSH,[10][11] and Tor.[12]
Views: 364 Thomas D
Lecture 7: Introduction to Galois Fields for the AES by Christof Paar
 
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For slides, a problem set and more on learning cryptography, visit www.crypto-textbook.com
On the Security of Supersingular Isogeny Cryptosystems
 
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Steven D. Galbraith and Christophe Petit and Barak Shani and Yan Bo Ti. Talk at Asiacrypt 2016. See http://www.iacr.org/cryptodb/data/paper.php?pubkey=27886
Views: 473 TheIACR
Quantum Cryptography Explained
 
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This episode is brought to you by Squarespace: http://www.squarespace.com/physicsgirl With recent high-profile security decryption cases, encryption is more important than ever. Much of your browser usage and your smartphone data is encrypted. But what does that process actually entail? And when computers get smarter and faster due to advances in quantum physics, how will encryption keep up? http://physicsgirl.org/ ‪http://twitter.com/thephysicsgirl ‪http://facebook.com/thephysicsgirl ‪http://instagram.com/thephysicsgirl http://physicsgirl.org/ Help us translate our videos! http://www.youtube.com/timedtext_cs_panel?c=UC7DdEm33SyaTDtWYGO2CwdA&tab=2 Creator/Editor: Dianna Cowern Writer: Sophia Chen Animator: Kyle Norby Special thanks to Nathan Lysne Source: http://gva.noekeon.org/QCandSKD/QCand... http://physicsworld.com/cws/article/n... https://epic.org/crypto/export_contro... http://fas.org/irp/offdocs/eo_crypt_9... Music: APM and YouTube
Views: 265465 Physics Girl
Cryptography in D - Amaury Séchet | DConf2017
 
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Abstract: "Don't roll your own crypto" may be one of the most common advice a programmer will get when asking about crypto. No matter how many time this is repeated, this advice proved as effective as telling teenagers to practice abstinence. This talk intends instead to provide developers with an understanding of major concepts in cryptography and a set of advice on how to implement and use these safely. The talk will be a tour of major symmetric cryptographic tools such as finite fields, hash functions, block and stream cyphers, and describe how to use them to create confidentiality, integrity and authenticity. Implementing these techniques safely is a much harder task than it may seems at first. This talk will discuss how to create fast, yet secure implementations by using techniques protecting against side channel attacks and why a system level programming language such as D is necessary to do so. The talk will then move on to public key cryptography using elliptic curve. Diffie-Hellman, digital signatures, ring signatures, hierarchical key derivation and homomorphic commitment will be explained. If these words sounds like gibberish to you, you are at the right place. Once again, safe and fast implementation techniques will be discussed. If time allows, we'll discuss how these techniques are used to secure cryptocurrencies. Slides (PDF): http://dconf.org/2017/talks/sechet.pdf Speaker Bio: Amaury SECHET is a LLVM committer and the main developer of SDC. After being a software developer at Facebook for four years, he recently started working in the cryptocurrency space. Amaury on Github: https://github.com/deadalnix
Views: 1038 Sociomantic
Mod-01 Lec-12 Arithmetic Implementation Strategies for VLSI -Part III
 
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Advanced VLSI Design by Prof. A.N. Chandorkar, Prof. D.K. Sharma, Prof. Sachin Patkar, Prof. Virendra Singh,Department of Electrical Engineering,IIT Bombay. For more details on NPTEL visit http://nptel.ac.in
Views: 868 nptelhrd
Stream Ciphers - Encryption/Decryption
 
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A beginner's guide to Stream Ciphers (Encryption/Decryption).
Views: 51578 Daniel Rees
Visual Cryptography
 
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Hiding your images in style since 1994. Copyright Protection Scheme for Digital Images Using Visual Cryptography and Sampling Methods Ching-Sheng Hsu Young-Chang Hou July 2005 RIT, IMGS-362 Image Processing & Computer Vision II
Views: 25825 Matt Donato
Secret Sharing Schemes
 
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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: 12388 nptelhrd
The BB84 Protocol
 
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A short video attempting to explain the Bennett & Brassard quantum cryptography protocol. I've omitted any mention of the particular details of quantum physics that would be involved in actual real-world implementations, such as particle polarization axes, spin, and so forth, instead replacing them with abstract "processes" and freakish mysterious "machines". The physical details (interesting though they are) are not needed to understand the basics of the protocol, and I'm no physicist, so I'd probably mess them up if I tried (assuming I haven't already!). Making these images has increased my affection for Microsoft PowerPoint, and putting them all into a video has hugely exacerbated my hatred for Windows Movie Maker. NOTE: An important missing piece of information: When Alice sends qubits to Bob, she chooses between process A and process B randomly for each qubit. NOTE 2: The following video explains BB84 as well, and gives more detail regarding the physics details: http://www.youtube.com/watch?v=7SMcf1MdOaQ NOTE 3: Here is another very interesting video about quantum cryptography. Any given real-world implementation, despite using the BB84 protocol, is bound to expose weaknesses that can be exploited. For example: http://www.youtube.com/watch?v=T0WnUlF2eAo
Views: 43754 Creature Mann
Moduli of Elliptic Curves
 
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An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/ Goals of Lecture 42: * To complete the proof of the fact that a suitable region in the upper half-plane, described in the previous lecture and shown there to be a fundamental region for the unimodular group, is also a fundamental region for the elliptic modular j-invariant function * In view of the above, we complete the proof of the theorem on the Moduli of Elliptic Curves: the natural Riemann surface structure, on the set of holomorphic isomorphism classes of complex 1-dimensional tori (complex algebraic elliptic curves) identified with the set of orbits of the unimodular group in the upper half-plane, is holomorphically isomorphic via the j-invariant to the complex plane Keywords for Lecture 42: Upper half-plane, quotient by the unimodular group, orbits of the unimodular group, representative of an orbit, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, automorphic function (or) automorphic form, weight two modular function (or) weight two modular form, full modular function (or) full modular form, period two modular form, congruence-mod-2 normal subgroup of the unimodular group, projective special linear group with mod-2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower half-plane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, j-invariant of a complex torus (or) j-invariant of an algebraic elliptic curve, Fundamental theorem of Algebra, complex field is algebraically closed, fundamental region for the full modular form, fundamental region for the unimodular group, ramified (or) branched covering, group-invariant holomorphic maps, fundamental region for a group-invariant holomorphic map, fundamental parallelogram associated to a lattice in the plane, Galois theory, Galois group, Galois extension of function fields of meromorphic functions on Riemann surfaces, symmetric group, Galois covering
Views: 1922 nptelhrd
Light-weight Cryptography: Asymmetric Encryption (ELLI)
 
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http://asecuritysite.com/encryption/elli
Views: 816 Bill Buchanan OBE
What Are Bulletproofs? w/ Jonathan Bootle (zk-SNARKS, Monero, Zcash, and PIVX)
 
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In this video, we sit down with Jon Bootle, who invented a faster and more efficient form of zero-knowledge proofs (one which does not require a trusted setup). Privacy coins from Monero to PIVX have voiced plans to integrate the new technology, which is a major security and speed upgrade. ➜ Coin Crunch: https://coincrunch.io ➜ Coin Crunch Mastermind Group: https://www.facebook.com/groups/coincrunch/ ➜ Our telegram channel: https://t.me/coincrunch ➜ Coin Crunch Announcements Channel - https://t.me/coincrunchannoucements ➜ The Crunch Podcast: https://itunes.apple.com/us/podcast/the-crunch/id1363950785 Based on a 2016 improvement in the space efficiency of discrete-log based zero knowledge proofs from Jonathan Bootle and others, Bulletproofs are an even more space efficient form of zero-knowledge proof. Importantly for our purposes, these proofs also have native support for committed values such as Pedersen commitments and public keys. This lets us implement things such as rangeproofs in this general zero-knowledge framework without implementing the heavy machinery of elliptic curve arithmetic in zero knowledge. Stronger. To limit the size of transactions, our old rangeproofs constrained outputs to be in a range of size 2^32. This limited outputs to about 43 BTC, though this could be increased by reducing the granularity of proofs from 1 satoshi to 10 or 100, or by increasing the minimum value from zero. These adjustments were possible but used explicit amounts, limiting the privacy provided by the system. These 32-bit rangeproofs were approximately 2.5 KiB in size. With Adam’s optimization they would have been 2 KiB in size. With Bulletproofs, they would have been 610 bytes. With such small sizes, we might as well double the range to 64 bits, eliminating the need for any non-private adjustments. This would increase the paltry 610 bytes to 1220, right? Nope. In fact, a 64-bit Bulletproof rangeproof is only 674 bytes. Smaller. The reason that we can double the range size while adding only 64 bytes to the proof size is that they grow logarithmically in size. This is done using a variant of a the inner product argument from the Bootle et al 2016 paper. (Jonathan Bootle also helped Benedikt and Dan develop Bulletproofs). Specifically, the logarithmically-sized inner product argument described in that paper was reduced even further in Bulletproofs from 6log(N) curvepoints to 2log(N). This same trick allows aggregation of multiple rangeproofs within a transaction into one, again with only a small size increase. An aggregate of two rangeproofs would be 738 bytes, of four would be 802, and of eight would be 866. Eight 64-bit classical rangeproofs would be over 40000 bytes! Faster. This space savings is great, but our initial analysis of the technique showed that verification would be slower than the old rangeproofs. It appeared that verification of a single 64-bit proof would require more than 200 scalar-point multiplications, each one an onerous 50-microsecond affair, while the old rangeproofs needed only 128. But after further analysis, we were able to combine many of the multiplications, reducing the total number to only 147. More importantly, we realized that unlike the old rangeproofs, none of these multiplications depended on each other, and we could do them all in one big batch. As part of our work on aggregate signatures, we knew how to batch-multiply very quickly. Pieter Wuille, Greg Maxwell, Jonas Nick, Peter Dettman and myself had spent several months on this problem and had reduced the speed of 147 multiplications to only 15.5 microseconds each, getting the total verification time of a Bulletproof down to 2.3 ms, vs 5.8 ms for the old proofs. This is already more than a doubling in speed, but because our batch-multiplication gets faster the more points you give it, the performance numbers for aggregates are even more impressive. An aggregate of eight 64-bit Bulletproofs can be verified in only 11.5 ms, vs 46.8 ms for the old proofs, more than quadrupling the speed. But it gets even better. Bulletproofs support an extremely efficient form of batch verification. Of the 147 multiplications we need to do, 130 of them use the same points in every Bulletproof, which means that during batch validation, these 130 multiplications can be combined, leaving only 17 new ones. In fact, this marginal cost increases only logarithmically: for aggregates of 2 ranges, each additional proof takes 19 extra points, and for aggregates of 4 each proof takes 21. ------------------------------------------ Sources: https://eprint.iacr.org/2017/1066 https://blockstream.com/2018/02/21/bulletproofs-faster-rangeproofs-and-much-more.html http://www0.cs.ucl.ac.uk/staff/J.Bootle/ https://pivx.org https://getmonero.org
Views: 1303 Coin Crunch
Introduction to the Post-Quantum Supersingular Isogeny Diffie-Hellman Protocol
 
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A talk given at the University of Waterloo on July 12th, 2016. The intended audience was mathematics students without necessarily any prior background in cryptography or elliptic curves. Apologies for the poor audio quality. Use subtitles if you can't hear.
Views: 1929 David Urbanik
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)

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