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: 395
UConn Mathematics

Solutions to some typical exam questions. See my other videos
https://www.youtube.com/channel/UCmtelDcX6c-xSTyX6btx0Cw/.

Views: 35553
Randell Heyman

Much of the research in number theory, like mathematics as a whole, has been inspired by hard problems which are easy to state. A famous example is 'Fermat's Last Theorem'. Starting in the 1970's number theoretic problems have been suggested as the basis for cryptosystems, such as RSA and Diffie-Hellman. In 1985 Koblitz and Miller independently suggested that the discrete logarithm problem on elliptic curves might be more secure than the 'conventional' discrete logarithm on multiplicative groups of finite fields. Since then it has inspired a great deal of research in number theory and geometry in an attempt to understand its security. I'll give a brief historical tour concerning the elliptic curve discrete logarithm problem, and the closely connected Weil Pairing algorithm.

Views: 1195
Microsoft Research

In this lecture series, you will be learning about cryptography basic concepts and examples related to it. Elliptic Curve (ECC) with example (ECC) with example.

Views: 18291
Eezytutorials

Mapping smooth elliptic curve in simple Weierstrass form over a prime finite field and then discarding all but rational points. You can find the accompanying article at https://trustica.cz/en/2018/03/01/elliptic-curves-over-finite-fields/

Views: 870
Trustica

Views: 3121
Internetwork Security

For slides, a problem set and more on learning cryptography, visit www.crypto-textbook.com

Views: 108425
Introduction to Cryptography by Christof Paar

Views: 1199
Jeff Suzuki

Views: 1842
Jeff Suzuki

Elliptic curve cryptography
Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.ECC requires smaller keys compared to non-ECC cryptography (based on plain Galois fields) to provide equivalent security.
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
https://www.youtube.com/watch?v=UTJ2jxuyL7g

Views: 481
WikiAudio

Vídeo original: https://youtu.be/iB3HcPgm_FI
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/blackboard1...
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/1JryT...
Here's the private key we use at the end:
42F615A574E9CEB29E1D5BD0FDE55553775A6AF0663D569D0A2E45902E4339DB
Public Address on Blockchain.info
https://blockchain.info/address/16iTd...
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: 5782
Fabio Carpi

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: 2787
Audiopedia

For slides, a problem set and more on learning cryptography, visit www.crypto-textbook.com

Views: 27894
Introduction to Cryptography by Christof Paar

This is lecture 2 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: 61
UConn Mathematics

A talk about the basics of Elliptic Curve Cryptography (ECC), its use and application today, strengths and weaknesses.

Views: 23149
mrdoctorprofessorsir

Multiplication and addition tables for GF(2^3), concept of generator in GF and operations using generator.

Views: 4132
DrVikasThada

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: 12683
nptelhrd

For slides, a problem set and more on learning cryptography, visit www.crypto-textbook.com
(Don't worry, I start in German but at minute 2:00 I am switiching to English for the remainder of the lecture :)

Views: 48458
Introduction to Cryptography by Christof Paar

Visualization of point addition on elliptic curve in simple Weierstrass form over real numbers and finite field. The underlying math is explained in next article of our elliptic curves' series: https://trustica.cz/en/2018/03/15/elliptic-curves-point-addition/

Views: 327
Trustica

Dan Boneh, Stanford University
Historical Papers in Cryptography Seminar Series
http://simons.berkeley.edu/crypto2015/historical-papers-seminar-series/Dan-Boneh-2015-07-13

Views: 10659
Simons Institute

Elliptic Curve Cryptography (ECC) is hot. Far better scalable than traditional encryption, more and more data and networks are being protected using ECC. Not many people know the gory details of ECC though, which given its increasing prevalence is a very bad thing. In this presentation I will turn all members of the audience into ECC experts who will be able to implement the relevant algorithms and also audit existing implementations to find weaknesses or backdoors. Actually, I won't. To fully understand ECC to a point where you could use it in practice, you would need to spend years inside university lecture rooms to study number theory, geometry and software engineering. And then you can probably still be fooled by a backdoored implementation. What I will do, however, is explain the basics of ECC. I'll skip over the gory maths (it will help if you can add up, but that's about the extent of it) and explain how this funny thing referred to as "point addition on curves" can be used to exchange a secret code between two entities over a public connection. I will also explain how the infamous backdoor in Dual_EC_DRGB (a random number generator that uses the same kind of maths) worked. At the end of the presentation, you'll still not be able to find such backdoors yourselves and you probably realise you never will. But you will be able to understand articles about ECC a little better. And, hopefully, you will be convinced it is important that we educate more people to become ECC-experts.

Views: 22091
Security BSides London

Scalar multiplication of points on elliptic curves over finite fields explained in article https://trustica.cz/2018/04/19/elliptic-curves-scalar-multiplication2/ is shown in this video.
Subscribe to our channel and follow us on Twitter: https://twitter.com/trusticacz

Views: 164
Trustica

This shows how mechanical computers can be assembled to guide the joints of a two link serial chain to draw an elliptic cubic curve. This is an example of Kempe's universality theorem and is the work of Yang Liu.

Views: 827
mechanicaldesign101

A short video I put together that describes the basics of the Elliptic Curve Diffie-Hellman protocol for key exchanges.

Views: 110403
Robert Pierce

This is episode one of the Math Behind Bitcoin. In an effort to understand the math behind bitcoin, I try to explain it to you guys. If there are any mistakes or suggestions, please put it in the comment section below. Thanks!
Resources
- https://www.coindesk.com/math-behind-bitcoin/
- https://eng.paxos.com/blockchain-101-foundational-math
- Mastering Bitcoin by Andreas Antonopoulos
- https://www.cryptocoinsnews.com/explaining-the-math-behind-bitcoin/
- https://en.wikipedia.org/wiki/Finite_field

Views: 1081
Kevin Su

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: 9392
Theoretically

Views: 2108
Jeff Suzuki

Demonstration of Elliptic Curve Diffie-Hellman key exchange described in article https://trustica.cz/2018/05/17/elliptic-curve-diffie-hellman-key-exchange/ shows the calculation of public points and shared secret on elliptic curve in simple Weierstrass form y²=x³-2x+15 over GF(23). Starring Alice and Bob - since 1978. Consider subscribing to our YouTube channel to see some interesting cryptography-related videos in the future and maybe follow us on Twitter https://twitter.com/trusticacz as well!

Views: 315
Trustica

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: 3309
Arthur Gleckler

Benjamin Bakker
New York University
May 2, 2014
Given an elliptic curve EE over a field kk, its p-torsion EpEp gives a 2-dimensional representation of the Galois group GkGk over 𝔽pFp. The Frey-Mazur conjecture asserts that for k=ℚk=Q and p13p13, EE is in fact determined up to isogeny by the representation EpEp. In joint work with J. Tsimerman, we prove a version of the Frey-Mazur conjecture over geometric function fields: for a complex curve CC with function field kCkC, any two elliptic curves over kCkC with isomorphic pp-torsion representations are isogenous, provided pp is larger than a constant only depending on the gonality of CC. The proof involves understanding the hyperbolic geometry of a modular surface.
For more videos, visit http://video.ias.edu

Views: 194
Institute for Advanced Study

Math 706 Section 10.5
Elliptic Curves as Abelian Groups

Views: 242
Todd Cochrane

Views: 1124
Jeff Suzuki

Recently, Gaudry and Diem have proposed an index calculus method for the resolution of the DLP on elliptic curves defined over extension fields. In this talk, I will first present a variant of this method that enables to decrease the asymptotic complexity of the DLP on E(Fqn) for a large range of q and n, then introduce a second improvement provided by the use of F4 traces for polynomial system solving. Finally, I will give a practical example of our index calculus variant to the oracle-assisted Static Diffie-Hellman Problem. This is a joint work with Antoine Joux.

Views: 240
Microsoft Research

Views: 1062
Jeff Suzuki

Video for article https://trustica.cz/2018/05/03/elliptic-curves-double-and-add/ showing the double-and-add algorithm in action on prime-order curve in simple Weierstrass form y²=x³-2x+15 over GF(23). This algorithm is used for performing fast scalar multiplication of points on elliptic curves. Please consider subscribing to our YouTube channel and following us on Twitter: https://twitter.com/trusticacz

Views: 206
Trustica

This is lecture on "Factoring with Elliptic Curves", by Jeremy Teitelbaum, 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: 99
UConn Mathematics