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: 30526
nptelhrd

Explore the history of counting points on elliptic curves, from ancient Greece to present day. Inaugural lecture of Professor Toby Gee.
For more information please visit http://bit.ly/1r3Lu8c

Views: 74886
Imperial College London

Bjorn Poonen
Massachusetts Institute of Technology
March 26, 2015
We prove that the probability that a curve of the form y2=f(x)y2=f(x) over ℚQ with degf=2g+1degf=2g+1 has no rational point other than the point at infinity tends to 1 as gg tends to infinity. This is joint work with Michael Stoll.
More videos on http://video.ias.edu

Views: 964
Institute for Advanced Study

I will demonstrate techniques to derive the addition law on an arbitrary elliptic curve. The derived addition laws are applied to provide methods for efficiently adding points. The contributions immediately find applications in cryptology such as the efficiency improvements for elliptic curve scalar multiplication and cryptographic pairing computations. In particular, contributions are made to case of the following five forms of elliptic curves: (a) Short Weierstrass form, y^2 = x^3 + ax + b, (b) Extended Jacobi quartic form, y^2 = dx^4 + 2ax^2 + 1, (c) Twisted Hessian form, ax^3 + y^3 + 1 = dxy, (d) Twisted Edwards form, ax^2 + y^2 = 1 + dx^2y^2, (e) Twisted Jacobi intersection form, bs^2 + c^2 = 1, as^2 + d^2 = 1. These forms are the most promising candidates for efficient computations and thus considered in this talk. Nevertheless, the employed methods are capable of handling arbitrary elliptic curves.

Views: 536
Microsoft Research

Elliptic curves E can be given by plane projective cubic curves and so seem to be very simple objects. A first hint for more structure is that there is an algebraic addition law for the rational points. In fact, there is a natural isomorphism of E with its Jacobian variety, and so E is at the same time a curve of low degree and an abelian variety of smallest possible dimension. This is the reason for a very rich and deep theory behind making elliptic curves to ideal objects for both theoretical and experimental investigations, always with a strong algorithmic aspect. As outcome we find an abundance of key conjectures of arithmetic geometry inspired (and even proven) by elliptic curves. It will be the purpose of the talk to explain some of these conjectures and results and, as important and rather astonishing side effect, state why these properties of elliptic curves make them to a most efficient and secure tool for public key crypto systems based on discrete logarithms.

Views: 411
Microsoft Research

Authors:
Giovanni Mingari Scarpello
Dipartimento di Matematica per le scienze economiche e socialiviale
Filopanti, 5, 40126 Bologna Italy
[email protected]
Daniele Ritelli
Dipartimento di Matematica per le scienze economiche e socialiviale
Filopanti, 5, 40126 Bologna Italy
[email protected]
Manuscript Number: JNT-D-08-00232R1

Views: 998
JournalNumberTheory

Legendre Hyperelliptic integrals, π new formulae and Lauricella functions through the elliptic singular moduli
Scarpello, Giovanni Mingari and Ritelli, Daniele*
*Dipartimento scienze statitistiche,
viale Filopanti, 5
40127 Bologna Italy
Email: [email protected], [email protected]
Manuscript Number: JNT-D-12-00309R1

Views: 298
JournalNumberTheory

The values of the elliptic modular function $j$ at imaginary quadratic numbers $\tau$ are called singular moduli. They are of fundamental importance in the study of elliptic curves and in algebraic number theory, including the study of elliptic curves over finite fields. The theorem of Gross and Zagier has provided striking congruences satisfied by these numbers. Various aspects of this theorem were generalized in recent years by Jan Bruinier and Tonghai Yang and by Kristin Lauter and the speaker. These may be viewed as concerning the theory of curves of genus 2 and their singular moduli that are obtained by evaluating the Igusa invariants at certain 2x2 complex matrices $\tau$. I will describe some of the recent ideas introduced in this area and, time allowing, describe some current projects.

Views: 52
Microsoft Research

We discuss the connection between degree and genus of the curve.
Ref: Daniel Perrin, Algebraic Geometry

Views: 761
Harpreet Bedi

The talk 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 work guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology.

Views: 112
Microsoft Research

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: 1169
Cloudflare

RETRO TRAX WEB RADIO
http://www.report2dancefloor.com

Views: 7627
Report2Dancefloor Radio

This is the second lecture of the 2014 Minerva Lecture series at the Princeton University Mathematics Department
October 15, 2014
An introduction to aspects of mathematical logic and the arithmetic of elliptic curves that make these branches of mathematics inspiring to each other. Specifically: algebraic curves - other than the projective line - over number fields tend to acquire no new rational points over many extension fields. This feature (which I call 'diophantine stability') makes elliptic curves, in particular, useful as vehicles to establish diophantine unsolvability for many large rings. To repay the debt, mathematical logic offers consequences to the arithmetic of elliptic curves over decidable rings. I will also discuss new results about diophantine stability.

Views: 750
princetonmathematics

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: 1657
Brown University

The Erdős number (Hungarian pronunciation: [ˈɛrdøːʃ]) describes the "collaborative distance" between a person and mathematician Paul Erdős, as measured by authorship of mathematical papers.
The same principle has been applied in other fields where a particular individual has collaborated with a large and broad number of peers. The American Mathematical Society provides a free online tool to determine the Erdős Number of every mathematical author listed in the Mathematical Reviews catalogue.
This video is targeted to blind users.
Attribution:
Article text available under CC-BY-SA
Creative Commons image source in video

Views: 312
Audiopedia

50 algebraic and polynomial functions
https://www.geogebra.org/material/show/id/bDQQnv6N

Views: 88
roman chijner

A video profile of the 2014 Fields medalist Manjul Bhargava, whose search for artistic truth and beauty has led to some of the most profound recent discoveries in number theory.
QUANTA MAGAZINE
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Manjul Bhargava is a professor of mathematics at Princeton University. Read more about the work that won him a 2014 Fields Medal: https://www.quantamagazine.org/20140812-the-musical-magical-number-theorist/. Video produced by the Simons Foundation, with the cooperation of the International Mathematical Union.
Quanta Magazine is an editorially independent publication launched by the Simons Foundation.

Views: 8888
Quanta Magazine