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
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
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 20, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent
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 Website: https://www.quantamagazine.org/ Facebook: https://www.facebook.com/QuantaNews Twitter: https://twitter.com/QuantaMagazine You can also sign up for our weekly newsletter: http://eepurl.com/6FnWj. 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