PPT-An Introduction to Elliptic Curves

w ith reference to Lyness cycles Jonny Griffiths UEA November 2010 a x by c 0 Straight line a x 2 bxy cy 2 dx ey f 0 Conics Circle ellipse parabola hyperbola . :. An . Empirical Study. Gary M. Weiss. Alexander . Battistin. Fordham University. Motivation. Classification performance related to amount of training data. Relationship visually represented by learning curve. Number Theory and Cryptography. A Pile of Cannonballs A Square of Cannonballs. 1. 4. 9. .. .. .. 1 + 4 + 9 + . . . + x. 2. . = x (x + 1) (2x + 1)/6. x=3:. 1 + 4 + 9 = 3(4)(7)/6 = 14. Sixth Edition. by William Stallings . Chapter 10. Other Public-Key Cryptosystems. “Amongst the tribes of Central Australia every man, woman, and child has a secret or sacred name which is bestowed by the older men upon him or her soon after birth, and which is known to none but the fully initiated members of the group. This secret name is never mentioned except upon the most solemn occasions; to utter it in the hearing of men of another group would be a most serious breach of tribal custom. When mentioned at all, the name is spoken only in a whisper, and not until the most elaborate precautions have been taken that it shall be heard by no one but members of the group. The native thinks that a stranger knowing his secret name would have special power to work him ill by means of magic.”. ATM Conference, Telford. Jonny Griffiths, April 2011. 10. 3. +9. 3. =12. 3. +1. 3. = 1729. x. 3. +y. 3. = 1729. Symmetrical about y = x. x. 3. +y. 3. =(. x+y. )(x. 2. -xy+y. 2. ). (1,12). (9,10). (10,9). Elliptic Curve Cryptography. CSCI 5857: Encoding and Encryption. Outline. Encryption as points on ellip. tic curves in space. Elliptic curves and modular arithmetic. Mathematical operations on elliptic curves. Kyungpook National University. Heavy Ion Meeting 2011-02, . Muju. Resort. Feb. 27-Mar. 1, 2011. Hadronic. . rescattering. in elliptic flow & Heavy quarks at RHIC. Contents . Introduction. Hadronic. Yan and Jean. -Yves . Ollitrault. CNRS, . Institut . de . Physique Théorique . de . Saclay. and Art Poskanzer. LBNL. Azimuthal Anisotropy Distributions:. The Elliptic Power Distribution. Main Point. Presented by Hans Georg Ritter. Sergei’s 60. th. Birthday. 16 Nov 13. Sergei at Work. 2. r. ecent at Wayne State. Happy . S. ergei. 3. 2008 at . J. aipur. 2002 at MSU. Sergei Exploring the Unknown. Assoc.Prof. .. Dr. . Ahmet . Zafer . Şenalp. e-mail: . azsenalp@gmail.com. Mechanical Engineering Department. Gebze. Technical University. ME 521. Computer. . Aided. . Design. . Curves are the basics for surfaces. & . ECC Diffie-Hellman. Presenter. : Le . Thanh. . Binh. Outline. What is . Elliptic Curve ?. Addition on an elliptic curve. Elliptic Curve Crypto (ECC). ECC Diffie–Hellman . Lets start with a puzzle…. A Pile of Cannonballs A Square of Cannonballs. 1. 4. 9. .. .. .. 1 4 9 . . . x. 2. . = x (x 1) (2x 1)/6. x=3:. 1 4 9 = 3(4)(7)/6 = 14. The number of cannonballs in x layers is. By . Abhijith. . Chandrashekar. . and . Dushyant. . Maheshwary. Introduction. What are Elliptic Curves?. Curve with standard form y. 2. = x. 3 . ax b a, b . ϵ ℝ. Characteristics of Elliptic Curve. . = 1.  . Elliptic . Cone: . +. .  . Hyperboloid . of one . sheet:. +. . . = 1.  . Hyperboloid of two . sheets: .  . Elliptic . paraboloid: .  . Hyperbolic . paraboloid: .  . Session 6 . – . Contents. Cryptography Basics. Elliptic Curve (EC) Concepts. Finite Fields. Selecting an Elliptic Curve. Cryptography Using EC. Digital Signature. Cryptography Basics. Security Services Security Mechanisms.