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 . edu EZRA BROWN Virginia Polytechnic Institute and State University Blacksburg VA 240610123 ezbrownmathvtedu After circles ellipses are probably the most familiar curves in all of mathematics Like circles they are a special subclass of the socalled co CLARK 01 Endomorphisms of elliptic curves Recall that a homomorphism of com plex elliptic curves is just a holomorphic map which preserves the origin It turns out that this condition is enough to force it to be a homomorphism of groups in the usual Endomorphisms of elliptic curves In this chapter we study the endomorphism ring of an elliptic curve We will assume throught out that is a 64257eld such that char unless otherwise stated This allows us to work only with elliptic curves in short Wei 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.”. Itay. . Khazon. Eyal. . Tolchinsky. Instructor: . Barukh. . Ziv. Introduction. Public key cryptography is based on the hardness of several mathematical problems such as factoring and DLP.. The public key protocols in use today are based on the discrete logarithm problem over . This problem can be solved in sub-exponential time.. Generation of Elliptic Curves (a.k.a. " NUMS" Curves) • Reduced customer confidence in NIST - standardized curves (FIPS 186 - 3) • Industry moving to Perfect Forward Secrecy (PFS) cip 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. Curves, Pairings, Cryptography. Elliptic Curves. Basic elliptic . cuves. :. Weierstrass. equation:. , with .  . The values . come from some set, usually a field.  . Part 1. Sets, Groups, Rings, Fields. UNIT V. Introduction. Objects in the real world may not always be made up of regular geometric shapes.. Surfaces are made up of curved surfaces & curved edges.. Curves are quite complicated to represent them in exact mathematical equations.. 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. Algorithms. draft-mcgrew-fundamental-ecc-02. mcgrew@cisco.. com. kmigoe@nsa.gov. Elliptic Curve Cryptography. Alternative to integer-based Key Exchange and Signature algorithms. Smaller keys and signatures. The Benefits of Reading Books,Most people read to read and the benefits of reading are surplus. But what are the benefits of reading. Keep reading to find out how reading will help you and may even add years to your life!.The Benefits of Reading Books,What are the benefits of reading you ask? Down below we have listed some of the most common benefits and ones that you will definitely enjoy along with the new adventures provided by the novel you choose to read.,Exercise the Brain by Reading .When you read, your brain gets a workout. You have to remember the various characters, settings, plots and retain that information throughout the book. Your brain is doing a lot of work and you don’t even realize it. Which makes it the perfect exercise!