1 / 22

Vorapong Suppakitpaisarn http://www-imai.is.s.u-tokyo.ac.jp/~mr_t_dtone/

Discrete Methods in Mathematical Informatics Lecture 1 : What is Elliptic Curve? 9 th October 2012. Vorapong Suppakitpaisarn http://www-imai.is.s.u-tokyo.ac.jp/~mr_t_dtone/ vorapong@mist.i.u-tokyo.ac.jp , Eng. 6 Room 363

dyan
Download Presentation

Vorapong Suppakitpaisarn http://www-imai.is.s.u-tokyo.ac.jp/~mr_t_dtone/

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Discrete Methods in Mathematical InformaticsLecture 1: What is Elliptic Curve?9th October 2012 Vorapong Suppakitpaisarn http://www-imai.is.s.u-tokyo.ac.jp/~mr_t_dtone/ vorapong@mist.i.u-tokyo.ac.jp, Eng. 6 Room 363 Download Slide at: https://www.dropbox.com/s/xzk4dv50f4cvs18/Lecture%201.pptx?m

  2. First Section of This Course [5 lectures] Lecture 1:What is Elliptic Curve? Lecture 2:Elliptic CurveCryptography Lecture 3-4:Fast Implementationfor Elliptic Curve Cryptography Lecture 5:Factoring and Primality Testing Recommended Reading Grading • L. C. Washington, “Elliptic Curves: Number Theory and Cryptography”, Chapman & Hall/CRC, 2003. • Lecture 1: Chapter 1, Chapter 2 (2.1, 2.2) • Lecture 2: Chapter 6 (6.1 – 6.6) • Lecture 5: Chapter 7 In each lecture, 1-2 exercises will be given, Choose 3 Problems out of them. Submit tovorapong@mist.i.u-tokyo.ac.jpbefore 31 Dec 2012 H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, F. Vercauteren, "Handbook of Elliptic and Hyperelliptic Curve Cryptography", Chapman & Hall/CRC, 2005. A. Cilardo, L. Coppolino, N. Mazzocca, L. Romano, "Elliptic Curve Cryptography Engineering", Proc. of IEEE Vol. 94, No. 2, pp. 395-406 (2006).

  3. First Section of This Course [5 lectures] Lecture 1:What is Elliptic Curve? Lecture 2: Elliptic CurveCryptography Lecture 3-4:Fast Implementationfor Elliptic Curve Cryptography Lecture 5:Factoring and Primality Testing Recommended Reading Grading • L. C. Washington, “Elliptic Curves: Number Theory and Cryptography”, Chapman & Hall/CRC, 2003. • Lecture 1: Chapter 1, Chapter 2 (2.1, 2.2) • Lecture 2: Chapter 6 (6.1 – 6.6) • Lecture 5: Chapter 7 In each lecture, 1-2 exercises will be given, Choose 3 Problems out of them. Submit tovorapong@mist.i.u-tokyo.ac.jpbefore 31 Dec 2012 H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, F. Vercauteren, "Handbook of Elliptic and Hyperelliptic Curve Cryptography", Chapman & Hall/CRC, 2005. A. Cilardo, L. Coppolino, N. Mazzocca, L. Romano, "Elliptic Curve Cryptography Engineering", Proc. of IEEE Vol. 94, No. 2, pp. 395-406 (2006).

  4. Problem 1: The Artillerymens Dilemma (is not a) Puzzle ? http://cashflowco.hubpages.com/ Height = 0: 0 Ball  Square Elliptic Curve Height = 1: 1 Ball  Square Height = 2: 1 + 4 = 5 Balls  Not Square Height = 3: 1 + 4 + 9 = 14 Balls  Not Square Height = 4: 1 + 4 + 9 + 16 = 30 Balls  Not Square

  5. Problem 1: The Artillerymens Dilemma (is not a) Puzzle (cont.) (1,1) (1/2,1/2) (0,0) y = x

  6. Problem 1: The Artillerymens Dilemma (is not a) Puzzle (cont.) (1,1) (1/2,1/2) (1/2,-1/2) (0,0) y = x y = 3x-2

  7. Problem 2: Right Triangle with Rational Sides We want to find a right triangle with rational sides in which area = 5 5 17 3 8 6 60 4 15 17/2 4 5 15 15/2

  8. Problem 2: Right Triangle with Rational Sides (cont.) c a ab/2 = 5 b Note Elliptic Curve

  9. Problem 2: Right Triangle with Rational Sides (cont.) (-4,6)

  10. Problem 2: Right Triangle with Rational Sides (cont.) (1681/144,62279/1728) (-4,6)

  11. Problem 2: Right Triangle with Rational Sides (cont.) (1681/144,62279/1728) (-4,6) 20/3 41/6 5 3/2

  12. Exercises Exercise 1 Exercise 2

  13. Problem 3: Fermat’s Last Theorem • Conjectured by Pierre de Fermat in Arithmetica (1637).“I have discovered a marvellous proof to this theorem, that this margin is too narrow to contain” • There are more than 1,000 attempts, butthe theorem is not proved until 1995 byAndrew Wiles. • One of his main tools is Elliptic Curve!!! http://wikipedia.com/

  14. Problem 3: Fermat’s Last Theorem (cont.) • Fermat kindly provided the proof for the case when n = 4 Elliptic Curve By several elliptic curves techniques, Fermat found that all rational solutions of the elliptic curve are (0,0), (2,0), (-2,0)

  15. Formal Definitions of Elliptic Curve Weierstrass Equation Elliptic Curve (1,1) Point Addition (1/2,1/2) (1/2,-1/2) (0,0) y = x

  16. Formal Definitions of Elliptic Curve (cont.) Point Addition

  17. Formal Definitions of Elliptic Curve (cont.) Point Addition (1681/144,62279/1728) (-4,6) (1/2,1/2) Point Double (1/2,-1/2) x = 1/2

  18. Formal Definitions of Elliptic Curve (cont.) Point Double

  19. First Section of This Course [5 lectures] Lecture 1:What is Elliptic Curve? Lecture 2:Elliptic CurveCryptography Lecture 3-4:Fast Implementationfor Elliptic Curve Cryptography Lecture 5:Factoring and Primality Testing Recommended Reading Grading • L. C. Washington, “Elliptic Curves: Number Theory and Cryptography”, Chapman & Hall/CRC, 2003. • Lecture 1: Chapter 1, Chapter 2 (2.1, 2.2) • Lecture 2: Chapter 6 (6.1 – 6.6) • Lecture 5: Chapter 7 In each lecture, 1-2 exercises will be given, Choose 3 Problems out of them. Submit tovorapong@mist.i.u-tokyo.ac.jpbefore 31 Dec 2012 H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, F. Vercauteren, "Handbook of Elliptic and Hyperelliptic Curve Cryptography", Chapman & Hall/CRC, 2005. A. Cilardo, L. Coppolino, N. Mazzocca, L. Romano, "Elliptic Curve Cryptography Engineering", Proc. of IEEE Vol. 94, No. 2, pp. 395-406 (2006).

  20. Exercises Exercise 1 Exercise 2

  21. Thank you for your attention Please feel free to ask questions or comment.

  22. Scalar Multiplication • Scalar Multiplication on Elliptic Curve S= P + P + … + P = rP whenr1 is positive integer, S,Pis a member of the curve • Double-and-add method • Let r = 14 = (01110)2 Compute rP = 14Pr = 14 = (0 1 1 1 0)2 r times Weight = 3 P 3P 7P 14P O 2P 6P 14P 3 – 1 =2Point Additions 4 – 1 = 3 Point Doubles

More Related