discrete methods in mathematical informatics lecture 1 what is elliptic curve 9 th october 2012 n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Vorapong Suppakitpaisarn http://www-imai.is.s.u-tokyo.ac.jp/~mr_t_dtone/ PowerPoint Presentation
Download Presentation
Vorapong Suppakitpaisarn http://www-imai.is.s.u-tokyo.ac.jp/~mr_t_dtone/

Loading in 2 Seconds...

play fullscreen
1 / 22

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


  • 128 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Vorapong Suppakitpaisarn http://www-imai.is.s.u-tokyo.ac.jp/~mr_t_dtone/' - dyan


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.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
discrete methods in mathematical informatics lecture 1 what is elliptic curve 9 th october 2012

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

first section of this course 5 lectures
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).

first section of this course 5 lectures1
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).

problem 1 the artillerymens dilemma is not a puzzle
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

problem 1 the artillerymens dilemma is not a puzzle cont1
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

problem 2 right triangle with rational sides
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

problem 2 right triangle with rational sides cont3
Problem 2: Right Triangle with Rational Sides (cont.)

(1681/144,62279/1728)

(-4,6)

20/3

41/6

5

3/2

exercises
Exercises

Exercise 1

Exercise 2

problem 3 fermat s last theorem
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/

problem 3 fermat s last theorem cont
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)

formal definitions of elliptic curve
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

formal definitions of elliptic curve cont1
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

first section of this course 5 lectures2
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).

exercises1
Exercises

Exercise 1

Exercise 2

thank you for your attention

Thank you for your attention

Please feel free to ask questions or comment.

scalar multiplication
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