Discrete Methods in Mathematical Informatics Lecture 1: What is Elliptic Curve? 9 th October 2012 Vorapong Suppakitpaisarn Eng. 6 Room 363 Download Slide at:

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

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

Problem 1: The Artillerymens Dilemma (is not a) Puzzle ? Height = 0: 0 Ball  Square Height = 1: 1 Ball  Square Height = 2: = 5 Balls  Not Square Height = 3: = 14 Balls  Not Square Height = 4: = 30 Balls  Not Square Elliptic Curve

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

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

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

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

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

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

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

Exercises Exercise 1 Exercise 2

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, but the theorem is not proved until 1995 by Andrew Wiles. One of his main tools is Elliptic Curve!!!

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 (0,0) (1,1) y = x (1/2,1/2) (1/2,-1/2) Weierstrass Equation Elliptic Curve Point Addition

Formal Definitions of Elliptic Curve (cont.) Point Addition

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

Formal Definitions of Elliptic Curve (cont.) Point Double

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

Exercises Exercise 1 Exercise 2

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

Scalar Multiplication Scalar Multiplication on Elliptic Curve S = P + P + … + P = rP when r 1 is positive integer, S,P is a member of the curve Double-and-add method Let r = 14 = (01110) 2 Compute rP = 14P r = 14 = ( ) 2 Weight = 3 P3P3P7P7P14P 6P6P2P2P 3 – 1 = 2 Point Additions 4 – 1 = 3 Point Doubles r times O