1. Lecture 9 Elliptic Curves

2. In 1984, Hendrik Lenstra described an ingenious algorithm for factoring integers that relies on properties of elliptic curves. This discovery prompted researchers to investigate other applications of elliptic curves in cryptography and computational number theory.

3. Elliptic curve cryptography (ECC) was discovered in 1985 by Neal Koblitz and Victor Miller. Elliptic curve cryptographic schemes are public-key mechanisms that provide the same functionality as RSA schemes. However, their security is based on the hardness of a different problem, namely the elliptic curve discrete logarithm problem (ECDLP).

4. Currently the best algorithms known to solve the ECDLP have fully exponential running time, in contrast to the subexponential-time algorithms known for the integer factorization problem. This means that a desired security level can be attained with significantly smaller keys in elliptic curve systems than is possible with their RSA counterparts.

5. For example, it is generally accepted that a 160-bit elliptic curve key provides the same level of security as a 1024-bit RSA key. The advantages that can be gained from smaller key sizes include speed and efficient use of power, bandwidth, and storage.

6. Outline  Weierstrass Equation  Elliptic Curves over R  Elliptic Curves over Finite Field  Elliptic Curve Cryptosystems  Factoring with Elliptic Curves

7. 1 Weierstrass Equation

9. 2 Elliptic Curves Over R 2.1 Simplified Weierstrass Equations

10. 2.2 Elliptic Curves over R

11. 2.3 Addition Law

12. Chord-and-Tangent Rule

13. Chord-and-Tangent Rule (Continued)

16. Algebraic Formulas

18. 3 Elliptic Curves over Finite Field 3.1 Elliptic Curves Mod p, p≠2,3 3.1.2 Addition Law

19. 3.1.2 Example

21. 3.2 Elliptic Curves over GF(2 n )

22. 3.2.1Simplified Weierstrass Equations

23. 3.2.2 Group law

27. 3.2.3 Example

28. 3.3 Number of Points

30. 3.4 Discrete Logarithms on Elliptic Curves

32. 4 Elliptic Curve Cryptosystems 4.1 Representing Plaintext

34. 4.2 An Elliptic Curve ElGamal Cryptosystem

38. 4.3 An Elliptic Curve Digital Signature Algorithm (ECDSA)

40. 5 Factoring with Elliptic Curves 5.1 The Elliptic Curve Factoring Algorithm

46. 5.2 Degenerate Curves

50. Thank You!