1. Lecture 7 Discrete Logarithms

2. In the RSA algorithm, we saw how the difficulty of factoring yields useful cryptosystem. There is another number theory problem, namely discrete logarithms, that has similar applications. According to Diffie, the discrete logarithm problem was suggested by Gill. The discrete logarithm problem is a major open question in public-key cryptography.

3. Outline Discrete Logarithms Computing Discrete Logs
The ElGamal Public Key Cryptosystem
Bit Commitment

4. 1 Discrete Logarithms

7. 2 Computing Discrete Logs
2.1 Exhaustive Search

8. 2.2 Baby-Step Giant-Step Algorithm

9. 2.2 Baby-Step Giant-Step Algorithm (Continued)

10. 2.2 Baby-Step Giant-Step Algorithm (Continued)

11. 2.2 Baby-Step Giant-Step Algorithm (Continued)

12. 2.3 Pollard’s Rho Algorithm

13. 2.3 Pollard’s Rho Algorithm (Continued)

14. 2.3 Pollard’s Rho Algorithm (Continued)

15. 2.3 Pollard’s Rho Algorithm (Continued)

16. 2.3 Pollard’s Rho Algorithm (Continued)

17. 2.3 Pollard’s Rho Algorithm (Continued)

18. 2.4 Pohlig-Hellman Algorithm

19. 2.4 Pohlig-Hellman Algorithm (Continued)

20. 2.4 Pohlig-Hellman Algorithm (Continued)

21. 2.4 Pohlig-Hellman Algorithm (Continued)

22. 2.4 Pohlig-Hellman Algorithm (Continued)

23. 2.5 The Index-Calculus Algorithm

24. 2.5 The Index-Calculus Algorithm (Continued)

25. 2.5 The Index-Calculus Algorithm (Continued)

26. 2.5 The Index-Calculus Algorithm (Continued)

27. 2.5 The Index-Calculus Algorithm (Continued)

28. 2.5 The Index-Calculus Algorithm (Continued)

29. 3 The ElGamal Public Key Cryptosystem
The security of the ElGamal public-key encryption scheme is relies on the intractability of the discrete logarithm problem and the Diffie-Hellman problem. The basic ElGamal encryption scheme is done by ElGamal in 1985.

30. 3.1 Description

31. 3.1 Description (Continued)

32. 3.1 Description (Continued)

33. 3.2 Example

34. 3.3 Efficiency of ElGamal Encryption

35. 3.4 Security of ElGamal Encryption

36. 3.4 Security of ElGamal Encryption (Continued)

37. 4 Bit Commitment 4.1 Scenarios
(1) Alice claims that she has a method to predict the outcome of football games. She wants to sell her method to Bob. Bob asks her method works by predicting the results of the games that will be played this weekend. “No way,” says Alice. “Then you will simply make your bets and not pay me. Why don’t I show you my predictions for last week’s game?”

38. 4.2 Requirements of Bit Commitment
Alice can send a bit b, which is either 0 or 1, to Bob. It require that
(1) Bob cannot determine the value of the bit without Alice’s help.
(2) Alice cannot change the bit once she send it.
Now, for each game, Alice sends a symbol b=1 if she predicts the team will win, a symbol b=0 if she predicts it will lose. After the game has been played, Alice reveals the bit to Bob.

39. 4.3 Computing Discrete Logs Modulo 4

40. 4.3 Computing Discrete Logs Modulo 4 (Continued)

41. 4.3 Computing Discrete Logs Modulo 4 (Continued)

42. 4.4 A Bit Commitment Scheme

43. Thank You!