1. Number Theory and Advanced Cryptography 2
Number Theory and Advanced Cryptography 2. Primes and Discrete Logarithms
Part I: Introduction to Number Theory
Part II: Advanced Cryptography
Chih-Hung Wang
Feb. 2011

2. The distribution of primes
The natural way of measuring the density of primes is to count the number of primes up to a bound x, where x is a real number. For a real number x ¸ 0, the function (x) is defined to be the number of primes up to x. Thus, (1) = 0, (2) = 1, (7.5) = 4, and so on.

3. Some values of (x)

4. The Sieve of Eratosthenes
This is an algorithm for generating all the primes up to a given bound k.

5. The prime number theorem

6. The error term in the prime number theory (1)

7. The error term in the prime number theory (2)

8. Sophie Germain primes

9. Probabilistic primality testing
Trial Division

10. Trial division

11. The Miller-Rabin test

12. Error parameter (1)

13. Error parameter (2)

14. Carmichael numbers

15. Good Primality testing (1)

16. Good Primality testing (2)

17. Error parameter

18. Generating random primes using the Miller-Rabin Test

19. Sieving up to a small bound

20. Generating a random k-bit prime

21. Perfect power testing (1)

22. Perfect power testing (2)

23. Perfect power testing (3)

24. Deterministic Primality Testing
The basic idea

25. AKS algorithm

26. Running time

27. Notes

28. Primality testing in Java
Public BigInteger ( int bitLength，int certainty，Random rnd )
Public boolean isProbablePrime (int certainty)

29. Cyclic groups
Order of group element

30. Order of group element

31. (Example)Powers of Integers, Modulo 19

32. Cyclic group & Group generator

33. Example of Cyclic Group

34. Theorem of Cyclic Group

35. Prime Order group

36. The Multiplicative Group Zn*

37. The Multiplicative Group Zn*

38. Example of The Multiplicative Group

39. Finding Primitive Root
Page 166

40. Application 1: Diffie-Hellman Key Exchange
Diffie and Hellman 1976
A number of commercial products employ this key exchange technique
This algorithm enables two users to exchange key securely

41. The Diffie-Hellman Key Exchange Protocol

42. Example of D-H Key Exchange (1)
q=97 =5
XA = 36 XB=58
YA=536=50 mod 97
YB=558=44 mod 97
K=(YB)XA mod 97 = 4436 = 75 nod 97
K=(YA)XB mod 97 = 5058 = 75 nod 97

43. Example of D-H Key Exchange (2)

44. Hybrid Encryption Diffie-Hellman based hybrid encryption system A B YA
K=(YB)xA
=(YA)xB
Mod q
SK=h(K) YB
ESK(M)
128 – 256 bits
SK can be a key of the
AES symmetric cryptosystem

45. The Man-in-the-Middle Attack (1)

46. The Man-in-the-Middle Attack (2)

47. The DH Problem and DL Problem (1)

48. The DH Problem and DL Problem (2)
Example: a = loggh = log3 5 mod 19 = 4

49. Importance of Arbitrary Instances for Intractability Assumptions
CRT
a=kiqi+ai
ri= g(p-1)/qi mod p
riai=ria (mod qi) = h(p-1)/qi mod p

50. Chinese Remainder Theorem (1)

51. Chinese Remainder Theorem (2)

52. Chinese Remainder Theorem (3)

53. Example of CRT

54. ElGamal (1)

55. ElGamal (2)

56. Meet-in-the-middle attack & Active attack of ElGamal
See Page 277 Example 8.8
Malice select
Malice sends (c1, c2’=rc2) to Alice
Alice returns rm to Malice