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Number Theory and Advanced Cryptography 2. Primes and Discrete Logarithms

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

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Number Theory and Advanced Cryptography 2. Primes and Discrete Logarithms

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  1. Number Theory and Advanced Cryptography2. Primes and Discrete Logarithms Chih-Hung Wang Feb. 2011 Part I: Introduction to Number Theory Part II: Advanced Cryptography

  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 ( intbitLength，intcertainty，Randomrnd ) • Public boolean isProbablePrime(intcertainty)

  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) =5 XA = 36 XB=58 q=97 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 YA B 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)

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