Cryptography

1. Cryptography

2. Symmetric Cryptosystems

3. Block Ciphers:Classical examples • Affine Cipher • Affine Linear and Linear Cipher • Vigenère • Hill

4. Block Ciphers:Remark Secure block ciphers must not be (affine) linear or easy to approximate by linear functions!!!

5. Remark Implementation of a (non-linear!) substitution often occurs through a look-up table, called S-box.

6. Block Ciphers:Advanced examples • DES – Feistel Cipher • AES – Rijndael

7. DES:Feistel Cipher An iterated block cipher is a block cipher involving the sequential repetition of an internal function called rounds. an iterated block cipher

8. DES:Feistel Cipher

9. DES:Feistel Cipher

10. DES:Algorithm

11. DES:Algorithm

12. DES:Algorithm

13. DES:Algorithm

14. DES:Algorithm

15. DES:Algorithm

16. DES:Algorithm

17. DES:Algorithm

18. DES:Algorithm

19. DES:S-Boxes

20. DES:Algorithm

21. DES:Algorithm

22. DES:Algorithm

23. DES:Algorithm

24. DES:Algorithm

25. DES:Algorithm

26. DES:Algorithm

27. AES:Rijndael Cipher We again need some algebra first!

28. Intermezzo:Polynomials over Rings

29. Example:Polynomials over Rings

30. Intermezzo:Polynomials over Rings

31. Example:Polynomials over Rings

32. Intermezzo:Polynomials over Fields

33. Intermezzo:Polynomials over Fields

34. Intermezzo:Polynomials over Fields

35. Intermezzo:Polynomials over Fields

36. Example:Polynomials over Fields

37. Intermezzo:Polynomials over Fields

38. Intermezzo:Polynomials over Fields

39. Example:Polynomials over Fields

40. Intermezzo:Finite Fields • Let R be a ring. If there is a least positive integer n such that nr=0 for all r in R, then we say that R has characteristicn and write char(R)=n. When no such integer exists, we set char(R)=0. • Let F be a field with char(F)>0, then char(F) is prime. • Any finite field F has char(F)=p, where p is prime. • Let F be a finite field, where char(F)=p, then |F|=pn, with n a strictly positive integer.

41. Intermezzo:Construction of Finite Fields Hence we can also denote it by GF(p). Note that char(GF(p))=p.

42. Intermezzo:Construction of Finite Fields

43. Intermezzo:Construction of Finite Fields 2

44. Intermezzo:Construction of Finite Fields

45. Intermezzo:Construction of Finite Fields For every prime p and positive integer n there is an irreducible polynomial of degree n in Zp[x] !

46. Intermezzo:Construction of Finite Fields Theorem Let p be a prime andf(x) an irreducible polynomial of degree n in Zp[x]. Then Zp[x]/ < f(x) > (or Zp[x] mod f(x) ) is a field with pn elements. ProofAs we can choose as coset representatives polynomials of the form a0 + a1x + a2x2 + ... + an-1xn-1 , we get a ring of order pn. As in Zn we use the analogue of the Extended Euclidean algorithm to find the inverse of an element.Let g(x) be a coset representative of a non-zero element of the ring. Since f(x) is irreducible it is not divisible by any lower degree polynomial and so the gcd(g(x), f(x)) = 1. Then by the analogue of the Extended Euclidean algorithm 1 = a(x)g(x) + b(x)f(x) for some polynomials a(x), b(x). Then a(x) is a coset representative for the inverse of g(x).

47. Example:Construction of Finite Fields

48. Example:Construction of Finite Fields

49. Intermezzo:Construction of Finite Fields Conclusion:For every prime p and positive integer nthe field GF(pn) exists!