2. SYMMETRIC CRYPTOSYSTEMS Symmetric Cryptosystems 20/10/2015 | pag. 2

3. Block Ciphers: Classical examples Symmetric Cryptosystems 20/10/2015 | pag. 3 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!!! Cryptography 20/10/2015 | pag. 4

5. Remark Cryptography 20/10/2015 | pag. 5 Implementation of a (non-linear!) substitution often occurs through a look-up table, called S-box.

6. Block Ciphers: Advanced examples Symmetric Cryptosystems 20/10/2015 | pag. 6 DES – Feistel Cipher AES – Rijndael

7. DES: Feistel Cipher Cryptography 20/10/2015 | pag. 7 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 Cryptography 20/10/2015 | pag. 8

9. DES: Feistel Cipher Cryptography 20/10/2015 | pag. 9

10. DES: Algorithm Cryptography 20/10/2015 | pag. 10

11. DES: Algorithm Cryptography 20/10/2015 | pag. 11

12. DES: Algorithm Cryptography 20/10/2015 | pag. 12

13. DES: Algorithm Cryptography 20/10/2015 | pag. 13

14. DES: Algorithm Cryptography 20/10/2015 | pag. 14

15. DES: Algorithm Cryptography 20/10/2015 | pag. 15

16. DES: Algorithm Cryptography 20/10/2015 | pag. 16

17. DES: Algorithm Cryptography 20/10/2015 | pag. 17

18. DES: Algorithm Cryptography 20/10/2015 | pag. 18

19. DES: S-Boxes Cryptography 20/10/2015 | pag. 19

20. DES: Algorithm Cryptography 20/10/2015 | pag. 20

21. DES: Algorithm Cryptography 20/10/2015 | pag. 21

22. DES: Algorithm Cryptography 20/10/2015 | pag. 22

23. DES: Algorithm Cryptography 20/10/2015 | pag. 23 Round number Number of left rotations 11 21 32 42 52 62 72 82 91 102 112 122 132 142 152 161

24. DES: Algorithm Cryptography 20/10/2015 | pag. 24

25. DES: Algorithm Cryptography 20/10/2015 | pag. 25

26. DES: Algorithm Cryptography 20/10/2015 | pag. 26

27. AES: Rijndael Cipher Cryptography 20/10/2015 | pag. 27 We again need some algebra first!

28. Intermezzo: Polynomials over Rings Cryptography 20/10/2015 | pag. 28

29. Example: Polynomials over Rings Cryptography 20/10/2015 | pag. 29

30. Intermezzo: Polynomials over Rings Cryptography 20/10/2015 | pag. 30

31. Example: Polynomials over Rings Cryptography 20/10/2015 | pag. 31

32. Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 32

33. Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 33

34. Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 34

35. Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 35

36. Example: Polynomials over Fields Cryptography 20/10/2015 | pag. 36

37. Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 37

38. Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 38

39. Example: Polynomials over Fields Cryptography 20/10/2015 | pag. 39

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 characteristic n 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|=p n, with n a strictly positive integer. Cryptography 20/10/2015 | pag. 40

41. Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 41 Hence we can also denote it by GF(p). Note that char(GF(p))=p.

42. Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 42

43. Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 43 2

44. Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 44

45. Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 45 For every prime p and positive integer n there is an irreducible polynomial of degree n in Z p [x] !

46. Intermezzo: Construction of Finite Fields Theorem Let p be a prime and f(x) an irreducible polynomial of degree n in Z p [x]. Then Z p [x]/ (or Z p [x] mod f(x) ) is a field with p n elements. Proof As we can choose as coset representatives polynomials of the form a 0 + a 1 x + a 2 x 2 +... + a n-1 x n-1, we get a ring of order p n. As in Z n 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). Cryptography 20/10/2015 | pag. 46

47. Example: Construction of Finite Fields Cryptography 20/10/2015 | pag. 47

48. Example: Construction of Finite Fields Cryptography 20/10/2015 | pag. 48

49. Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 49 Conclusion: For every prime p and positive integer n the field GF(p n ) exists!