1. COMP 170 L2 Page 1 Part 2 of Course Chapter 2 of Textbook

2. COMP 170 L2 Page 2 Part 2 of Course l Objective: Application of Number Theory in Computer security. l Number theory has a long history n E.g.: Chinese Remainder Theorem: 2300 years old l Regarded as useless until recently

3. COMP 170 L2 Page 3 Part 2 of Course l Part 2 of course: n Show how to make e-commerce secure using Number theory. n Three logic lectures: L04-L06

4. COMP 170 L2 Page 4 L04: Intro to Crypto and Modulus l Objective: n Basic mathematical concepts for Part 2 n Introduction to Cryptography l Outline n Modular Arithmetic: mod n n Operations on the set n Introduction Cryptography  Private-Key Cryptography  Caesar cipher: Using addition mod n  Crypto using multiplication mod n  Public-Key Cryptography

5. COMP 170 L2 Page 5 Modular Arithmetic

6. COMP 170 L2 Page 6 Euclid’s Division Theorem l If m = n q’ + r’, 0<= r’ <n l Then q’=q, r’=r l Examples n m=25, n=4  25 = 4 x 6 +1  q=6, r=1 n m=-25, n=4  -25 = 4 x (-7) +3  q=-7, r=3 l Will be proved later

7. COMP 170 L2 Page 7 Modular Arithmetic l Applies also to the case when m is negative.

8. COMP 170 L2 Page 8 Modular Arithmetic l Applies also to the case when m is negative.

9. COMP 170 L2 Page 9 Modular Arithmetic/Simple Properties l Note n [-25 mod 4] = 4 - [25 mod 4] l In general Example: 5 mod 4 = 1, (-5) mod 4 = 3 6 mod 4 = 2, (-6) mod 4 = 2

10. COMP 170 L2 Page 10 Modular Arithmetic/Properties

11. COMP 170 L2 Page 11 Modular Arithmetic/Properties l Examples

12. COMP 170 L2 Page 12 Intuition l Adding multiples of n to i changes the quotient, but not the remainder.

13. COMP 170 L2 Page 13

14. COMP 170 L2 Page 14

15. COMP 170 L2 Page 15 l Lemma 2.3 has a second part

16. COMP 170 L2 Page 16 L04: Intro to Crypto and Modulus l Modular Arithmetic: mod n l Operations on the set l Introduction Cryptography n Private-Key Cryptography  Caesar cipher: Using addition mod n  Cryto using multiplication mod n n Public-Key Cryptography

17. COMP 170 L2 Page 17 Modulo Arithmetic on the Set l Operations on

18. COMP 170 L2 Page 18

19. COMP 170 L2 Page 19 Laws of Arithmetic over Real Numbers

20. COMP 170 L2 Page 20 Properties of Operations on

21. COMP 170 L2 Page 21

22. COMP 170 L2 Page 22

23. COMP 170 L2 Page 23 l Does each n Has additive inverse?  Yes. -x mod n n Has multiplicative inverse?  Major question to be discussed later. Properties of Operations on

24. COMP 170 L2 Page 24 L04: Intro to Crypto and Modulus l Modular Arithmetic: mod n l Operations on the set l Introduction Cryptography n Private-Key Cryptography  Caesar cipher: Using addition mod n  Cryto using multiplication mod n n Public-Key Cryptography

25. COMP 170 L2 Page 25

26. COMP 170 L2 Page 26 L04: Intro to Crypto and Modulus l Modular Arithmetic: mod n l Modulo arithmetic on the set l Introduction Cryptography n Private-Key Cryptography  Caesar cipher: Using addition mod n  Crypto using multiplication mod n n Public-Key Cryptography

27. COMP 170 L2 Page 27 Private-Key Cryptography

28. COMP 170 L2 Page 28 Caeser Cipher and Mod 26

29. COMP 170 L2 Page 29 Caeser Cipher and Mod 26 l Encrypting l Decrypting: l E.G. s=2 n Plaintext message: SEA 18 4 0 n Cipher text: 20 6 2 n Decrypted message: 18 4 0

30. COMP 170 L2 Page 30 Caeser Cipher and Mod 26

31. COMP 170 L2 Page 31 Encrypting/Decrypting as Functions

32. COMP 170 L2 Page 32 L04: Intro to Crypto and Modulus l Modular Arithmetic: mod n l Operations on the set l Introduction Cryptography n Private-Key Cryptography  Caesar cipher: Using addition mod n  Crypto using multiplication mod n n Public-Key Cryptography

33. COMP 170 L2 Page 33 Cryptography with Multiplication mod n l Also possible to implement crypto system using multiplication mod n l Need to deal with an important new issue. l Plaintext: 5 7 8 l Ciphertext: 1 11 4

34. COMP 170 L2 Page 34 Cryptography with Multiplication mod n

35. COMP 170 L2 Page 35 Cryptography with Multiplication mod n

36. COMP 170 L2 Page 36 Multiplicative Inverse Exists?

37. COMP 170 L2 Page 37 Multiplicative Inverse Exists?

38. COMP 170 L2 Page 38 Multiplicative Inverse Exists?

39. COMP 170 L2 Page 39 Multiplicative Inverse Exists?

40. COMP 170 L2 Page 40 L04: Intro to Crypto and Modulus l Modular Arithmetic: mod n l Operations on the set l Introduction Cryptography n Private-Key Cryptography  Caesar cipher: Using addition mod n  Crypto using multiplication mod n n Public-Key Cryptography

41. COMP 170 L2 Page 41 Drawback of Private-Key Cryptosystem

42. COMP 170 L2 Page 42 Public-Key Cryptosystem

43. COMP 170 L2 Page 43 Public-Key Cryptosystem

44. COMP 170 L2 Page 44 Public-Key Cryptosystem

45. COMP 170 L2 Page 45 Public-Key Cryptosystem

46. COMP 170 L2 Page 46

47. COMP 170 L2 Page 47 Is Public-Key Cryptosystem Possible? l Need a function whose inverse is DIFFICULT to compute without private key. Sounds almost impossible. l In 1970’s, Rivest, Shamir and Adelman figured out how to do this using modular arithmetic l The result: RSA public-key crypto-system. L06.

48. COMP 170 L2 23-02-2010: Recap Page 48

49. COMP 170 L2 23-02-2010: Recap Page 49

50. COMP 170 L2 25-02-2010: Recap

51. COMP 170 L2 25-02-2010: Recap l Example of Private-Key cryptosystem n Caeser Cipher: cryptosystem using addition mod n

52. COMP 170 L2 25-02-2010: Recap l L04: Examples on multiplicative inverse l L05: n When does multiplicative inverse exist? n How to find it?