1. Lecture 2-3 Basic Number Theory and Algebra

2. In modern cryptographic systems, the messages are represented by numerical values prior to being encrypted and transmitted. The encryption processes are mathematical operations that turn the input numerical values into output numerical values. Building, analyzing, and attacking these cryptosystem requires mathematical tools. The most important of these is number theory, especially the theory of congruences.

3. Outline Basic Notions Congruence Quadratic Residues Primitive Root
Inverting Matrices Mod n
Groups Rings Fields

4. 1 Basic Notions
1.1 Divisibility

5. 1.1 Divisibility (Continued)

6. 1.1 Divisibility (Continued)

7. 1.2 Prime
The primes less than 200:

8. 1.2 Prime (Continued)

9. 1.2 Prime (Continued)

10. 1.2 Prime (Continued)

11. 1.3 Greatest Common Divisor

12. 1.3 Greatest Common Divisor (Continued)

13. 1.3 Greatest Common Divisor (Continued)

14. 1.3 Greatest Common Divisor (Continued)

15. 1.3 Greatest Common Divisor (Continued)

16. 1.4 The Fundamental Theorem of Arithmetic

17. 1.4 The Fundamental Theorem of Arithmetic (Continued)

18. 1.4 The Fundamental Theorem of Arithmetic (Continued)

19. 1.4 The Fundamental Theorem of Arithmetic (Continued)

20. 1.4 The Fundamental Theorem of Arithmetic (Continued)

21. 1.5 Linear Diophantine Equations

22. 2 Congruences
2.1 Introduction to Congruences

23. 2.1 Introduction to Congruences (Continued)

24. 2.1 Introduction to Congruences (Continued)

25. 2.1 Introduction to Congruences (Continued)

26. 2.1 Introduction to Congruences (Continued)

27. 2.2 Linear Congruences

28. 2.2 Linear Congruences (Continued)

29. 2.2 Linear Congruences (Continued)

30. 2.2 Linear Congruences (Continued)

31. 2.3 The Chinese Remainder Theorem

32. 2.3 The Chinese Remainder Theorem (Continued)

33. 2.3 The Chinese Remainder Theorem (Continued)

34. 2.4 Polynomial Modulo Prime

35. 2.5 Fermat’s Little Theorem and Euler’s Theorem

36. 2.5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

37. 2.5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

38. 2.5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

39. 2.5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

40. 2.5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

41. 3 Quadratic Residues
3.1 Quadratic Residues and Nonresidues

42. 3.1 Quadratic Residues and Nonresidues (Continued)

43. 3.1 Quadratic Residues and Nonresidues (Continued)

44. 3.2 Modular Square Roots

45. 3.2 Modular Square Roots (Continued)

46. 4 Primitive Root
4.1 The Order of an Integer

47. 4.1 The Order of an Integer (Continued)

48. 4.1 The Order of an Integer (Continued)

49. 4.2 Primitive Root

50. 4.2 Primitive Root (Continued)

51. 5 Inverting Matrices Mod n

52. 5 Inverting Matrices Mod n (Continued)

53. 5 Inverting Matrices Mod n (Continued)

54. 6 Groups, Rings, Fields
6.1 Groups

55. 6.1 Groups (Continued)

56. 6.2 Rings

57. 6.2 Rings (Continued)

58. 6.3 Fields

59. Thank you!