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 value 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)

7. 1.2 Prime The primes less than 200: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199

8. 1.2 Prime (Continued)

11. 1.3 Greatest Common Divisor

12. 1.3 Greatest Common Divisor (Continued)

16. 1.4 The Fundamental Theorem of Arithmetic

17. 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)

27. 2.2 Linear Congruences

28. 2.2 Linear Congruences (Continued)

31. 2.3 The Chinese Remainder Theorem

32. 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)

41. 3 Quadratic Residues 3.1 Quadratic Residues and Nonresidues

42. 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)

49. 4.2 Primitive Root

50. 4.2 Primitive Root (Continued)

51. 5 Inverting Matrices Mod n

52. 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!