1. Lecture 2 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  Solving ax+by=d=gcd(a,b)  Congruence  The Chinese Remainder Theorem  Fermat’s Little Theorem and Euler’s Theorem  Primitive Root  Inverting Matrices Mod n  Square Roots 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. 2 Solving ax+by=d=gcd(a,b)

17. 3 Congruences

19. 3.1 Addition, Subtraction, Multiplication

20. 3.1 Addition, Subtraction, Multiplication (Continued)

21. 3.2 Division

22. 3.2 Division (Continued)

24. 3.3 Division (Continued)

25. 3.2 Division (Continued)

26. 4 The Chinese Remainder Theorem

27. 4 The Chinese Remainder Theorem (Continued)

31. 5 Fermat’s Little Theorem and Euler’s Theorem

32. 5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

37. 6 Primitive Root

38. 6 Primitive Root (Continued)

39. 7 Inverting Matrices Mod n

40. 7 Inverting Matrices Mod n (Continued)

42. 8 Square Roots Mod n

43. 8 Square Roots Mod n (Continued)

46. 9 Groups, Rings, Fields 9.1 Groups

47. 9.1 Groups (Continued)

48. 9.2 Rings

49. 9.2 Rings (Continued)

50. 9.3 Fields

51. Thank you!