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Lecture 2 Basic Number Theory and Algebra

Lecture 2 Basic Number Theory and Algebra.

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Lecture 2 Basic Number Theory and Algebra

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  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)

  6. 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)

  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. 2 Solving ax+by=d=gcd(a,b)

  17. 3 Congruences

  18. 3.1 Addition, Subtraction, Multiplication

  19. 3.1 Addition, Subtraction, Multiplication (Continued)

  20. 3.2 Division

  21. 3.2 Division (Continued)

  22. 3.2 Division (Continued)

  23. 3.3 Division (Continued)

  24. 3.2 Division (Continued)

  25. 4 The Chinese Remainder Theorem

  26. 4 The Chinese Remainder Theorem (Continued)

  27. 4 The Chinese Remainder Theorem (Continued)

  28. 4 The Chinese Remainder Theorem (Continued)

  29. 4 The Chinese Remainder Theorem (Continued)

  30. 5 Fermat’s Little Theorem and Euler’s Theorem

  31. 5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

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

  33. 5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

  34. 5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

  35. 5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

  36. 6 Primitive Root

  37. 6 Primitive Root (Continued)

  38. 7 Inverting Matrices Mod n

  39. 7 Inverting Matrices Mod n (Continued)

  40. 7 Inverting Matrices Mod n (Continued)

  41. 8 Square Roots Mod n

  42. 8 Square Roots Mod n (Continued)

  43. 8 Square Roots Mod n (Continued)

  44. 8 Square Roots Mod n (Continued)

  45. 9 Groups, Rings, Fields 9.1 Groups

  46. 9.1 Groups (Continued)

  47. 9.2 Rings

  48. 9.2 Rings (Continued)

  49. 9.3 Fields

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