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Midterm 1 Review: Counting Principles and Number Theory Overview

This review covers key concepts from Midterm 1, focusing on Counting Principles and Number Theory. In Part I, we delve into fundamental counting techniques including the Sum and Product Principles, Bijection Principle, and methods of counting objects like lists, functions, and permutations. Part II shifts to Number Theory and Cryptography, exploring modular arithmetic, the GCD, and the Extended GCD algorithm. Gain insights into ensuring secure e-commerce through these mathematical principles as you prepare for your midterm.

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Midterm 1 Review: Counting Principles and Number Theory Overview

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Presentation Transcript

  1. Review for Midterm 1 • Part I: Counting • L01-L03 • Part II: Number Theory and Cryptography • L04, L05

  2. Why counting? Counting

  3. Principles • Sum principle, Product Principle, Bijection Principle Objects to count • Lists, functions, subsets, permutations, partitions Counting Overview

  4. Sum Principle

  5. Product Principle Si and Sj are disjoint, |Si| = n S = S1 U S2 U … U Sm |S| = m |Si| = mn

  6. Product Principle

  7. Bijection Principle

  8. Principles • Sum principle, Product Principle, Bijection Principle Objects to count • Lists, functions, permutations, subsets, partitions Counting Overview

  9. Counting Lists

  10. Counting Functions

  11. Counting Functions

  12. Counting Permutations • Number of k-element permutations • Number of permutations of a set of size n

  13. k-element subsets/k-elemen permutations

  14. Page 14 Counting Subsets

  15. Counting Subsets

  16. Avoid Double Counting Exco Members: Year 1: 4; Year 2: 5; Year 3: 3 • WRONG ANSWER: • First choose 1 from each year • Then pick 3 from remaining 9 members • Answer

  17. Counting Partitions/Labelings

  18. Review for Midterm 1 • Part I: Counting • L01-L03 • Part II: Number Theory and Cryptography • L04, L05

  19. Part II of Course: Objective • Show how to make e-commerce secure using Number theory. • Three logic lectures: L04-L06 • L04-05 covered in Midterm 1

  20. Addition and multiplication mod n • Basic properties Multiplicative inverse • GCD • Extended GCD algorithm Introduction to cryptography L04-L05 Overview

  21. Modular Arithmetic

  22. Proved: Euclid’s Division Theorem • Proof technique • Proof by contradiction • Proof by smallest counter example

  23. Basic Properties

  24. Addition and multiplication mod n • Basic properties Multiplicative inverse • GCD • Extended GCD algorithm Introduction to cryptography L04-L05 Overview

  25. Link to GCD

  26. GCD Algorithm

  27. The Extended GCD Algorithm

  28. a has multiplicative inverse in Zn iff gcd(a, n) =1 In that case, inverse of a = x mod n. Multiplicative Inverse

  29. Addition and multiplication mod n • Basic properties Multiplicative inverse • GCD • Extended GCD algorithm Introduction to cryptography L04-L05 Overview

  30. Introduction to Cryptography

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