Review for Midterm 1

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