COMP 170 L2 Page 1 Review for Midterm 1 l Part I: Counting n L01-L03 l Part II: Number Theory and Cryptography n L04, L05

COMP 170 L2 Counting l Why counting? Page 2

COMP 170 L2 Counting Overview l Principles n Sum principle, Product Principle, Bijection Principle l Objects to count n Lists, functions, subsets, permutations, partitions Page 3

COMP 170 L2 Page 4 Sum Principle

COMP 170 L2 Page 5 Product Principle l Si and Sj are disjoint, |Si| = n l S = S1 U S2 U … U Sm l |S| = m |Si| = mn

COMP 170 L2 Page 6 Product Principle

COMP 170 L2 Page 7 Bijection Principle

COMP 170 L2 Counting Overview l Principles n Sum principle, Product Principle, Bijection Principle l Objects to count n Lists, functions, permutations, subsets, partitions Page 8

COMP 170 L2 Page 9 Counting Lists

COMP 170 L2 Page 10 Counting Functions

COMP 170 L2 Counting Functions Page 11

COMP 170 L2 Page 12 Counting Permutations l Number of k-element permutations l Number of permutations of a set of size n

COMP 170 L2 Page 13 k-element subsets/k-elemen permutations

COMP 170 L2 Counting Subsets Page 14

COMP 170 L2 Page 15 Counting Subsets

COMP 170 L2 Avoid Double Counting Page 16 l Exco Members: Year 1: 4; Year 2: 5; Year 3: 3 l WRONG ANSWER: l First choose 1 from each year l Then pick 3 from remaining 9 members l Answer

COMP 170 L2 Page 17 Counting Partitions/Labelings

COMP 170 L2 Page 18 Review for Midterm 1 l Part I: Counting n L01-L03 l Part II: Number Theory and Cryptography n L04, L05

COMP 170 L2 Part II of Course: Objective Page 19 n Show how to make e-commerce secure using Number theory. n Three logic lectures: L04-L06 n L04-05 covered in Midterm 1

COMP 170 L2 L04-L05 Overview l Addition and multiplication mod n n Basic properties l Multiplicative inverse n GCD n Extended GCD algorithm l Introduction to cryptography Page 20

COMP 170 L2 Modular Arithmetic Page 21

COMP 170 L2 Euclid’s Division Theorem l Proved: Page 22 l Proof technique l Proof by contradiction l Proof by smallest counter example

COMP 170 L2 Basic Properties Page 23

COMP 170 L2 L04-L05 Overview l Addition and multiplication mod n n Basic properties l Multiplicative inverse n GCD n Extended GCD algorithm l Introduction to cryptography Page 24

COMP 170 L2 Link to GCD Page 25

COMP 170 L2 GCD Algorithm Page 26

COMP 170 L2 Page 27 The Extended GCD Algorithm

COMP 170 L2 Multiplicative Inverse Page 28 l a has multiplicative inverse in Zn iff gcd(a, n) =1 l In that case, inverse of a = x mod n.

COMP 170 L2 L04-L05 Overview l Addition and multiplication mod n n Basic properties l Multiplicative inverse n GCD n Extended GCD algorithm l Introduction to cryptography Page 29

COMP 170 L2 Introduction to Cryptography