COMP 170 L2
Page 1
Review for Midterm 1

Part I: Counting


L01-L03
Part II: Number Theory and Cryptography

L04, L05
COMP 170 L2
Page 2
Counting

Why counting?
COMP 170 L2
Page 3
Counting Overview

Principles


Sum principle, Product Principle, Bijection Principle
Objects to count

Lists, functions, subsets, permutations, partitions
COMP 170 L2
Page 4
Sum Principle
COMP 170 L2
Page 5
Product Principle

Si and Sj are disjoint, |Si| = n

S = S1 U S2 U … U Sm

|S| = m |Si| = mn
COMP 170 L2
Page 6
Product Principle
COMP 170 L2
Page 7
Bijection Principle
COMP 170 L2
Page 8
Counting Overview

Principles


Sum principle, Product Principle, Bijection Principle
Objects to count

Lists, functions, permutations, subsets, partitions
COMP 170 L2
Page 9
Counting Lists
COMP 170 L2
Page 10
Counting Functions
COMP 170 L2
Page 11
Counting Functions
COMP 170 L2
Page 12
Counting Permutations

Number of k-element permutations

Number of permutations of a set of size n
COMP 170 L2
Page 13
k-element subsets/k-elemen permutations
COMP 170 L2
Page 14
Counting Subsets
COMP 170 L2
Page 15
Counting Subsets
COMP 170 L2
Page 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
COMP 170 L2
Page 17
Counting Partitions/Labelings
COMP 170 L2
Page 18
Review for Midterm 1

Part I: Counting


L01-L03
Part II: Number Theory and Cryptography

L04, L05
COMP 170 L2
Page 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
COMP 170 L2
Page 20
L04-L05 Overview

Addition and multiplication mod n



Basic properties
Multiplicative inverse

GCD

Extended GCD algorithm
Introduction to cryptography
COMP 170 L2
Page 21
Modular Arithmetic
COMP 170 L2
Page 22
Euclid’s Division Theorem

Proved:

Proof technique

Proof by contradiction

Proof by smallest counter example
COMP 170 L2
Page 23
Basic Properties
COMP 170 L2
Page 24
L04-L05 Overview

Addition and multiplication mod n



Basic properties
Multiplicative inverse

GCD

Extended GCD algorithm
Introduction to cryptography
COMP 170 L2
Page 25
Link to GCD
COMP 170 L2
Page 26
GCD Algorithm
COMP 170 L2
Page 27
The Extended GCD Algorithm
COMP 170 L2
Page 28
Multiplicative Inverse

a has multiplicative inverse in Zn iff gcd(a, n) =1

In that case,
inverse of a = x mod n.
COMP 170 L2
Page 29
L04-L05 Overview

Addition and multiplication mod n



Basic properties
Multiplicative inverse

GCD

Extended GCD algorithm
Introduction to cryptography
COMP 170 L2
Introduction to Cryptography