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COMP 170 L2Page 1
Review for Midterm 1
Part I: Counting L01-L03
Part II: Number Theory and Cryptography L04, L05
COMP 170 L2
Counting
Why counting?
Page 2
COMP 170 L2
Counting Overview
Principles Sum principle, Product Principle, Bijection Principle
Objects to count Lists, functions, subsets, permutations, partitions
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Sum Principle
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Product Principle
Si and Sj are disjoint, |Si| = n
S = S1 U S2 U … U Sm
|S| = m |Si| = mn
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Product Principle
COMP 170 L2Page 7
Bijection Principle
COMP 170 L2
Counting Overview
Principles Sum principle, Product Principle, Bijection Principle
Objects to count Lists, functions, permutations, subsets, partitions
Page 8
COMP 170 L2Page 9
Counting Lists
COMP 170 L2Page 10
Counting Functions
COMP 170 L2
Counting FunctionsPage 11
COMP 170 L2Page 12
Counting Permutations
Number of k-element permutations
Number of permutations of a set of size n
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k-element subsets/k-elemen permutations
COMP 170 L2
Counting SubsetsPage 14Page 14
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Counting Subsets
COMP 170 L2
Avoid Double CountingPage 16
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
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Counting Partitions/Labelings
COMP 170 L2Page 18
Review for Midterm 1
Part I: Counting L01-L03
Part II: Number Theory and Cryptography L04, L05
COMP 170 L2
Part II of Course: ObjectivePage 19
Show how to make e-commerce secure using Number theory. Three logic lectures: L04-L06 L04-05 covered in Midterm 1
COMP 170 L2
L04-L05 Overview
Addition and multiplication mod n Basic properties
Multiplicative inverse GCD
Extended GCD algorithm
Introduction to cryptography
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COMP 170 L2
Modular ArithmeticPage 21
COMP 170 L2
Euclid’s Division Theorem
Proved:
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Proof technique
Proof by contradiction
Proof by smallest counter example
COMP 170 L2
Basic PropertiesPage 23
COMP 170 L2
L04-L05 Overview
Addition and multiplication mod n Basic properties
Multiplicative inverse GCD
Extended GCD algorithm
Introduction to cryptography
Page 24
COMP 170 L2
Link to GCDPage 25
COMP 170 L2
GCD AlgorithmPage 26
COMP 170 L2Page 27
The Extended GCD Algorithm
COMP 170 L2
Multiplicative InversePage 28
a has multiplicative inverse in Zn iff gcd(a, n) =1
In that case, inverse of a = x mod n.
COMP 170 L2
L04-L05 Overview
Addition and multiplication mod n Basic properties
Multiplicative inverse GCD
Extended GCD algorithm
Introduction to cryptography
Page 29
COMP 170 L2
Introduction to Cryptography
