COMP 170 L2 Page 1 L05: Inverses and GCDs l Objective: n When does have an inverse? n How to compute...

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COMP 170 L2Page 1

L05: Inverses and GCDs

Objective: When does have an inverse? How to compute the inverse?

Need: Greatest common dividers (GCDs) Results will be used in L06.

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Inverses and GCDs

Greatest Common Divisors (GCDs)

Definitions

Euclid’s Division Theorem

The GCD Algorithm

Multiplicative Inverses

Definition and Properties

Link to GCD

The extended GCD algorithm.

Computing inverses

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Divisors of an Integer

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Prime Numbers

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Common Divisors

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Relatively Prime

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How to Find GCD?

How to calculate gcd(m, n)?

Need Euclid’s division theorem

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Inverses and GCDs

Greatest Common Divisors (GCDs)

Definitions

Euclid’s Division Theorem

The GCD Algorithm

Multiplicative Inverses

Definition and Properties

Link to GCD

The extended GCD algorithm.

Computing inverses

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Euclid’s Division Theorem

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Euclid’s Division Theorem

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Proof by Smallest Counter Example

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Inverses and GCDs

Greatest Common Divisors (GCDs)

Definitions

Euclid’s Division Theorem

The GCD Algorithm

Multiplicative Inverses

Definition and Properties

Link to GCD

The extended GCD algorithm.

Computing inverses .

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A Lemma

This lemma gives us a way to calculate GCDs.

Example

10 = 4 * 2 + 2

gcd(10, 4) = 2 = gcd(4, 2)

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Euclid’s GCD algorithm

The 2nd argument is nonnegative

Decreases in each recursive call

Becomes 0 in a finite number of steps

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Euclid’s GCD algorithm

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Euclid’s GCD algorithm

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Inverses and GCDs

Greatest Common Divisors (GCDs)

Definitions

Euclid’s Division Theorem

The GCD Algorithm

Multiplicative Inverses

Definition and Properties

Link to GCD

The extended GCD algorithm.

Computing inverses

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Multiplicative Inverse mod n

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Determine Inverses from Multiplication Table

Which nonzero elements of have multiplicative inverses?

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Determine Inverses from Multiplication Table

Which nonzero elements of have multiplicative inverses?

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It seems determining inverses is simple

Just scan multiplication tables

But do you see a problem with this method?

Yes, too complex…

For e-commerce, we need to determine inverse of integers of

more 200 or 300 digits…

Computationally .

Next:

Show a way to prove inverse does not exist.

Develop efficient way to calculate inverses if they exist.

Determine Inverses from Multiplication Table

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Multiplicative Inverse mod n

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Multiplicative Inverse mod n

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Proving Inverse does not Exist

Gives us a way to prove that inverse does not exist

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Proving Inverse does not Exist

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Proving Inverse does not Exist

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Proving Inverse does not Exist

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Inverses and GCDs

Greatest Common Divisors (GCDs)

Definitions

Euclid’s Division Theorem

The GCD Algorithm

Multiplicative Inverses

Definition and Properties

Link to GCD

The extended GCD algorithm.

Computing inverses

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Link to GCD

Objective:

Show the following two important results

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Modular Equations and Normal Equations

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Inverse and Normal Equations

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Clue on Finding Inverse Second part of the proof of Lemma 2.8:

Does this give us a way to find the inverse of a?

Yes,

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Clue on Finding Inverse

So, given a and n, if we can find x and y, such that

a x + ny =1 (*)

Then, we find inverse of a, i.e., x mod n

Given a and n, how do we find x and y, to satisfy (*)?

Link to GCD

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Link to GCD

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Link to GCD/Summary

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Inverses and GCDs

Greatest Common Divisors (GCDs)

Definitions

Euclid’s Division Theorem

The GCD Algorithm

Multiplicative Inverses

Definition and Properties

Link to GCD

The extended GCD algorithm.

Computing inverses

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The GCD algorithm Revisited

Computes GCD, but does not give x and y such thata x + ny =1

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The Extended GCD Algorithm/Base Case

Base case: k=jq

gcd(k, j) = j

j * 1 + k * 0 = gcd(k, j), x=1, y=0

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The Extended GCD Algorithm/Induction

Induction: k \= jq

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The Extended GCD Algorithm

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The Extended GCD Algorithm/Example

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Easy Manual Way to Find x and y

Without remembering:

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The Extended GCD AlgorithmPage 52

Proved by induction already.

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Related Results

Together, those two imply:

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Related ResultsPage 54

Together, those two imply:

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Inverses and GCDs

Greatest Common Divisors (GCDs)

Definitions

Euclid’s Division Theorem

The GCD Algorithm

Multiplicative Inverses

Definition and Properties

Link to GCD

The extended GCD algorithm.

Computing inverses

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Extend GCD Algo and InversesPage 57Page 57

Together, those two imply:

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Finding Inverse/Example

Find the

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Manual Way to Find InversePage 59

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02-03-2010: Recap

Proved:

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Proof technique

Proof by contradiction

Proof by smallest counter example

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02-03-2010: RecapPage 62

COMP 170 L2

02-03-2010: Today

Greatest Common Divisors (GCDs)

Definitions

Euclid’s Division Theorem

The GCD Algorithm

Multiplicative Inverses

Definition and Properties

Link to GCD (only results, proofs later)

The extended GCD algorithm.

Computing inverses

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04-03-2010: RecapPage 64

a has multiplicative inverse in Zn iff gcd(a, n) =1

In that case, inverse of a = x mod n.

Today: prove correctness

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A Note on an Old Exam QuestionPage 65

Question: Should I first try to run extended GCD or try to show

equation with no solution? How to figure “3” in second case?

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A Note on an Old Exam Question

Answer: Factorize the numbers into product of primes

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53 cannot be divided by 2, 3, 5, 7, 11, 13, 17, 23, 29

So, it is a prime number.

Inverse exist. Run extended GCD.

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A Note on an Old Exam QuestionPage 67

Answer: Factorize the numbers into product of primes

12 = 3 * 4

147 = 3 * 49

= 3 * 7 * 7

12 and 147 have common divisor 3

Left divisible by 3, but not right.

Write proof

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