Cryptography Applied to Cryptography Applied to Linear FunctionsLinear Functions

Louena L. ManluctaoLouena L. ManluctaoEast Early College High SchoolEast Early College High SchoolHouston Independent School Houston Independent School

DistrictDistrict

Dr. Guofei GuDr. Guofei GuAssistant ProfessorAssistant Professor

College of Computer Science and College of Computer Science and EngineeringEngineering

Texas A & M UniversityTexas A & M University

RoadmapRoadmap

• IntroductionIntroduction

- Bridging Research to Lesson- Bridging Research to Lesson

- STAAR/TEKS Objective- STAAR/TEKS Objective

• The Classroom ProjectThe Classroom Project

• Samples of Pre-Test/Post-TestSamples of Pre-Test/Post-Test

• AcknowledgementsAcknowledgements

• Q & A Q & A

Research InterestResearch Interest

•Network and system security such as Internet malware detection, defense, and analysis

• Intrusion detection, anomaly detection

• Network security

• Web and social networking security

Relevance of the ResearchRelevance of the Research

• http://www.youtube.com/watch?http://www.youtube.com/watch?v=Mh1tfKyC-bYv=Mh1tfKyC-bY

Sony Pictures hacked by Luiz Security, 1,000,000 passwords claimed stolen (update)

http://consumerist.com/2011/06/how-hackers-stole-200000-citi-accounts-by-exploiting-basic-browser-vulnerability.html

Bridging Research to Lesson in Bridging Research to Lesson in Algebra 1Algebra 1

Research Lab Algebra 1 Research Lab Algebra 1 CryptographyCryptography Probability and StatisticsProbability and Statistics Linear Linear

Number TheoryNumber Theory FunctionsFunctions Abstract AlgebraAbstract Algebra and and

SequencesSequences ProgrammingProgramming

STAAR/TEKS OBJECTIVESSTAAR/TEKS OBJECTIVES

A1.3 & A1.4 (C) Linear A1.3 & A1.4 (C) Linear Functions:Functions:(A)  use symbols to represent unknowns and

variables; and(B)  look for patterns and represent eneralizations algebraically.

((C)  connect equation notation with function C)  connect equation notation with function notation, such as y = x + 1 and f(x) = x + notation, such as y = x + 1 and f(x) = x + 1.1.

STAAR/TEKS OBJECTIVESTAAR/TEKS OBJECTIVE

(7)  Linear functions. The student (7)  Linear functions. The student formulates equations and inequalities formulates equations and inequalities based on linear functions, based on linear functions,

(A)  analyze situations involving (A)  analyze situations involving linear functions and formulate linear linear functions and formulate linear equations or inequalities to solve equations or inequalities to solve problems;problems;

THE CLASSROOM PROJECT : THE CLASSROOM PROJECT : CRYPTOGRAPHY APPLIED TO LINEAR CRYPTOGRAPHY APPLIED TO LINEAR FUNCTIONFUNCTION

• Day 1:Day 1:

• Pre-TestPre-Test

• HookHook

• Power Point Presentation on Computer Power Point Presentation on Computer Network Security and CryptographyNetwork Security and Cryptography

• Explain ( How to encrypt and decrypt)Explain ( How to encrypt and decrypt)

• Worksheets on finding the function key Worksheets on finding the function key

HOOKHOOK• Find the sum of 5 sets of 4-digit numbersFind the sum of 5 sets of 4-digit numbers

without using the calculator.without using the calculator.

• I am texting this message, -4, -13, 2, -22, -1, I am texting this message, -4, -13, 2, -22, -1, -46, -46, -73-46, -46, -73

what do you think is it?what do you think is it?

• Do you think your computers are safe from Do you think your computers are safe from cyber attack in your own homes?cyber attack in your own homes?

Power Point on Power Point on CryptographyCryptography

• What is cryptography?What is cryptography?

• Different Kinds of CryptographyDifferent Kinds of Cryptography

• How to Encrypt and DecryptHow to Encrypt and Decrypt

Encrypt/DecryptEncrypt/Decrypt

Finding The Key (Linear Finding The Key (Linear Equation)Equation)

• The key will be the equation from a given The key will be the equation from a given sequence:sequence:

• Consider: 3, 6, 9, 12, 15Consider: 3, 6, 9, 12, 15

5,9,13,17,215,9,13,17,21

9,7,5,3,1,-19,7,5,3,1,-1

-3,3,9,15,21-3,3,9,15,21

Cont.Cont.• The common difference is the multiplier The common difference is the multiplier

of n (the position of the term)of n (the position of the term)

• To find out what should be added or To find out what should be added or subtracted, find the zero termsubtracted, find the zero term

• We are now going to develop a formulaWe are now going to develop a formula

• First let us define our variables:First let us define our variables:

• T = the nth term z = the zero T = the nth term z = the zero termterm

• d= the common differenced= the common difference

• n = position in the sequencen = position in the sequence

Cont.Cont.Sequence Difference

of termsDifference Multiplied by Position n

The zero term(1st term-d)

The Equation

3,6,9,12,15 3 3n 3-3=0 T = 3n+0

5,9,13,17,21 4 4n 5-4=1 T = 4n+1

9,7,5,3,1,-1 -2 -2n 9—2=11 T = -2n+11

-3,3,9,15,21 6 6n -3-6=-9 T = 6n - 9

The equation is T = dn + z or y = mx + b

Write an equation to Write an equation to describe the following describe the following sequencessequences1.1. 9, 8, 7, 6, 59, 8, 7, 6, 5

2.2.13, 17 , 21, 25, 2913, 17 , 21, 25, 29

3.3.6, 11, 16, 21, 266, 11, 16, 21, 26

4.4.112, 100, 88, 76112, 100, 88, 76

Day 2: Day 2:

• Review of Previous LessonReview of Previous Lesson

• Discussion of Finding the Inverse of Discussion of Finding the Inverse of the equation of the line.the equation of the line.

• Decoding ActivityDecoding Activity

• Post-TestPost-Test

To Find the Inverse of an To Find the Inverse of an EquationEquation

• Exchange x and yExchange x and y

• Solve for y in terms of xSolve for y in terms of x

Ex. Y = 3xEx. Y = 3x

x = 3yx = 3y

x/3 = yx/3 = y

Find the inverse of the Find the inverse of the following equationsfollowing equations

• 1. y = 3x + 51. y = 3x + 5

• 2. y = -2x -42. y = -2x -4

• 3. y = 4x – 63. y = 4x – 6

• 4. y = 3/4x + 24. y = 3/4x + 2

The Decoding ActivityThe Decoding Activity

• Students will be grouped by two’sStudents will be grouped by two’s

• Each pair will be given a sequence where Each pair will be given a sequence where they will need to find the key.they will need to find the key.

• Use the key to find the encrypted codeUse the key to find the encrypted code

• Decipher using inverse equation.Decipher using inverse equation.

• Use : Space = 0 ; A = 1; B=2; C=3…Use : Space = 0 ; A = 1; B=2; C=3…Z=26Z=26

• Give the message.Give the message.

What is the text message?What is the text message? -4, -13, 2, -22, -1, -46, -46, -73 -4, -13, 2, -22, -1, -46, -46, -73

• Key: -1, -4, -7, -10, . . . Key: -1, -4, -7, -10, . . .

• What is the encrypted key? y = -3x+2What is the encrypted key? y = -3x+2

• What is the decrypted key? Y = -(x-What is the decrypted key? Y = -(x-2)/32)/3

• What is the deciphered message?What is the deciphered message?

2, 5 , 0, 8, 1, 16, 16, 252, 5 , 0, 8, 1, 16, 16, 25

Plaintext message isPlaintext message is: : BE HAPPY BE HAPPY

Decoding Activity: What is the message?Decoding Activity: What is the message?

• -21 3 -35 -11 5 -13 -33 5 - 37 -33 -5 -7 -37 -21 3 -35 -11 5 -13 -33 5 - 37 -33 -5 -7 -37 -15 5 3 -23 -3 5 -7 -37 -23-15 5 3 -23 -3 5 -7 -37 -23

• KEY: 3, 1, -1, -3, …KEY: 3, 1, -1, -3, …

• What is the encrypted key? ( y = mx+b) What is the encrypted key? ( y = mx+b) ______________________________________

• What is the decrypted key? ________What is the decrypted key? ________

• What is the deciphered message?What is the deciphered message?____________________

• Remember : Space = 0 A = 1 B =2 C Remember : Space = 0 A = 1 B =2 C =3… Z = 26=3… Z = 26

• Plaintext Message: Plaintext Message: __________________________________________________________________

Assessment: Pre-Test/Post TestAssessment: Pre-Test/Post TestSample QuestionsSample Questions

1. The nth term of a sequence is defined 1. The nth term of a sequence is defined to be 3n +7. The 35to be 3n +7. The 35thth term is how much term is how much less than the 39less than the 39thth term? term?

A)4 B) 12 C) 19 D) 33A)4 B) 12 C) 19 D) 33

2. Given the sequence -5, -2, 1, 4 … 2. Given the sequence -5, -2, 1, 4 … which equation best represents the which equation best represents the sequence?sequence?

A) y =3x-5 C) y = 3x-8 A) y =3x-5 C) y = 3x-8

B) y = -3x-5 D) y = -3x-8B) y = -3x-5 D) y = -3x-8

AcknowledgementsAcknowledgements

Dr. Robin Autenrieth, Dr. Cheryl Page, Dr. Arun Dr. Robin Autenrieth, Dr. Cheryl Page, Dr. Arun

Srinivasa Matthew Pariyothorn, Ashwin Rao, Srinivasa Matthew Pariyothorn, Ashwin Rao, Roberto Dimaliwat, Stephen HudsonRoberto Dimaliwat, Stephen Hudson

Dr. Guofei Gu, Chao Dr. Guofei Gu, Chao Yang, Yang,

Jialong Zhang , Jialong Zhang , Wilber RivasWilber Rivas

Question and Answer

Thanks to Chao Yang’s Presentation