APPLICATIONS OF POLYNOMIALS IN CRYPTOGRAPHY

Presented byJaveria Faisal

Mathematics DepartmentDACW Phase VIII

Applications of mathematical equations & formulae

from human body to the universe,

from simple machines to space crafts,

from house hold to complex

business modeling.

INTRODUCTION

Mathematics is called as the MOTHER OF ALL SCIENCES

Introduction

Objective of the presentation

Topic chosen for analysis- Polynomials

Evaluation of types of polynomials

Creativity-Application of polynomials in cryptography

Encryption-decryption algorithms

Practical examples

SCHEME OF PRESENTATION

To apprise the

students about

practical applications

of mathematic

s, in general.

To educate the students about the variety of

polynomials

To develop power of analysis and evaluation

among the students

To inform the

students about

general working

philosophy of

cryptographyOBJECTIVE

LONG TERM GOAL

To develop a strategy based on 21st century approaches and to apprise the students about latest trends & applications of Mathematics in

the best possible way

SHORT TERM GOALS

Develop the analyzing capabilities of the students in the field of cryptography

Update the students about the latest applications of Mathematics

Adopt 21st century approaches in my instructional techniques

TOPIC TO BE DISCUSSEDPOLYNOMIALS

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in one variable (i.e., a univariate polynomial) with constant coefficients is given by

TYPES OF POLYNOMIALSPo

lyno

mia

l of

1 te

rm MONOMIAL

3xy2

Poly

nom

ial o

f

2

term

s

BINOMIAL

5x-1

Poly

nom

ial o

f

3

term

s

TRINOMIAL

4+3x + 5y2

DEGREE OF A POLYNOMIAL

The degree of a polynomial with only one variable is the largest exponent of that

variable.For example degree of given polynomial is

“3”

4x3-x+5

POLYNOMIAL ORDER POLYNOMIAL NAME

1 Linear

2 Quadratic

3 Cubic

4 Quartic

5 Quintic

6 Sextic

NOMENCLATURE OF POLYNOMIALS

CRYPTOGRAPHY

Definitionthe science or study of the techniques of

secret writing, especially code and

cipher systems, methods, and the

like.

the procedures, processes, methods, etc., of making and using secret writing, as codes or ciphers.

anything written in a secret code, cipher,

or the like.

Cryptography: The cryptographic practices are in use by the mankind since the ancient times due to the basic instinct of human beings to keep their communication secret and on the other hand their curiosity about what others are talking. In older times the cryptographic practices were limited to particular organizations specially military and secret services. Initially the symmetric key cryptography concept was adapted where the keys were forwarded to the recipient of the message prior sending the message. These keys were onwards used to cipher and decipher the messages. Some of these symmetric systems were Ceaser’s cipher, Enigma and other rotor machines. Symmetric systems are still considered as the most rapid way of cryptographic communication if the joint key exchange is established successfully.

CRYPTOGRAPHY

Types of cryptography

Symmetric key

cryptography

Asymmetric key

cryptography

CRYPTOGRAPHY FLOW DIAGRAM

How are you 4729845923829852

4729845923829852 How are you

USE

R A

USER B

COMMUNICATION MEDIUM

ENCR

YPTI

OND

ECRYPTION

ALGORITHM

Definition

a set of rules for solving a problem in a finite number of

steps.

a logical arithmetical or computational procedure that if correctly applied

ensures the solution of a problem

a recursive procedure whereby an infinite sequence

of terms can be generated

Original Text

Coded Text

Encrypted Text

ENCRYPTION

Encrypted Text

Coded Text

Original Text

DECRYPTION

CRYPTOGRAPHY ALGORITHMS

OriginalText

• A• B• C• D• E• F• G• H• I• J• K• L• M

CodedText - x

• 01• 02• 03• 04• 05• 06• 07• 08• 09• 10• 11• 12• 13

Encrypted Textf(x)=x2 + 3

• 004• 007• 012• 019• 028• 039• 052• 067• 084• 103• 124• 147• 172

CRYPTOGRAPHY ALGORITHMS

OriginalText

• N• O• P• Q• R• S• T• U• V• W• X• Y• Z

CodedText - x

• 14• 15• 16• 17• 18• 19• 20• 21• 22• 23• 24• 25• 26

Encrypted Textf(x)=x2 + 3

• 199• 228• 259• 292• 327• 364• 403• 444• 487• 532• 579• 628• 679

CRYPTOGRAPHY ALGORITHMS

•HOW ARE YOUOriginal

Text•081523

2701180527251521

CodedText-x

•067228532732004327028732728228444

EncryptedText-f(x)

CRYPTOGRAPHY ALGORITHMS

OriginalText

• A• B• C• D• E• F• G• H• I• J• K• L• M

CodedText - x

• 30• 29• 28• 27• 26• 25• 24• 23• 22• 21• 20• 19• 18

Encrypted Textf(x)=3x2 -1

• 2699• 2522• 2351• 2186• 2027• 1874• 1727• 1586• 1451• 1322• 1199• 1082• 0971

CRYPTOGRAPHY ALGORITHMS

OriginalText

• N• O• P• Q• R• S• T• U• V• W• X• Y• Z• space

CodedText - x

• 17• 16• 15• 14• 13• 12• 11• 10• 09• 08• 07• 06• 05• 04

Encrypted Textf(x)=y=3x2 -1

• 0866• 0767• 0674• 0587• 0506• 0431• 0362• 0299• 0242• 0191• 0146• 0107• 0074• 0047

CRYPTOGRAPHY ALGORITHMSEXERCISE # 1

LETS TRY TO ENCRYPT YOUR OWN NAME WITH THIS ALGORITHM

CRYPTOGRAPHY ALGORITHMS

•JAVERIA FAISAL

OriginalText

•2130092613223004253022123019

CodedText-x

•1322269902422027050614512699

•0047187426991451043126991082

EncryptedText-f(x)

CRYPTOGRAPHY ALGORITHMSEXERCISE # 2

LETS TRY TO DECRYPT THIS1451 0047 2699 0971 0047 0341

0971 2699 0506 0362

CRYPTOGRAPHY ALGORITHMSEXERCISE # 3

LETS TRY TO CREATE YOUR OWN ALGORITHM FROM TH EQUATION

f(x)=x2 +x+1WITH INCREMENTAL CODING OF ORDER 1 IN ASCENDING ORDER

21st Century approaches

21st Century skills Mode/ technique

Visualizing skills Understanding Relating in the real

scenario Comparison Demonstrating

Communicating skills

Computing skills Problem Solving Result oriented

approach Critical Thinking

Internet based research

Activity based Relating with daily

life

INSTRUCTIONAL STRATEGIES

REFERENCES

• “Handbook of Elliptic and Hyper Elliptic Curve Cryptography” by Henri Cohen & Gerhard Frey

• “Handbook of Applied Cryptography” by Alfred J. Menezes, Paul C. van Oorschot & Scott A. Vanstone

• www.wikipedia.org/wiki/Cryptography• www.wikipedia.org/wiki/Polynomial • www.mathworld.wolfram.com › Algebra ›

Polynomials

YOU

THANK