APPLICATIONS OF POLYNOMIALS IN CRYPTOGRAPHY Presented by Javeria Faisal Mathematics Department DACW Phase VIII
Jan 04, 2016
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