1. APPLICATIONS OF POLYNOMIALS IN CRYPTOGRAPHY
Presented by
Javeria Faisal
Mathematics Department
DACW Phase VIII

2. INTRODUCTION Applications of mathematical equations & formulae
Mathematics is called as the MOTHER OF ALL SCIENCES
Applications of mathematical equations & formulae
from human body to the universe,
from simple machines to space crafts,
from house hold to complex business modeling.

3. SCHEME OF PRESENTATION
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

4. OBJECTIVE
To apprise the students about practical applications of mathematics, 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 cryptography

5. 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

6. SHORT TERM GOALS
Adopt 21st century approaches in my instructional techniques
Update the students about the latest applications of Mathematics
Develop the analyzing capabilities of the students in the field of cryptography

7. TOPIC TO BE DISCUSSED POLYNOMIALS
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

8. TYPES OF POLYNOMIALS BINOMIAL 5x-1 TRINOMIAL 4+3x + 5y2 MONOMIAL 3xy2
Polynomial of term
MONOMIAL
3xy2
Polynomial of terms
BINOMIAL
5x-1
Polynomial of terms
TRINOMIAL
4+3x + 5y2

9. For example degree of given polynomial is “3”
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

10. NOMENCLATURE OF POLYNOMIALS
POLYNOMIAL ORDER
POLYNOMIAL NAME 1
Linear 2
Quadratic 3
Cubic 4
Quartic 5
Quintic 6
Sextic

11. anything written in a secret code, cipher, or the like.
CRYPTOGRAPHY
Definition
the 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.

12. 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.

13. Asymmetric key cryptography
Types of cryptography
Symmetric
key cryptography
Asymmetric key cryptography

14. CRYPTOGRAPHY FLOW DIAGRAM
How are you
USER A
ENCRYPTION
COMMUNICATION MEDIUM
USER B
DECRYPTION

15. a set of rules for solving a problem in a finite number of steps.
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

16. ENCRYPTION
Original Text
Coded Text
Encrypted Text

17. DECRYPTION
Encrypted Text
Coded Text
Original Text

18. CRYPTOGRAPHY ALGORITHMS
Original
Text A B C D E F G H I J K L M
Coded
Text - x 01 02 03 04 05 06 07 08 09 10 11 12 13
Encrypted Text
f(x)=x2 + 3
004
007
012
019
028
039
052
067
084
103
124
147
172

19. CRYPTOGRAPHY ALGORITHMS
Original
Text N O P Q R S T U V W X Y Z
Coded
Text - x 14 15 16 17 18 19 20 21 22 23 24 25 26
Encrypted Text
f(x)=x2 + 3
199
228
259
292
327
364
403
444
487
532
579
628
679

20. CRYPTOGRAPHY ALGORITHMS
Original
Text
HOW ARE YOU
Text-x
Coded
Text-f(x)
Encrypted

21. CRYPTOGRAPHY ALGORITHMS
Original
Text A B C D E F G H I J K L M
Coded
Text - x 30 29 28 27 26 25 24 23 22 21 20 19 18
Encrypted Text
f(x)=3x2 -1
2699
2522
2351
2186
2027
1874
1727
1586
1451
1322
1199
1082
0971

22. CRYPTOGRAPHY ALGORITHMS
Original
Text N O P Q R S T U V W X Y Z
space
Coded
Text - x 17 16 15 14 13 12 11 10 09 08 07 06 05 04
Encrypted Text
f(x)=y=3x2 -1
0866
0767
0674
0587
0506
0431
0362
0299
0242
0191
0146
0107
0074
0047

23. CRYPTOGRAPHY ALGORITHMS EXERCISE # 1
LETS TRY TO ENCRYPT YOUR OWN NAME WITH THIS ALGORITHM

24. CRYPTOGRAPHY ALGORITHMS
Original
Text
JAVERIA FAISAL
Text-x
Coded
Text-f(x)
Encrypted

25. CRYPTOGRAPHY ALGORITHMS EXERCISE # 2
LETS TRY TO DECRYPT THIS

26. f(x)=x2 +x+1 CRYPTOGRAPHY ALGORITHMS EXERCISE # 3
LETS TRY TO CREATE YOUR OWN ALGORITHM FROM TH EQUATION
f(x)=x2 +x+1
WITH INCREMENTAL CODING OF ORDER 1 IN ASCENDING ORDER

27. INSTRUCTIONAL STRATEGIES
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

28. 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
› Algebra › Polynomials

29. YOU
THANK