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

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.

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

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

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 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

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

TYPES OF POLYNOMIALS BINOMIAL 5x-1 TRINOMIAL 4+3x + 5y2 MONOMIAL 3xy2 Polynomial of 1 term MONOMIAL 3xy2 Polynomial of 2 terms BINOMIAL 5x-1 Polynomial of 3 terms TRINOMIAL 4+3x + 5y2

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

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

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.

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.

Asymmetric key cryptography Types of cryptography Symmetric key cryptography Asymmetric key cryptography

CRYPTOGRAPHY FLOW DIAGRAM How are you 4729845923829852 USER A ENCRYPTION COMMUNICATION MEDIUM USER B DECRYPTION

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

ENCRYPTION Original Text Coded Text Encrypted Text

DECRYPTION Encrypted Text Coded Text Original Text

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

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

CRYPTOGRAPHY ALGORITHMS Original Text HOW ARE YOU Text-x Coded 0815232701180527251521 Text-f(x) Encrypted 067228532732004327028732728228444

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

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

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

CRYPTOGRAPHY ALGORITHMS Original Text JAVERIA FAISAL Text-x Coded 2130092613223004253022123019 Text-f(x) Encrypted 1322269902422027050614512699 0047187426991451043126991082

CRYPTOGRAPHY ALGORITHMS EXERCISE # 2 LETS TRY TO DECRYPT THIS 1451 0047 2699 0971 0047 0341 0971 2699 0506 0362

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

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

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

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