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Applied Number Theory

Applied Number Theory

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KALIPRASAD RATH

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Course Name :

Applied Number Theory

 

Code(Credit) : CUTM1534(3-1-0)

Course Objectives

  • To analyze, evaluate, or solve problems with in given a set of circumstances or data.
  • To understand and utilize mathematical functions and empirical principles and processes.
  • Enhance and reinforce the student’s understanding of concepts through the use of technology when appropriate.

Learning Outcomes

Upon successful completion of this course a students will:

  • Demonstrate knowledge and understanding of topics including  applications.
  • Learn methods and techniques used in number theory.
  • Use mathematical induction and other types of proof writing techniques and programming to compute number theoretic problems.

Course Syllabus

Module-I (T-4 hr P-2 hr P-0 hr)

Divisibility, Representations of Integers,Computer Operations with Integers,Prime Numbers

Practice-1: Write a program to decide whether an integer is prime using trial division of the integer by all primes not exceeding its square root.

Module-II (T-6 hr P-4 hr P-0 hr)

Greatest common divisor,Euclidean Algorithm,Modified Euclidean Algorithm, Prime factorization,Factorization of Integers

Practice-2: Write a program to find the greatest common divisor of two integers using the Euclidean algorithm.

Practice-3:Find the prime factorization of a positive integer.

Module-III (T-5 hr P-2 hr P-0 hr)

Congruence’s, Properties of Congruence’s,System linear Congruence’s

Chinese Remainder Theorem.

Practice -4: Write a program to solves systems of linear congruence

Module-IV (T-5 hr P-0 hr P-0 hr)

Wilson's Theorem,Fermat's Little Theorem,Pseudo prime, Carmichael number

Module-V (T-4 hr P-2 hr P-0 hr)

Euler's Theorem, Euler Phi-function, Perfect Numbers, Mersenne Primes

Practice -5: Write programs to find values of the Euler phi-function

Module-VI (T-3 hr P-2 hr P-0 hr)

Character Ciphers, Block Ciphers, Exponentiation ciphers, Public-Key Cryptography (RSA Cryptosystem).

Practice-6: Write a program for RSA crypto system/Algorithm.

Module-VII (T-3 hr P-0 hr P-0 hr)

Knapsack ciphers, Some applications to computer science.

Reference books

  1. Elementary Number Theory and lts Applications by Kenneth H. Rosen, ADDISON-WESLEY PUBLISHING COMPANY ISBN  0-201-06561c chapter- 1(1.2-1.5), 2(2.1-2.4), 3,5,6(6.1-6.3),7
  2. Elementary Number Theory by David M. Burton, fifth edition, McGraw-Hill Publication, ISBN- 0-07-232569-0
  3. A Course in Number Theoretic Cryptography by Neal Koblitz, Springer Verlag, GTM

Session Plan

Session 1

Session 2

Representations of Integers

PPT link: https://www.slideshare.net/gavhays/integer-representation

Session 3

Computer Operations with Integers

PPT link: Computer Operations with Integers

Session 5 (Practice-1, 2hr)

Write programs to decide whether an integer is prime using trial division of the integer by all primes not exceeding its square root.

 

Session 6

Greatest common Divisor

PPT Link: Number theory-Greatest common Divisor

Youtube link : https://youtu.be/IuyU9Ng75Xc

Session 8

Modified Euclidean Algorithm

Video link: https://youtu.be/fz1vxq5ts5I

Session 9 (Practice-2, 2 hr)

Write programs to find the greatest common divisor of two integers using the Euclidean algorithm

Session 10

Session 11

Factorization of Integers

Session 12 (Practice-3, 2hr)

Write program to find the prime factorization of a positive integer.

Session 13

Fermat Numbers

Session 14

Session 15

Properties of Congruence’s

PPT link: https://www.slideshare.net/sangeethas20/modular-arithmetic-78720842

 

 

Session 16

System linear Congruence’s

PPT link: https://www.slideshare.net/viaspe9/linear-cong-slide-2

Video link: https://youtu.be/KwxslifHITg

 

Session 17

Chinese Remainder Theorem

Video link: https://youtu.be/XoTEKjS61kI

 

Session 18

System linear Congruence’s Using Chinese Remainder Theorem

Video link:https://youtu.be/zIFehsBHB8o

Session 19 (Practice-4, 2hr)

Write programs to solve systems of linear congruence.

Session 23

Theorems on Pseudo prime

 

Session 24

Carmichael number

PPT link : https://slideplayer.com/slide/12536174/

Video link: https://youtu.be/ddvkJF1-pL0

 

Session 26

Euler Phi-function

PPT link :https://slideplayer.com/slide/10256039/

Video link: https://youtu.be/ddvkJF1-pL0

Session 27 (Practice-5, 2hr)

Write programs to find values of the Euler phi-function

Session 28

Perfect Numbers

Video link: https://youtu.be/q8n15q1v4Xo

Session 32

Exponentiation ciphers

PPT link: https://slideplayer.com/slide/5156161/

You tube link: https://youtu.be/UMMoPAw4c2M

Session 34 (Practice-6, 2hr)

Write programs for RSA Cryptosystem/Algorithm

Session 35

Session 36

Applications to computer science

youtube link: https://youtu.be/4n-lB4iclqs

Case Studies

Case Studies

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    Our Main Teachers

    Kali Prasad Rath received his M.Tech. in Computer Science and  & M.Sc. in Mathematics from Berhampur University.This is his 10th year at Centurion University of Technology and Management and his 17th year in teaching profession. He is associated with qualitative study on theory of difference equation.