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Applied Number Theory
Teacher
KALIPRASAD RATH
Category
Core Courses
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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.
Course 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.
| COs | Course outcomes | Mapping Cos with POs (High-3, Medium-2, Low-1) |
| CO1 | · Demonstrate knowledge and understanding of topics including applications.
|
PO1(3), PO2(2) |
| CO3 | · Learn methods and techniques used in number theory.
|
PO2(2) |
| CO5 | · Use mathematical induction and other types of proof writing techniques and programming to compute number theoretic problems.
|
PO4(2), PO9(2) |
Course Outlines
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
- 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
- Elementary Number Theory by David M. Burton, fifth edition, McGraw-Hill Publication, ISBN- 0-07-232569-0
- A Course in Number Theoretic Cryptography by Neal Koblitz, Springer Verlag, GTM
Session Plan
Session 1
Division Algorithm
PPT link: https://www.slideshare.net/nikhilsairam1/introduction-to-division-algorithm
youtube link: https://youtu.be/uemTUnFxG_Q
Session 2
Representations of Integers
PPT link: https://www.slideshare.net/gavhays/integer-representation
Session 4
Prime Numbers
PPT link: https://www.slideshare.net/taleese/prime-numbers-472705
Video link: https://youtu.be/u7cBLb0b7pk
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 7
Euclidean Algorithm
PPT link: https://www.slideshare.net/amrinderarora/euclids-algorithm-for-greatest-common-divisor
Video link: https://youtu.be/NdqwT9kfquY
Session 9 (Practice-2, 2 hr)
Write programs to find the greatest common divisor of two integers using the Euclidean algorithm
Session 10
Prime factorization
PPT link: https://www.slideshare.net/swiftsonya7/prime-factorisation
Video link: https://youtu.be/XGbOiYhHY2c
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
Introduction to Congruence’s
PPT link: https://www.slideshare.net/sangeethas20/modular-arithmetic-78720842
youtube link: https://youtu.be/lJ3CD9M3nEQ
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 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 20
Wilson's Theorem
PPT link: https://www.slideshare.net/kath091583/report-in-math-830
Video link: https://youtu.be/Rmtjry3HUMM
Session 21
Fermat's Little Theorem
PPT link: https://www.slideshare.net/ANKU3686/euler-and-fermat-theorem-14977572
Video link: https://youtu.be/oT7kRlh1nVQ
Session 22
Pseudo prime
PPT link: https://www.slideshare.net/PranshuBhatnagar/introduction-to-the-aks-primality-test
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 25
Euler's Theorem
PPT link : https://www.slideshare.net/ANKU3686/euler-and-fermat-theorem-14977572
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 29
Mersenne Primes
PPT link : https://www.slideshare.net/pasarhizne/perfect-numbers-and-mersenne-primes
Video link: https://youtu.be/q8n15q1v4Xo
Session 30
Character Ciphers
PPT link: https://www.slideshare.net/TusharSwami1/cryptography-72969031
youtube link: https://youtu.be/1P8Xpxm76e8
Session 31
Block Ciphers
PPT link: https://www.slideshare.net/AsadAli108/l-3block-ciphers
Video link: https://youtu.be/-Gk9kaFoBxU
Session 32
Exponentiation ciphers
PPT link: https://slideplayer.com/slide/5156161/
You tube link: https://youtu.be/UMMoPAw4c2M
Session 33
Public-Key Cryptography
PPT link: https://www.slideshare.net/gopalsakarkar/public-key-cryptography-39319369
Video link: https://youtu.be/ERp8420ucGs
Session 34 (Practice-6, 2hr)
Write programs for RSA Cryptosystem/Algorithm
Session 35
Knapsack ciphers
PPT link: https://www.slideserve.com/alpha/knapsack-cipher
youtube link:https://youtu.be/F5cPdjZTeHo
Case Studies
Case Studies
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.