   Free

Home Courses Applied Number Theory

Applied Number Theory Teacher Category

Course Attendees

Still no participant

Still no reviews

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 1

Division Algorithm

Session 2

Representations of Integers

Session 3

Computer Operations with Integers

PPT link: Computer Operations with Integers

Prime Numbers

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

Session 7

Euclidean Algorithm

Session 8

Modified Euclidean Algorithm

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

Session 11

Factorization of Integers

Session 12 (Practice-3, 2hr)

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

Fermat Numbers

Session 14

Introduction to Congruence’s

Session 15

Properties of Congruence’s

Session 16

System linear Congruence’s

Session 17

Chinese Remainder Theorem

Session 18

System linear Congruence’s Using Chinese Remainder Theorem

Session 19 (Practice-4, 2hr)

Write programs to solve systems of linear congruence.

Wilson's Theorem

Session 21

Fermat's Little Theorem

Pseudo prime

Session 23

Theorems on Pseudo prime

Session 24

Carmichael number

Euler's Theorem

Session 26

Euler Phi-function

Session 27 (Practice-5, 2hr)

Write programs to find values of the Euler phi-function

Perfect Numbers

Mersenne Primes

Session 30

Character Ciphers

Block Ciphers

Session 32

Exponentiation ciphers

Session 33

Public-Key Cryptography

Session 34 (Practice-6, 2hr)

Write programs for RSA Cryptosystem/Algorithm

Knapsack ciphers

Session 36

Applications to computer science