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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.

## 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