• To understand the number theoretic foundations of modern cryptography and the principles behind their security.

Course Outcomes

• SL NO	DESCRIPTION	PO MAPPING
  CO1	Understand practice of hiding information	PO1, PO2, PO5, PO6, PO11, PO12
  CO2	Understand  to converting some secret information to not readable texts	PO1, PO2, PO4, PO5, PO11
  CO3	Understand the cryptography, Understand the role of encryption algorithm.	PO1,PO2, PO5, PO6, PO7

Course Syllabus

MODULE (I):                                                                                                      

NUMBER THEORY: Introduction-Divisibility-Greatest common divisor -Prime numbers - Fundamental theorem of arithmetic - Mersenne primes -Fermat numbers - Euclidean algorithm -
Fermat’s theorem - Euler totient function - Euler’s theorem. Congruences: Definition - Basic properties of congruences - Residue classes - Chinese remainder theorem.

MODULE (II):                                                                                                     

ALGEBRAIC STRUCTURES: Groups - Cyclic groups, Cosets, Modulo groups -Primitive roots - Discrete logarithms. Rings – Sub rings, ideals and quotient rings, Integral domains. Fields - Finite fields – GF(Pⁿ), GF(2ⁿ) - Classification -Structure of finite fields. Lattice, Lattice as Algebraic system, sub lattices, some special lattices.

MODULE (III):                                                                                                    

PROBABILITY THEORY: Introduction – Concepts of Probability – Conditional Probability - Baye’s Theorem - Random Variables – discrete and continuous central Limit Theorem-Stochastic Process Markov Chain.

MODULE (IV):                                                                                                    

CODING THEORY: Introduction - Basic concepts: codes, minimum distance, equivalence of codes, Linear codes - Linear codes - Generator matrices and parity check matrices - Syndrome decoding – Hamming codes - Hadamard Code –Goppa codes.

PSEUDORANDOM NUMBER GENERATION: Introduction and examples -
Indistinguishability of Probability Distributions - Next Bit Predictors - The Blum
Blum-Shub Generator – Security of the BBS Generator.

Session Plan

Session 1

Introduction to Number theory

Introduction to Number theory

Session 2

Greatest common divisor

GCD

Session 3

Prime numbers

Prime numbers

Session 4

Fundamental theorem of arithmetic-

Fundamental theorem of arithmetic

Session 5

Mersenne primes

Mersenne primes

Session 6

Fermat numbers, Fermat’s theorem

Fermat numbers

Session 7

Euclidean algorithm

Euclidean algorithm

Session 8

Euler totient function - Euler’s theorem

Euler totient function

Session 9

Congruences: Definition - Basic properties of congruences

Congruences

Session 10

Residue classes - Chinese remainder theorem

Residue classes

Session 11

Groups - Cyclic groups

Groups

Session 12

Cosets

Cosets

Session 13

Modulo groups

Modulo groups

Session 14

Primitive roots

Primitive roots

Session 15

Discrete logarithms

Discrete logarithms

Session 16

Rings – Sub rings

Rings

Session 17

ideals and quotient rings

Ideals and quotient rings

Session 18

Fields - Finite fields – GF(Pⁿ), GF

Fields

Session 19

Classification -Structure of finite fields-

Classification

Session 20

Lattice, Lattice as Algebraic system, sublattices, some special lattices-

Lattice

Session 21

Introduction PROBABILITY THEORY

Introduction PROBABILITY THEORY

Session 22

Concepts of Probability

Concepts of Probability

Session 23

Conditional Probability-

Conditional Probability

Session 24

Baye’s Theorem-

Baye

Session 25

Random Variables

Random Variables

Session 26

discrete and continuous central Limit Theorem

Discrete and continuous central Limit Theorem

Session 27

Stochastic Process Markov Chain-

Stochastic Process Markov Chain

Session 28

Introduction - Basic concepts: codes-

Basic codes

Session 29

minimum distance

minimum distance

Session 30

equivalence of codes

Equivalence of codes

Session 31

Linear codes-

Linear codes

Session 32

Generator matrices and parity check matrices

Generator matrices and parity check matrices

Session 33

Syndrome decoding

Syndrome decoding

Session 34

Hamming codes - Hadamard Code

Hamming codes

Session 35

Goppa codes

Goppa codes

Session 36

PSEUDORANDOM NUMBER GENERATION

PSEUDORANDOM NUMBER GENERATION

Session 37

Indistinguishability of Probability Distributions

Indistinguishability of Probability Distributions

Session 38

Next Bit Predictors

Next Bit Predictors

Session 39

The Blum- Blum-Shub Generator

Security of the BBS Generator

Session 40

Security of the BBS Generator-

Security of the BBS Generator