NUMBER THEORY & CRYPTOGRAPHY

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Sangram Routray

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Core Courses

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Course Name : NUMBER THEORY & CRYPTOGRAPHY

 

Code(Credit) :CUTM1623(4-0-0)

Course Objectives:

Covers fundamental algorithms for integer arithmetic, greatest common divisor calculation, modular arithmetic, and other number-theoretic computations. Algorithms are derived, implemented, and analyzed for primality testing and integer factorization. Applications to cryptography are explored including symmetric and public-key cryptosystems. A cryptosystem will be implemented and methods of attack investigated. To be able to implement and analyze algorithms for integer factorization and primality testing. To be able to use a system like Maple to explore concepts and theorems from number theory. To understand fundamental algorithms from symmetric-key and public-key cryptography.

Learning Outcomes

  • To understand fundamental number-theoretic algorithms such as the Euclidean algorithm, the Chinese Remainder algorithm, binary powering, and algorithms for integer arithmetic.
  • To understand fundamental algorithms for symmetric key and public-key cryptography.
  • To understand the number-theoretic foundations of modern cryptography and the principles behind their security.
  • To implement and analyze cryptographic and number-theoretic algorithms.
  • To be able to use Maple to explore mathematical concepts and theorems.

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

https://www.youtube.com/watch?v=Oyw5OmOd9B8

Session 2

Session 4

Fundamental theorem of arithmetic

https://www.youtube.com/watch?v=5o48Bn4KzfA

Session 7

Session 8

Session 9

Congruences: Definition - Basic properties of congruences

https://www.youtube.com/watch?v=MNj_e-t9tIs

https://www.youtube.com/watch?v=boSgvgmkLvw

 

Session 10

Session 11

Session 14

Session 15

Session 19

Classification -Structure of finite fields

https://www.youtube.com/watch?v=Sw8FH87J1XA&t=259s

Session 21

Introduction PROBABILITY THEORY

https://www.youtube.com/watch?v=cp7_ZF2kNi4

Session 22

Session 23

Session 24

Session 25

Session 26

Session 27

Stochastic Process Markov Chain

https://www.youtube.com/watch?v=sb4jo4P4ZLI

Session 28

Introduction - Basic concepts: codes

https://www.youtube.com/watch?v=VbxVIj-pnBw

Session 29

Session 30

Session 32

Generator matrices and parity check matrices

https://www.youtube.com/watch?v=wjdBfcLSrEY

Session 33

Session 36

PSEUDORANDOM NUMBER GENERATION

https://www.youtube.com/watch?v=4sYawx70iP4

Session 37

Indistinguishability of Probability Distributions

https://www.youtube.com/watch?v=b9a27XN_6tg

Session 38

Session 39

The Blum- Blum-Shub Generator

https://www.youtube.com/watch?v=M2VOfZJyk_o

Session 40

Security of the BBS Generator

https://www.youtube.com/watch?v=ScAcS2oBUXQ

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