 # NUMBER THEORY & CRYPTOGRAPHY Teacher

##### Sangram Routray Category

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# 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 1

Introduction to Number theory

## Session 4

Fundamental theorem of arithmetic

## Session 9

Congruences: Definition - Basic properties of congruences

## Session 19

Classification -Structure of finite fields

## Session 21

Introduction PROBABILITY THEORY

## Session 27

Stochastic Process Markov Chain

## Session 28

Introduction - Basic concepts: codes

## Session 32

Generator matrices and parity check matrices

## Session 36

PSEUDORANDOM NUMBER GENERATION

## Session 37

Indistinguishability of Probability Distributions

## Session 39

The Blum- Blum-Shub Generator

## Session 40

Security of the BBS Generator