MTRA-105	Number Theory and Cryptography
Lecture	Tutorial	Practical	Credit	Major Test	Minor Test	Total	Time
4	0	0	4	60	40	100	3Hrs.
Program Objective (PO)	To introduce the concepts and methodology used in the Number Theory and Cryptography.
Course Outcomes (CO)
CO1	To introduce the mathematical fundamentals involve in cryptography.
CO2	To describe the process of primality testing and factorization
CO2	To understand the strength and weakness of cryptosystems
CO3	To introduce the elliptic curve cryptography.

### Unit I

Elementary Number Theory: Divisibility, Division Algorithm, Euclidean Algorithm; Congruences, Complete Residue systems, Reduced Residue systems; Fermat's little theorem, Euler's Generalization, Wilson's Theorem; Chinese Remainder Theorem, Generalized Chinese Remainder Theorem-Euler Phi-function, multiplicative property; Finite Fields, Primitive Roots; Quadratic Residues, Legendre Symbol, Jacobi Symbol; Gauss's lemma, Quadratic Reciprocity Law.

### Unit II

Primality Testing and Factorization: Primality Tests; Pseudo primes, Carmichael Numbers; Fermat's pseudoprimes, Euler pseudo primes; Factorization by Pollard's Rho method; Simple Continued Fraction, simple infinite continued fractions; Approximation to irrational numbers using continued fractions; Continued Fraction method for factorization.

### Unit III

Public Key Cryptosystems: Traditional Cryptosystem, limitations; Public Key Cryptography; Diffie Hellmann key exchange; Discrete Logarithm problem; One-way functions, Trapdoor functions; RSA cryptosystem; Digital signature schemes; Digital signature standards; RSA signature schemes; Knapsack problem; El Gamal Public Key Cryptosystem; Attacks on RSAcryptosystem: Common modulus attack; Homomorphism attack, timing attack; Forging of digital signatures; Strong primes, Safe primes, Gordon's algorithm for generating strong primes.

### Unit IV

Elliptic Curve Cryptography: Cubic Curves, Singular points, Discriminant; Introduction to Elliptic Curves, Geometry of elliptic curves over reals; Weier strass normal form, point at infinity; Addition of two points; Bezout's theorem, associativity; Group structure, Points of finite order; Elliptic Curves over finite fields, Discrete Log problem for Elliptic curves; Elliptic Curve Cryptography; Factorization using Elliptic Curve; Lenstra's algorithm; ElGamal Public Key Cryptosystem for elliptic curves.

### Reference Books :

- 1. A Course in Number Theory and Cryptography, Neal Koblitz, (Springer 2006).
- 2. An Introduction to Mathematical Cryptography, Jill Pipher, Jeffrey Hoffstein, Joseph H.Silverman (Springer, 2008).
- 3. An Introduction to theory of numbers, Niven, Zuckerman and Montgomery, (Wiley 2006).
- 4. Elliptic curves: Number theory and cryptography, Lawrence C. Washington, (Chapman & Hall/CRC 2003).
- 5. An Introduction to Cryptography, R.A. Mollin (Chapman & Hall, 2001).
- 6. Rational Points on Elliptic Curves, Silverman and Tate (Springer 2005).
- 7. Guide to elliptic curve cryptography Hankerson, Menezes, Vanstone (Springer, 2004).
- 8. Elementary Number Theory, Jones and Jones (Springer, 1998).

Note for paper setter for paper setter : Eight question will be set in all. Two questions from each unit. The candidate will be required to attempt four question in all and selecting one question from each unit.