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Core/Elective:Elective Semester: 1 Credits: 3

The course provides an introduction to basic number theory, where the focus is on computational aspects with applications in cryptography. Applications to cryptography are explored including symmetric and public-key cryptosystems. Modern cryptographic methods are also discussed.


To understand the number theoretic foundations of modern cryptography
To implement and analyze cryptographic and number theoretic algorithms
To understand public key cryptosystems
To understand modern cryptographic techniques


1. 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, Euler Phi-function, multiplicative property, Finite Fields, Primitive Roots, Quadratic Residues, Legendre Symbol, Jacobi Symbol, Gauss's lemma, Quadratic Reciprocity Law

2. Primality Tests, Pseudoprimes, Carmichael Numbers, Fermat’s pseudoprimes, Euler pseudoprimes, 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.

3. 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, ElGamal Public Key Cryptosystem, Attacks on RSA Cryptosystem: Common modulus attack, Homomorphism attack, timing attack, Forging of digital signatures, Strong primes, Safe primes, Gordon's algorithm for generating strong primes.

4. Cubic Curves, Singular points, Discriminant, Introduction to Elliptic Curves, Geometry of elliptic curves over reals, Weierstrass normal form, point at infinity, Addition of two points, Bezout's theorem, associativity, Group structure, Points of finite order

5. 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


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)

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