Syllabus CST 292 Number Theory

NUMBER THEORY- SYLLABUS

Module 1:Divisibility and Modular Arithmetic:

Finite Fields – Groups, Rings and Fields.
Divisibility - Divisibility and Division Algorithms, Well ordering Principle,Bezout’s Identity.
Modular Arithmetic- Properties, Euclid's algorithm for the greatest common divisor, Extended Euclid’s
Algorithm, Least Common multiple, Solving Linear Diophantine Equations, Modular Division.

Module 2:Primes and Congruences:

Prime Numbers-Prime Numbers andprime-powerfactorization, Fermat and Mersenne primes.,Primality testing and factorization.

Congruences-Linear congruences, Simultaneous linear congruences, Chinese Remainder Theorem,

Fermat’s little theorem, Wilson's theorem.

Module 3:Congruences with a Prime-Power Modulus&Euler's Function:

Congruences with a Prime-Power Modulus-Arithmetic modulo p, Pseudoprimes and Carmichael numbers, Solving congruences modulo prime powers.

Euler's Function-Euler’s Totient function, Applications of Euler’s Totient function, Traditional Cryptosystem, Limitations.

The Group of units- The group Un,Primitive roots, Existence of primitive roots, Applications of primitive roots.

Module 4:Quadratic Residues & Arithmetic Functions :

Quadratic Residues- Quadratic Congruences, The group of Quadratic residues, Legendre symbol,
Jacobi Symbol, Quadratic reciprocity.

Arithmetic Functions- Definition and examples, Perfect numbers, Mobius function and its properties,
Mobius inversion formula, The Dirichlet Products

Module 5:Sum of Squares and Continued Fractions:

Sum of Squares- Sum of two squares, The Gaussian Integers, Sum of three squares, Sum of four squares.

Continued Fractions -Finite continued fractions, Infinite continued fractions, Pell's Equation, Solution of Pell’s equation by continued fractions.

Text Books

1. G.A. Jones & J.M. Jones, Elementary Number Theory, Springer UTM, 2007.
2. Joseph Silverman, A Friendly introduction to Number Theory, Pearson Ed. 2009.

Reference Books

1.William Stallings, Cryptography and Network Security Principles and Practice, Pearson Ed.
2. Tom M.Apostol, ‘Introduction to Analytic Number Theory’, Narosa Publishing House Pvt. Ltd,
New Delhi, (1996).
3. Neal Koblitz, A course in Number Theory and Cryptography, 2nd Edition, Springer ,2004.

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Sum of Squares

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