Subject Name
Mathematical Foundations of Information Security
Subject code
14M22CI121
Module
No.
Subtitle of the
Module
Topics
1.
Probability Theory
Random Experiments, Sample Space, Events, Mathematical, Statistical and
Axiomatic Definitions of Probability; Conditional Probability, Independent
Events; Total Probability Theorem; Bayes’ Theorem
2.
Random Variables
Discrete Random Variable, Probability Function, Continuous Random
Variable, Probability Density Function; Distribution Function; Statistical
Averages
3
Divisibility
Euclidean
Algorithm
Divisibility; Prime numbers; Fundamental Theorem of Arithmetic; Series of
reciprocals of the primes; Euclidean algorithm and Continued fractions;
Euclid’s theorem and the Sieve of Eratosthenes; Pseudo-primes.
4
Arithmetical
Functions
Mobius function
5
Congruences
Basic properties of congruences; Residue classes and complete residue
systems; Linear congruences; Reduced residue systems and Euler-Fermat
theorem; Polynomial congruences modulo p, Lagrange’s theorem and
applications; Chinese remainder theorem and its applications; Polynomial
congruences with prime power moduli.
6
Finite Fields
Basics of Groups. Rings, Fields. Finite Fields and Existence of Finite
Fields, Polynomials, Unique Factorization
7
Theorems
on
Distribution
of
Prime Numbers
and
 ( n); Euler totient function  (n); Connection between
Mobius and Totient functions; Product formula for Totient function;
Dirichlet product of arithmetical functions; Dirichlet inverses and the
Mobius inversion formula; Mangoldt function; multiplicative functions;
multiplicative functions and Dirichlet multiplication  (n)
Chebychev’s functions and their relations; Prime number theorem; Shapiro’s
Tauberian theorem and its applications; Asymptotic formula for partial sum
of
 (1 / p) . Partial sums of Mobius function; Elementary proof of the
p x
prime number theorem; Selberg’s asymptotic formula.
8
Quadratic Residues
and
Quadratic
Reciprocity
Quadratic Residues, Legendre’s Symbol and its properties; Gauss Lema;
Law of Quadratic Reciprocity, Binary Quadratic Forms, Equivalence and
Reduction of Binary Quadratic Forms, Gauss Sum and Quadratic reciprocity
law, Positive Definite Binary Quadratic Forms
9
Applications
cryptography
Public key cryptography, RSA, Discrete Log Problem (Diffie Hellman Key
Exchange).
10
Graph Theory
to
Introduction to graph theory, vertex cover problem, graph coloring problem,
Basic introduction to graph connectivity and network flow
Recommended Reading: Author(s), Title, Edition, Publisher, Year of Publication etc. in IEEE format
1
J.J. Schiller, M.R. Spiegel, R.A. Srinivasan, Probability and Statistics, Schaum Series, 3rd edition
2
Tom M. Apostol: Introduction to Analytic Number Theory, Narosa Publishing House, 1998
3
Melvyan B. Nathanson: Elementary Methods in Number Theory (Graduate Texts in Math), Springer,
2000
4
Neal Koblitz: A Course in Number Theory and Cryptography, Second Edition, Springer, 1994
5
Graham Everest & Thomas Ward: An Introduction to Number Theory, Springer, 2005
5
T. Verarajan, Probability theory, Statistics and Random Processes, TMH