MATH 4383 - Number Theory and Cryptography - University of Houston

# MATH 4383 - Number Theory and Cryptography

***This is a course guideline.  Students should contact instructor for the updated information on current course syllabus, textbooks, and course content*

Prerequisites: MATH 3330 or MATH 2305.

Catalog Course Description: Divisibility theory, primes and their distribution, theory of congruences and application in security, integer representations, Fermat's Little Theorem and Euler's Theorem, primitive roots, quadratic reciprocity, and introduction to cryptography

Textbook: Instructor's lecture notes

Course Content:

Number theory, once considered the purest of subjects, has become an essential tool in providing computer and Internet security. This course will cover the topics in the standard one semester introduction to number theory, as well its applications in computer science and cryptography. It includes:  Divisibility theory, primes and their distribution, theory of congruences and its application in security, Integer representations (binary and base expansions, base conversion algorithm), Fermat's Little Theorem and Euler's Theorem, primitive roots, quadratic reciprocity, and introduction to cryptography (classical cryptography, public key cryptography, RSA cryptosystem, cryptographic protocols).

Syllabus:

Chapter 1: Preliminaries

1.1 The number system and the Well-Ordering Principle
1.2 Mathematical Induction

Chapter 2:  Divisibility and Factorization

2.1 Divisibility, Greatest Common Divisors, Euclidean Algorithm
2.2 Least Common Multiple
2.3 Representations of integers (Decimal Representation and Binary Representation of integers)

Chapter 3: Solving Linear Diophantine Equations

Chapter 4: Primes

4.1 Prime Numbers
4.2 Unique Prime Factorization
4.3 Test of Primality by Trial Division

Chapter 5: The Theory of Congruences

5.1 The concept of congruences
5.2 Congruence Classes
5.3 Applications of Congruences: Check digits

Chapter 6: Solving Linear Congruences

6.1 Solving (single) linear congruence
6.2 Solving system of linear congruences, the Chinese Remainder Theorem

Chapter 7: Fermat's Theorem and Euler's Generalization

7.1 Fermat's Little Theorem
7.2 The general case: Euler's theorem

Chapter 8: Primitive Roots

8.1 The multiplicative order
8.2 Promitive Roots (mod n)
8.3 The modulus n which does not have primitive roots
8.4 The Existence Theorems
8.5 Applications: The use of primitive roots

9.1 Euler's Criterion
9.2 The Legendre Symbol and its properties
9.3 Examples of computing the Legendre symbol
9.4 Jacobi Symbol
9.5 Quadratic Residues and Primitive Roots

Chapter 10: Cryptography

10.1 Introduction
10.2 Symmetric-key cryptography
10.3 Assymetric Key or public key cryptography