An Introduction to Number Theory with Cryptography
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Book Description
Number theory has a rich history. For many years it was one of the purest areas of pure mathematics, studied because of the intellectual fascination with properties of integers. More recently, it has been an area that also has important applications to subjects such as cryptography. An Introduction to Number Theory with Cryptography presents number theory along with many interesting applications. Designed for an undergraduatelevel course, it covers standard number theory topics and gives instructors the option of integrating several other topics into their coverage. The "Check Your Understanding" problems aid in learning the basics, and there are numerous exercises, projects, and computer explorations of varying levels of difficulty.
Table of Contents
Introduction
Diophantine Equations
Modular Arithmetic
Primes and the Distribution of Primes
Cryptography
Divisibility
Divisibility
Euclid's Theorem
Euclid's Original Proof
The Sieve of Eratosthenes
The Division Algorithm
The Greatest Common Divisor
The Euclidean Algorithm
Other Bases
Linear Diophantine Equations
The Postage Stamp Problem
Fermat and Mersenne Numbers
Chapter Highlights
Problems
Unique Factorization
Preliminary Results
The Fundamental Theorem of Arithmetic
Euclid and the Fundamental Theorem of Arithmetic
Chapter Highlights
Problems
Applications of Unique Factorization
A Puzzle
Irrationality Proofs
The Rational Root Theorem
Pythagorean Triples
Differences of Squares
Prime Factorization of Factorials
The Riemann Zeta Function
Chapter Highlights
Problems
Congruences
Definitions and Examples
Modular Exponentiation
Divisibility Tests
Linear Congruences
The Chinese Remainder Theorem
Fractions mod m
Fermat's Theorem
Euler's Theorem
Wilson's Theorem
Queens on a Chessboard
Chapter Highlights
Problems
Cryptographic Applications
Introduction
Shift and Affine Ciphers
Secret Sharing
RSA
Chapter Highlights
Problems
Polynomial Congruences
Polynomials Mod Primes
Solutions Modulo Prime Powers
Composite Moduli
Chapter Highlights
Problems
Order and Primitive Roots
Orders of Elements
Primitive Roots
Decimals
Card Shuffling
The Discrete Log Problem
Existence of Primitive Roots
Chapter Highlights
Problems
More Cryptographic Applications
DiffieHellman Key Exchange
Coin Flipping over the Telephone
Mental Poker
The ElGamal Public Key Cryptosystem
Digital Signatures
Chapter Highlights
Problems
Quadratic Reciprocity
Squares and Square Roots Mod Primes
Computing Square Roots Mod p
Quadratic Equations
The Jacobi Symbol
Proof of Quadratic Reciprocity
Chapter Highlights
Problems
Primality and Factorization
Trial Division and Fermat Factorization
Primality Testing Factorization
Coin Flipping over the Telephone
Chapter Highlights
Problems
Geometry of Numbers
Volumes and Minkowski's Theorem
Sums of Two Squares
Sums of Four Squares
Pell's Equation
Chapter Highlights
Problems
Arithmetic Functions
Perfect Numbers
Multiplicative Functions
Chapter Highlights
Problems
Continued Fractions
Rational Approximations; Pell's Equation
Basic Theory
Rational Numbers
Periodic Continued Fractions
Square Roots of Integers
Some Irrational Numbers
Chapter Highlights
Problems
Gaussian Integers
Complex Arithmetic
Gaussian Irreducibles
The Division Algorithm
Unique Factorization
Applications
Chapter Highlights
Problems
Algebraic Integers
Quadratic Fields and Algebraic Integers
Units
Z[√2]
Z[√3]
Nonunique Factorization
Chapter Highlights
Problems
Analytic Methods
Σ1/p Diverges
Bertrand's Postulate
Chebyshev's Approximate Prime Number Theorem
Chapter Highlights
Problems
Epilogue: Fermat's Last Theorem
Introduction
Elliptic Curves
Modularity
Supplementary Topics
Geometric Series
Mathematical Induction
Pascal’s Triangle and the Binomial Theorem
Fibonacci Numbers
Problems
Answers and Hints for OddNumbered Exercises
Index
Reviews
"… provides a fine history of number theory and surveys its applications. Collegelevel undergrads will appreciate the number theory topics, arranged in a format suitable for any standard course in the topic, and will also appreciate the inclusion of many exercises and projects to support all the theory provided. In providing a foundation text with stepbystep analysis, examples, and exercises, this is a top teaching tool recommended for any cryptography student or instructor."
—California Bookwatch, January 2014
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