Cryptography. Number Theory with AN INTRODUCTION TO. James S. Kraft. Lawrence C. Washington. CRC Press

Transcription

1 AN INTRODUCTION TO Number Theory with Cryptography James S Kraft Gilman School Baltimore, Maryland, USA Lawrence C Washington University of Maryland College Park, Maryland, USA CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business A CHAPMAN & HALL BOOK

2 Contents Preface xv 0 Introduction 1 01 Diophantine Equations 2 02 Modular Arithmetic 4 03 Primes and the Distribution of Primes 5 04 Cryptography 7 1 Divisibility 9 11 Divisibility 9 12 Euclid's Theorem Euclid's Original Proof The Sieve of Eratosthenes The Division Algorithm A Cryptographic Application The Greatest Common Divisor The Euclidean Algorithm The Extended Euclidean Algorithm Other Bases Linear Diophantine Equations The Postage Stamp Problem Fermat and Mersenne Numbers Chapter Highlights Problems Exercises Projects Computer Explorations 55 vii

3 viii Contents 1134 Answers to "Check Your Understanding" 57 2 Unique Factorization Preliminary Results The Fundamental Theorem of Arithmetic Euclid and the Fundamental Theorem of Arithmetic Chapter Highlights Problems Exercises Projects Answers to "Check Your Understanding" 70 3 Applications of Unique Factorization A Puzzle Irrationality Proofs Four More Proofs That y/2 Is Irrational The Rational Root Theorem Pythagorean Triples Differences of Squares Prime Factorization of Factorials The Riemann Zeta Function Chapter Highlights Problems Exercises Projects Computer Explorations Answers to "Check Your Understanding" Congruences Definitions and Examples Modular Exponentiation Divisibility Tests Linear Congruences The Chinese Remainder Theorem 127

4 ix 46 Fractions mod m Fermat's Theorem Euler's Theorem Wilson's Theorem Queens on a Chessboard Chapter Highlights Problems Exercises Projects Computer Explorations Answers to "Check Your Understanding" Cryptographic Applications Introduction Shift and Affine Ciphers Secret Sharing RSA Chapter Highlights Problems Exercises Projects Computer Explorations Answers to "Check Your Understanding" Polynomial Congruences Polynomials Mod Primes Solutions Modulo Prime Powers Composite Moduli Chapter Highlights Problems Exercises Projects Computer Explorations 205

5 x Contents 654 Answers to "Check Your Understanding" Order and Primitive Roots Orders of Elements Fermat Numbers Mersenne Numbers Primitive Roots Decimals Midy's Theorem Card Shuffling The Discrete Log Problem Baby Step-Giant Step Method The Index Calculus Existence of Primitive Roots Chapter Highlights Problems Exercises Projects Computer Explorations Answers to "Check Your Understanding" More Cryptographic Applications Diffie-Hellman Key Exchange Coin Flipping over the Telephone Mental Poker The ElGamal Public Key Cryptosystem Digital Signatures Chapter Highlights Problems Exercises Projects Computer Explorations Answers to "Check Your Understanding" 260

6 - 1 xi 9 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 Exercises Projects Answers to "Check Your Understanding" Primality and Factorization Trial Division and Fermat Factorization Primality Testing Pseudoprimes The Pocklington-Lehmer Primality Test The AKS Primality Test Fermat Numbers Mersenne Numbers Factorization x2 = y Factoring Pseudoprimes and Factoring Us ing RSA Exponents Pollard's p Method The Quadratic Sieve Coin Flipping over the Telephone Chapter Highlights Problems Exercises Projects Computer Explorations Answers to "Check Your Understanding" 334

7 xii Contents 11 Geometry of Numbers Volumes and Minkowski's Theorem Sums of Two Squares Algorithm for Writing p = 1 (mod 4) as a Sum of Two Squares Sums of Four Squares Pell's Equation Bhaskara's Chakravala Method Chapter Highlights Problems Exercises Projects Answers to "Check Your Understanding" Arithmetic Functions Perfect Numbers Multiplicative Functions Chapter Highlights Problems Exercises Projects Computer Explorations Answers to "Check Your Understanding" Continued Fractions Rational Approximations; Pell's Equation Evaluating Continued Fractions Pell's Equation Basic Theory Rational Numbers Periodic Continued Fractions Purely Periodic Continued Fractions Eventually Periodic Continued Fractions 409

8 xiii 135 Square Roots of Integers Some Irrational Numbers Chapter Highlights Problems Exercises Projects Computer Explorations Answers to "Check Your Understanding" Gaussian Integers Complex Arithmetic Gaussian Irreducibles The Division Algorithm Unique Factorization Applications Sums of Two Squares Pythagorean Triples y2 = x Chapter Highlights Problems Exercises Projects Computer Explorations Answers to "Check Your Understanding" Algebraic Integers Quadratic Fields and Algebraic Integers Units Z[y^2] Z[y/3] The Lucas-Lehmer Test Non-unique Factorization Chapter Highlights 474

9 xiv Contents 157 Problems Exercises Projects Answers to "Check Your Understanding" Analytic Methods Y, l/p Diverges Bertrand's Postulate Chebyshev's Approximate Prime Number Theorem Chapter Highlights Problems Exercises Projects Computer Explorations Epilogue: Fermat's Last Theorem Introduction Elliptic Curves Modularity 510 A Supplementary Topics 513 Al Geometric Series 513 A2 Mathematical Induction 515 A3 Pascal's Triangle and the Binomial Theorem 521 A4 Fibonacci Numbers 526 A5 Problems 530 A51 Exercises 530 A52 Answers to "Check Your Understanding" 532 B Answers and Hints for Odd-Numbered Exercises 535 Index 549