Primality Testing And Integer Factorization In Public-Key Cryptography
by Yan, Song Y.Edition: 2nd
Format: Hardcover
Pub. Date: 2008-11-01
Publisher(s): Springer Verlag
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Summary
The Primality Testing Problem (PTP) has now proved to be solvable in deterministic polynomial-time (P) by the AKS (Agrawal-Kayal-Saxena) algorithm, whereas the Integer Factorization Problem (IFP) still remains unsolvable in (P). There is still no polynomial-time algorithm for IFP. Many practical public-key cryptosystems and protocols such as RSA (Rivest-Shamir-Adleman) rely their security on computational intractability of IFP.Primality Testing and Integer Factorization in Public Key Cryptography, Second Edition, provides a survey of recent progress in primality testing and integer factorization, with implications to factoring based public key cryptography. Notable new features are the comparison of Rabin-Miller probabilistic test in RP, Atkin-Morain elliptic curve test in ZPP and AKS deterministic test.This volume is designed for advanced level students in computer science and mathematics, and as a secondary text or reference book; suitable for practitioners and researchers in industry.First edition was very positively reviewed by Prof Samuel Wagstaff at Purdue University in AMS Mathematical Reviews (See MR2028480 2004j:11148), and by Professor J.T. Ayuso of University of Simon Bolivar in the European Mathematical Society's review journal Zentralblatt für Mathematik (see Zbl 1048.11103).
Table of Contents
| Preface to the Second Edition | p. ix |
| Preface to the First Edition | p. xi |
| Number-Theoretic Preliminaries | p. 1 |
| Problems in Number Theory | p. 1 |
| Groups, Rings and Fields | p. 13 |
| Divisibility Properties | p. 23 |
| Euclid's Algorithm and Continued Fractions | p. 34 |
| Arithmetic Functions ¿(n), ¿(n), ¿(n), ¿(n), ¿(n) | p. 50 |
| Linear Congruences | p. 63 |
| Quadratic Congruences | p. 85 |
| Primitive Roots and Power Residues | p. 103 |
| Arithmetic of Elliptic Curves | p. 113 |
| Chapter Notes and Further Reading | p. 124 |
| Primality Testing and Prime Generation | p. 127 |
| Computing with Numbers and Curves | p. 127 |
| Riemann ¿ and Dirichlet L Functions | p. 139 |
| Rigorous Primality Tests | p. 149 |
| Compositeness and Pseudoprimality Tests | p. 157 |
| Lucas Pseudoprimality Test | p. 168 |
| Elliptic Curve Primality Tests | p. 172 |
| Superpolynomial-Time Tests | p. 177 |
| Polynomial-Time Tests | p. 182 |
| Comparison of General Purpose Primality Tests | p. 188 |
| Primality Tests for Special Numbers | p. 192 |
| Prime Number Generation | p. 201 |
| Chapter Notes and Further Reading | p. 207 |
| Integer Factorization and Discrete Logarithms | p. 209 |
| Introduction | p. 209 |
| Simple Factoring Methods | p. 212 |
| Elliptic Curve Method (ECM) | p. 221 |
| General Factoring Congruence | p. 226 |
| Continued FRACtion Method (CFRAC) | p. 230 |
| Quadratic Sieve (QS) | p. 234 |
| Number Field Sieve (NFS) | p. 239 |
| Quantum Factoring Algorithm | p. 251 |
| Discrete Logarithms | p. 257 |
| kth Roots | p. 270 |
| Elliptic Curve Discrete Logarithms | p. 278 |
| Chapter Notes and Further Reading | p. 285 |
| Number-Theoretic Cryptography | p. 287 |
| Public-Key Cryptography | p. 287 |
| RSA Cryptosystem | p. 292 |
| Security and Cryptanalysis of RSA | p. 301 |
| Rabin Cryptography | p. 314 |
| Quadratic Residuosity Cryptography | p. 320 |
| Discrete Logarithm Cryptography | p. 326 |
| Elliptic Curve Cryptography | p. 331 |
| Zero-Knowledge Techniques | p. 338 |
| Deniable Authentication | p. 341 |
| Non-Factoring Based Cryptography | p. 346 |
| Chapter Notes and Further Reading | p. 351 |
| Bibliography | p. 353 |
| Index | p. 367 |
| About the Author | p. 373 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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