An Introduction to Mathematical Cryptography
by Hoffstein, Jeffrey; Pipher, Jill; Siverman, Joseph H.Format: Hardcover
Pub. Date: 2008-07-01
Publisher(s): Springer Verlag
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Summary
"This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic filter algebra is requited of the reader; techniques from algebra, number theory, and probability are introduced and developed as required." "This book is an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online."--BOOK JACKET.
Author Biography
Dr. Jeffrey Hoffstein has been a professor at Brown University since 1989 and has been a visiting professor and tenured professor at several other universities since 1978. His research areas are number theory, automorphic forms, and cryptography. He has authored more than 50 publications.Dr. Jill Pipher has been a professor at Brown Univesity since 1989. She has been an invited lecturer and has received numerous awards and honors. Her research areas are harmonic analysis, elliptic PDE, and cryptography. She has authored over 40 publications.Dr. Joseph Silverman has been a professor at Brown University 1988. He served as the Chair of the Brown Mathematics department from 2001-2004. He has received numerous fellowships, grants and awards and is a frequently invited lecturer. His research areas are number theory, arithmetic geometry, elliptic curves, dynamical systems and cryptography. He has authored more than120 publications and has had more than 20 doctoral students.
Table of Contents
| Preface | p. v |
| Introduction | p. xi |
| An Introduction to Cryptography | p. 1 |
| Simple substitution ciphers | p. 1 |
| Divisibility and greatest common divisors | p. 10 |
| Modular arithmetic | p. 19 |
| Prime numbers, unique factorization, and finite fields | p. 26 |
| Powers and primitive roots in finite fields | p. 29 |
| Cryptography before the computer age | p. 34 |
| Symmetric and asymmetric ciphers | p. 36 |
| Exercises | p. 47 |
| Discrete Logarithms and Diffie-Hellman | p. 59 |
| The birth of public key cryptography | p. 59 |
| The discrete logarithm problem | p. 62 |
| Diffie-Hellman key exchange | p. 65 |
| The ElGamal public key cryptosystem | p. 68 |
| An overview of the theory of groups | p. 72 |
| How hard is the discrete logarithm problem? | p. 75 |
| A collision algorithm for the DLP | p. 79 |
| The Chinese remainder theorem | p. 81 |
| The Pohlig-Hellman algorithm | p. 86 |
| Rings, quotients, polynomials, and finite fields | p. 92 |
| Exercises | p. 105 |
| Integer Factorization and RSA | p. 113 |
| Euler's formula and roots modulo pq | p. 113 |
| The RSA public key cryptosystem | p. 119 |
| Implementation and security issues | p. 122 |
| Primality testing | p. 124 |
| Pollard's p - 1 factorization algorithm | p. 133 |
| Factorization via difference of squares | p. 137 |
| Smooth numbers and sieves | p. 146 |
| The index calculus and discrete logarithms | p. 162 |
| Quadratic residues and quadratic reciprocity | p. 165 |
| Probabilistic encryption | p. 172 |
| Exercises | p. 176 |
| Combinatorics, Probability, and Information Theory | p. 189 |
| Basic principles of counting | p. 190 |
| The Vigenere cipher | p. 196 |
| Probability theory | p. 210 |
| Collision algorithms and meet-in-the-middle attacks | p. 227 |
| Pollard's [rho] method | p. 234 |
| Information theory | p. 243 |
| Complexity Theory and P versus NP | p. 258 |
| Exercises | p. 262 |
| Elliptic Curves and Cryptography | p. 279 |
| Elliptic curves | p. 279 |
| Elliptic curves over finite fields | p. 286 |
| The elliptic curve discrete logarithm problem | p. 290 |
| Elliptic curve cryptography | p. 296 |
| The evolution of public key cryptography | p. 301 |
| Lenstra's elliptic curve factorization algorithm | p. 303 |
| Elliptic curves over F[subscript 2] and over F[subscript 2 superscript k] | p. 308 |
| Bilinear pairings on elliptic curves | p. 315 |
| The Weil pairing over fields of prime power order | p. 325 |
| Applications of the Weil pairing | p. 334 |
| Exercises | p. 339 |
| Lattices and Cryptography | p. 349 |
| A congruential public key cryptosystem | p. 349 |
| Subset-sum problems and knapsack cryptosystems | p. 352 |
| A brief review of vector spaces | p. 359 |
| Lattices: Basic definitions and properties | p. 363 |
| Short vectors in lattices | p. 370 |
| Babai's algorithm | p. 379 |
| Cryptosystems based on hard lattice problems | p. 383 |
| The GGH public key cryptosystem | p. 384 |
| Convolution polynomial rings | p. 387 |
| The NTRU public key cryptosystem | p. 392 |
| NTRU as a lattice cryptosystem | p. 400 |
| Lattice reduction algorithms | p. 403 |
| Applications of LLL to cryptanalysis | p. 418 |
| Exercises | p. 422 |
| Digital Signatures | p. 437 |
| What is a digital signature? | p. 437 |
| RSA digital signatures | p. 440 |
| ElGamal digital signatures and DSA | p. 442 |
| GGH lattice-based digital signatures | p. 447 |
| NTRU digital signatures | p. 450 |
| Exercises | p. 458 |
| Additional Topics in Cryptography | p. 465 |
| Hash functions | p. 466 |
| Random numbers and pseudorandom number generators | p. 468 |
| Zero-knowledge proofs | p. 470 |
| Secret sharing schemes | p. 473 |
| Identification schemes | p. 474 |
| Padding schemes and the random oracle model | p. 476 |
| Building protocols from cryptographic primitives | p. 479 |
| Hyperelliptic curve cryptography | p. 480 |
| Quantum computing | p. 483 |
| Modern symmetric cryptosystems: DES and AES | p. 485 |
| List of Notation | p. 489 |
| References | p. 493 |
| Index | p. 501 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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