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Introduction to Mathematical Cryptography [Kietas viršelis]

4.23/5 (136 ratings by Goodreads)
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  • Formatas: Hardback, 544 pages, aukštis x plotis x storis: 234x156x30 mm, weight: 2060 g, 1, black & white illustrations
  • Serija: Undergraduate Texts in Mathematics
  • Išleidimo metai: 12-Aug-2008
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 0387779930
  • ISBN-13: 9780387779935
Kitos knygos pagal šią temą:
Introduction to Mathematical CryptographyDidesnis paveikslėlis
  • Formatas: Hardback, 544 pages, aukštis x plotis x storis: 234x156x30 mm, weight: 2060 g, 1, black & white illustrations
  • Serija: Undergraduate Texts in Mathematics
  • Išleidimo metai: 12-Aug-2008
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 0387779930
  • ISBN-13: 9780387779935
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780387779935.html
Raktažodžiai:

An Introduction to Mathematical Cryptography provides an introduction to public key cryptography and underlying mathematics that is required for the subject. Each of the eight chapters expands on a specific area of mathematical cryptography and provides an extensive list of exercises.

It is a suitable text for advanced students in pure and applied mathematics and computer science, or the book may be used as a self-study. This book also provides a self-contained treatment of mathematical cryptography for the reader with limited mathematical background.



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. It includes exercises and examples at the end of each section.

Recenzijos

From the reviews: "The book is devoted to public key cryptography, whose principal goal is to allow two or more people to exchange confidential information ... . The material is very well organized, and it is self-contained: no prerequisites in higher mathematics are needed. In fact, everything is explained and carefully covered ... . there is abundance of examples and proposed exercises at the end of each chapter. ... This book is ideal as a textbook for a course aimed at undergraduate mathematics or computer science students." (Fabio Mainardi, The Mathematical Association of America, October, 2008) "This book focuses on public key cryptography ... . Hoffstein, Pipher, and Silverman ... provide a thorough treatment of the topics while keeping the material accessible. ... The book uses examples throughout the text to illustrate the theorems, and provides a large number of exercises ... . The volume includes a nice bibliography. ... Summing Up: Highly recommended. Upper-division undergraduate through professional collections." (C. Bauer, Choice, Vol. 46 (7), March, 2009) "For most undergraduate students in mathematics or computer science (CS), mathematical cryptography is a challenging subject. ... it is written in a way that makes you want to keep reading. ... The authors officially targeted the book for advanced undergraduate or beginning graduate students. I believe that this audience is appropriate. ... it could even be used with students who are just learning how to execute rigorous mathematical proofs. ... I strongly believe that it finds the right tone for today's students ... ." (Burkhard Englert, ACM Computing Reviews, March, 2009) "The exercises and text would make an excellent course for undergraduate independent study. ... This is an excellent book. Hoffstein, Pipher and Silverman have written as good a book as is possible to explain public key cryptography. ... This book would probably be best suited for a graduate course that focused on public key cryptography, for undergraduate independent study, or for the mathematician who wants to see how mathematics is used in public key cryptography." (Jintai Ding and Chris Christensen, Mathematical Reviews, Issue 2009 m)

Preface v
Introduction xi
1 An Introduction to Cryptography 1
1.1 Simple substitution ciphers
1
1.2 Divisibility and greatest common divisors
10
1.3 Modular arithmetic
19
1.4 Prime numbers, unique factorization, and finite fields
26
1.5 Powers and primitive roots in finite fields
29
1.6 Cryptography before the computer age
34
1.7 Symmetric and asymmetric ciphers
36
Exercises
47
2 Discrete Logarithms and Diffie—Hellman 59
2.1 The birth of public key cryptography
59
2.2 The discrete logarithm problem
62
2.3 Diffie—Hellman key exchange
65
2.4 The EIGamal public key cryptosystem
68
2.5 An overview of the theory of groups
72
2.6 How hard is the discrete logarithm problem?
75
2.7 A collision algorithm for the DLP
79
2.8 The Chinese remainder theorem
81
2.9 The Polllig—Hellman algorithm
86
2.10 Rings, quotients, polynomials, and finite fields
92
Exercises
105
3 Integer Factorization and RSA 113
3.1 Euler's formula and roots modulo pq
113
3.2 The RSA public key cryptosystem
119
3.3 Implementation and security issues
122
3.4 Primality testing
124
3.5 Pollard's p — 1 factorization algorithm
133
3.6 Factorization via difference of squares
137
3.7 Smooth numbers and sieves
146
3.8 The index calculus and discrete logarithms
162
3.9 Quadratic residues and quadratic reciprocity
165
3.10 Probabilistic encryption
172
Exercises
176
4 Combinatorics, Probability, and Information Theory 189
4.1 Basic principles of counting
190
4.2 The Vigenere cipher
196
4.3 Probability theory
210
4.4 Collision algorithms and meet-in-the-middle attacks
227
4.5 Pollard's p method
234
4.6 Information theory
243
4.7 Complexity Theory and P versus NP
258
Exercises
262
5 Elliptic Curves and Cryptography 279
5.1 Elliptic curves
279
5.2 Elliptic curves over finite fields
286
5.3 The elliptic curve discrete logarithm problem
290
5.4 Elliptic curve cryptography
296
5.5 The evolution of public key cryptography
301
5.6 Lenstra's elliptic curve factorization algorithm
303
5.7 Elliptic curves over F2 and over F2k
308
5.8 Bilinear pairings on elliptic curves
315
5.9 The Weil pairing over fields of prime power order
325
5.10 Applications of the Weil pairing
334
Exercises
339
6 Lattices and Cryptography 349
6.1 A congruential public key cryptosystem
349
6.2 Subset-sum problems and knapsack cryptosystems
352
6.3 A brief review of vector spaces
359
6.4 Lattices: Basic definitions and properties
363
6.5 Short vectors in lattices
370
6.6 Babai's algorithm
379
6.7 Cryptosystems based on hard lattice problems
383
6.8 The GGH public key cryptosystem
384
6.9 Convolution polynomial rings
387
6.10 The NTRU public key cryptosystem
392
6.11 NTRU as a lattice cryptosystem
400
6.12 Lattice reduction algorithms
403
6.13 Applications of LLL to cryptanalysis
418
Exercises
422
7 Digital Signatures 437
7.1 What is a digital signature?
437
7.2 RSA digital signatures
440
7.3 ElGamal digital signatures and DSA
442
7.4 GGH lattice-based digital signatures
447
7.5 NTRU digital signatures
450
Exercises
458
8 Additional Topics in Cryptography 465
8.1 Hash functions
466
8.2 Random numbers and pseudorandom number generators
468
8.3 Zero-knowledge proofs
470
8.4 Secret sharing schemes
473
8.5 Identification schemes
474
8.6 Padding schemes and the random oracle model
476
8.7 Building protocols from cryptographic primitives
479
8.8 Hyperelliptic curve cryptography
480
8.9 Quantum computing
483
8.10 Modern symmetric cryptosystems: DES and AES
485
List of Notation 489
References 493
Index 501
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.