An edition of An introduction to mathematical cryptography (2008)

An introduction to mathematical cryptography

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An edition of An introduction to mathematical cryptography (2008)

An introduction to mathematical cryptography

by

  • ★★★★ 4.0 (1 rating)
  • 8 Want to read
  • 1 Currently reading

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 linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include: * classical cryptographic constructions, such as Diffie-Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures; * fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms; * an in-depth treatment of important recent cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem. 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.

Publish Date
Publisher
Springer
Language
English
Pages
523

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Previews available in: English

Subjects

Mathematics, Number theory, Algebra, Cryptography, Coding theory, Cryptographie, Mathématiques, Cryptage, Data encryption (Computer science), Data structures (Computer science), Cryptography--mathematics, Qa268 .h64 2008, 652.80151, 003.54
Edition Availability
Cover of: An Introduction to Mathematical Cryptography
An Introduction to Mathematical Cryptography
Sep 10, 2016, Springer
paperback
Cover of: An Introduction to Mathematical Cryptography
An Introduction to Mathematical Cryptography
Sep 11, 2014, Springer
paperback
Cover of: An Introduction to Mathematical Cryptography
An Introduction to Mathematical Cryptography
Sep 11, 2014, Springer
hardcover in English
Cover of: An Introduction to Mathematical Cryptography
An Introduction to Mathematical Cryptography
Dec 01, 2010, Springer
paperback
Cover of: An introduction to mathematical cryptography
An introduction to mathematical cryptography
2008, Springer
in English

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Book Details

Table of Contents

Preface
Page v
Introduction
Page xi
Chapter 1. An Introduction to Cryptography
Page 1
1.1. Simple substitution ciphers
Page 1
1.2. Divisibility and greatest common divisors
Page 10
1.3. Modular arithmetic
Page 19
1.4. Prime numbers, unique factorization, and finite fields
Page 26
1.5. Powers and primitive roots in finite fields
Page 29
1.6. Cryptography before the computer age
Page 34
1.7. Symmetric and asymmetric ciphers
Page 36
Exercises
Page 47
Chapter 2. Discrete Logarithms and Diffie-Hellman
Page 59
2.1. The birth of public key cryptography
Page 59
2.2. The discrete logarithm problem
Page 62
2.3. Diffie-Hellman key exchange
Page 65
2.4. The ElGamal public key cryptosystem
Page 68
2.5. An overview of the theory of groups
Page 72
2.6. How hard is the discrete logarithm problem?
Page 75
2.7. A collision algorithm for the DLP
Page 79
2.8. The Chinese remainder theorem
Page 81
2.9. The Pohlig-Hellman algorithm
Page 86
2.10. Rings, quotients, polynomials, and finite fields
Page 92
Exercises
Page 105
Chapter 3. Integer Factorization and RSA
Page 113
3.1. Euler's formula and roots modulo pq
Page 113
3.2. The RSA public key cryptosystem
Page 119
3.3. Implementation and security issues
Page 122
3.4. Primality testing
Page 124
3.5. Pollard's p-1 factorization algorithm
Page 133
3.6. Factorization via difference of squares
Page 137
3.7. Smooth numbers and sieves
Page 146
3.8. The index calculus and discrete logarithms
Page 162
3.9. Quadratic residues and quadratic reciprocity
Page 165
3.10. Probabilistic encryption
Page 172
Exercises
Page 176
Chapter 4. Combinatorics, Probability, and Information Theory
Page 189
4.1. Basic principles of counting
Page 190
4.2. The Vigenère cipher
Page 196
4.3. Probability theory
Page 210
4.4. Collision algorithms and meet-in-the-middle attacks
Page 227
4.5. Pollard's p method
Page 234
4.6. Information theory
Page 243
4.7. Complexity Theory and P versus NP
Page 258
Exercises
Page 262
Chapter 5. Elliptic Curves and Cryptography
Page 279
5.1. Elliptic curves
Page 279
5.2. Elliptic curves over finite fields
Page 286
5.3. The elliptic curve discrete logarithm problem
Page 290
5.4. Elliptic curve cryptography
Page 296
5.5. The evolution of public key cryptography
Page 301
5.6. Lenstra's elliptic curve factorization algorithm
Page 303
5.7. Elliptic curves over \(\mathbb{F}_2\) and over \(\mathbb{F}_{2^k}\)
Page 308
5.8. Bilinear pairings on elliptic curves
Page 315
5.9. The Weil pairing over fields of prime power order
Page 325
5.10. Applications of the Weil pairing
Page 334
Exercises
Page 339
Chapter 6. Lattices and Cryptography
Page 349
6.1. A congruential public key cryptosystem
Page 349
6.2. Subset-sum problems and knapsack cryptosystems
Page 352
6.3. A brief review of vector spaces
Page 359
6.4. Lattices: Basic definitions and properties
Page 363
6.5. Short vectors in lattices
Page 370
6.6. Babai's algorithm
Page 379
6.7. Cryptosystems based on hard lattice problems
Page 383
6.8. The GGH public key cryptosystem
Page 384
6.9. Convolution polynomial rings
Page 387
6.10. The NTRU public key cryptosystem
Page 392
6.11. NTRU as a lattice cryptosystem
Page 400
6.12. Lattice reduction algorithms
Page 403
6.13. Applications of LLL to cryptanalysis
Page 418
Exercises
Page 422
Chapter 7. Digital Signatures
Page 437
7.1. What is a digital signature?
Page 437
7.2. RSA digital signatures
Page 440
7.3. ElGamal digital signatures and DSA
Page 442
7.4. GGH lattice-based digital signatures
Page 447
7.5. NTRU digital signatures
Page 450
Exercises
Page 458
Chapter 8. Additional Topics in Cryptography
Page 465
8.1. Hash functions
Page 466
8.2. Random numbers and pseudorandom number generators
Page 468
8.3. Zero-knowledge proofs
Page 470
8.4. Secret sharing schemes
Page 473
8.5. Identification schemes
Page 474
8.6. Padding schemes and the random oracle model
Page 476
8.7. Building protocols from cryptographic primitives
Page 479
8.8. Hyperelliptic curve cryptography
Page 480
8.9. Quantum computing
Page 483
8.10. Modern symmetric cryptosystems: DES and AES
Page 485
List of Notation
Page 489
References
Page 493
Index
Page 501

Edition Notes

Includes bibliographical references (p. 493-500) and index.

Published in
New York, London
Series
Undergraduate texts in mathematics

Classifications

Dewey Decimal Class
652.80151, 003.54
Library of Congress
QA268 .H64 2008

The Physical Object

Pagination
xv, 523 p. ;
Number of pages
523

Edition Identifiers

Open Library
OL23169651M
Internet Archive
introductiontoma00hoff_722
ISBN 10
0387779930
ISBN 13
9780387779935
LCCN
2008923038
OCLC/WorldCat
226280654
LibraryThing
7241500
Deutsche National Bibliothek
987817833
Goodreads
3884171

Work Identifiers

Work ID
OL13764896W

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May 15, 2025 Edited by MARC Bot import existing book
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