Atnaujinkite slapukų nuostatas

Introduction to Mathematical Cryptography Second Edition 2014 [Minkštas viršelis]

4.22/5 (82 ratings by Goodreads)
, ,
  • Formatas: Paperback / softback, 538 pages, aukštis x plotis: 235x155 mm, weight: 8307 g, 32 Illustrations, black and white; XVII, 538 p. 32 illus., 1 Paperback / softback
  • Serija: Undergraduate Texts in Mathematics
  • Išleidimo metai: 10-Sep-2016
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1493939386
  • ISBN-13: 9781493939381
Kitos knygos pagal šią temą:
Introduction to Mathematical Cryptography Second Edition 2014Didesnis paveikslėlis
  • Formatas: Paperback / softback, 538 pages, aukštis x plotis: 235x155 mm, weight: 8307 g, 32 Illustrations, black and white; XVII, 538 p. 32 illus., 1 Paperback / softback
  • Serija: Undergraduate Texts in Mathematics
  • Išleidimo metai: 10-Sep-2016
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1493939386
  • ISBN-13: 9781493939381
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781493939381.html
Raktažodžiai:
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. This text provides 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.

The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include:









classical cryptographic constructions, such as DiffieHellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, anddigital signatures;









fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms;









an in-depth treatment of important cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.





The second edition of An Introduction

to Mathematical Cryptography includes a significant revision of the material on digital signatures, including an earlier introduction to RSA, Elgamal, and DSA signatures, and new material on lattice-based signatures and rejection sampling. Many sections have been rewritten or expanded for clarity, especially in the chapters on information theory, elliptic curves, and lattices, and the chapter of additional topics has been expanded to include sections on digital cash and homomorphic encryption. Numerous new exercises have been included.

Recenzijos

This book explains the mathematical foundations of public key cryptography in a mathematically correct and thorough way without omitting important practicalities. I would like to emphasize that the book is very well written and quite clear. Topics are well motivated, and there are a good number of examples and nicely chosen exercises. To me, this book is still the first-choice introduction to public-key cryptography. (Klaus Galensa, Computing Reviews, March, 2015)

This is a text for an upper undergraduate/lower graduate course in mathematical cryptography. It is very well written and quite clear. Topics are well-motivated, and there are a good number of examples and nicely chosen exercises. An instructor of a fairly sophisticated undergraduate course in cryptography who wants to emphasize public key cryptography should definitely take a look at this book. (Mark Hunacek, MAA Reviews, October, 2014)

Preface v
Introduction xiii
1 An Introduction to Cryptography
1(60)
1.1 Simple Substitution Ciphers
1(9)
1.1.1 Cryptanalysis of Simple Substitution Ciphers
4(6)
1.2 Divisibility and Greatest Common Divisors
10(9)
1.3 Modular Arithmetic
19(7)
1.3.1 Modular Arithmetic and Shift Ciphers
23(1)
1.3.2 The Fast Powering Algorithm
24(2)
1.4 Prime Numbers, Unique Factorization, and Finite Fields
26(3)
1.5 Powers and Primitive Roots in Finite Fields
29(5)
1.6 Cryptography Before the Computer Age
34(3)
1.7 Symmetric and Asymmetric Ciphers
37(24)
1.7.1 Symmetric Ciphers
37(2)
1.7.2 Encoding Schemes
39(1)
1.7.3 Symmetric Encryption of Encoded Blocks
40(1)
1.7.4 Examples of Symmetric Ciphers
41(3)
1.7.5 Random Bit Sequences and Symmetric Ciphers
44(2)
1.7.6 Asymmetric Ciphers Make a First Appearance
46(1)
Exercises
47(14)
2 Discrete Logarithms and Dime--Hellman
61(56)
2.1 The Birth of Public Key Cryptography
61(3)
2.2 The Discrete Logarithm Problem
64(3)
2.3 Diffie---Hellman Key Exchange
67(3)
2.4 The Elgamal Public Key Cryptosystem
70(4)
2.5 An Overview of the Theory of Groups
74(3)
2.6 How Hard Is the Discrete Logarithm Problem?
77(4)
2.7 A Collision Algorithm for the DLP
81(2)
2.8 The Chinese Remainder Theorem
83(5)
2.8.1 Solving Congruences with Composite Moduli
86(2)
2.9 The Pohlig-Hellman Algorithm
88(6)
2.10 Rings, Quotients, Polynomials, and Finite Fields
94(23)
2.10.1 An Overview of the Theory of Rings
95(1)
2.10.2 Divisibility and Quotient Rings
96(2)
2.10.3 Polynomial Rings and the Euclidean Algorithm
98(4)
2.10.4 Polynomial Ring Quotients and Finite Fields
102(5)
Exercises
107(10)
3 Integer Factorization and RSA
117(76)
3.1 Euler's Formula and Roots Modulo pq
117(6)
3.2 The RSA Public Key Cryptosystem
123(3)
3.3 Implementation and Security Issues
126(2)
3.4 Primality Testing
128(9)
3.4.1 The Distribution of the Set of Primes
133(3)
3.4.2 Primality Proofs Versus Probabilistic Tests
136(1)
3.5 Pollard's p -- 1 Factorization Algorithm
137(4)
3.6 Factorization via Difference of Squares
141(9)
3.7 Smooth Numbers and Sieves
150(16)
3.7.1 Smooth Numbers
150(5)
3.7.2 The Quadratic Sieve
155(7)
3.7.3 The Number Field Sieve
162(4)
3.8 The Index Calculus and Discrete Logarithms
166(3)
3.9 Quadratic Residues and Quadratic Reciprocity
169(8)
3.10 Probabilistic Encryption
177(16)
Exercises
180(13)
4 Digital Signatures
193(14)
4.1 What Is a Digital Signature?
193(3)
4.2 RSA Digital Signatures
196(2)
4.3 Elgamal Digital Signatures and DSA
198(9)
Exercises
203(4)
5 Combinatorics, Probability, and Information Theory
207(92)
5.1 Basic Principles of Counting
208(6)
5.1.1 Permutations
210(1)
5.1.2 Combinations
211(2)
5.1.3 The Binomial Theorem
213(1)
5.2 The Vigenere Cipher
214(14)
5.2.1 Cryptanalysis of the Vigenere Cipher: Theory
218(5)
5.2.2 Cryptanalysis of the Vigenere Cipher: Practice
223(5)
5.3 Probability Theory
228(18)
5.3.1 Basic Concepts of Probability Theory
228(5)
5.3.2 Bayes's Formula
233(3)
5.3.3 Monte Carlo Algorithms
236(2)
5.3.4 Random Variables
238(6)
5.3.5 Expected Value
244(2)
5.4 Collision Algorithms and Meet-in-the-Middle Attacks
246(7)
5.4.1 The Birthday Paradox
246(1)
5.4.2 A Collision Theorem
247(3)
5.4.3 A Discrete Logarithm Collision Algorithm
250(3)
5.5 Pollard's ρ Method
253(10)
5.5.1 Abstract Formulation of Pollard's ρ Method
254(5)
5.5.2 Discrete Logarithms via Pollard's ρ Method
259(4)
5.6 Information Theory
263(15)
5.6.1 Perfect Secrecy
263(6)
5.6.2 Entropy
269(6)
5.6.3 Redundancy and the Entropy of Natural Language
275(2)
5.6.4 The Algebra of Secrecy Systems
277(1)
5.7 Complexity Theory and P Versus NP
278(21)
Exercises
282(17)
6 Elliptic Curves and Cryptography
299(74)
6.1 Elliptic Curves
299(7)
6.2 Elliptic Curves over Finite Fields
306(4)
6.3 The Elliptic Curve Discrete Logarithm Problem
310(6)
6.3.1 The Double-and-Add Algorithm
312(3)
6.3.2 How Hard Is the ECDLP?
315(1)
6.4 Elliptic Curve Cryptography
316(5)
6.4.1 Elliptic Diffie--Hellman Key Exchange
316(3)
6.4.2 Elliptic Elgamal Public Key Cryptosystem
319(2)
6.4.3 Elliptic Curve Signatures
321(1)
6.5 The Evolution of Public Key Cryptography
321(3)
6.6 Lenstra's Elliptic Curve Factorization Algorithm
324(5)
6.7 Elliptic Curves over F2 and over F2κ
329(7)
6.8 Bilinear Pairings on Elliptic Curves
336(11)
6.8.1 Points of Finite Order on Elliptic Curves
337(1)
6.8.2 Rational Functions and Divisors on Elliptic Curves
338(2)
6.8.3 The Weil Pairing
340(3)
6.8.4 An Efficient Algorithm to Compute the Weil Pairing
343(3)
6.8.5 The Tate Pairing
346(1)
6.9 The Weil Pairing over Fields of Prime Power Order
347(9)
6.9.1 Embedding Degree and the MOV Algorithm
347(3)
6.9.2 Distortion Maps and a Modified Weil Pairing
350(2)
6.9.3 A Distortion Map on y2 = x3 + x
352(4)
6.10 Applications of the Weil Pairing
356(17)
6.10.1 Tripartite Diffie--Hellman Key Exchange
356(2)
6.10.2 ID-Based Public Key Cryptosystems
358(3)
Exercises
361(12)
7 Lattices and Cryptography
373(98)
7.1 A Congruential Public Key Cryptosystem
373(4)
7.2 Subset-Sum Problems and Knapsack Cryptosystems
377(7)
7.3 A Brief Review of Vector Spaces
384(4)
7.4 Lattices: Basic Definitions and Properties
388(7)
7.5 Short Vectors in Lattices
395(8)
7.5.1 The Shortest and the Closest Vector Problems
395(1)
7.5.2 Hermite's Theorem and Minkowski's Theorem
396(4)
7.5.3 The Gaussian Heuristic
400(3)
7.6 Babai's Algorithm
403(4)
7.7 Cryptosystems Based on Hard Lattice Problems
407(2)
7.8 The GGH Public Key Cryptosystem
409(3)
7.9 Convolution Polynomial Rings
412(4)
7.10 The NTRU Public Key Cryptosystem
416(9)
7.10.1 NTRUEncrypt
417(5)
7.10.2 Mathematical Problems for NTRUEncrypt
422(3)
7.11 NTRUEncrypt as a Lattice Cryptosystem
425(3)
7.11.1 The NTRU Lattice
425(2)
7.11.2 Quantifying the Security of an NTRU Lattice
427(1)
7.12 Lattice-Based Digital Signature Schemes
428(8)
7.12.1 The GGH Digital Signature Scheme
428(2)
7.12.2 Transcript Analysis
430(1)
7.12.3 Rejection Sampling
431(2)
7.12.4 Rejection Sampling Applied to an Abstract Signature Scheme
433(1)
7.12.5 The NTRU Modular Lattice Signature Scheme
434(2)
7.13 Lattice Reduction Algorithms
436(14)
7.13.1 Gaussian Lattice Reduction in Dimension 2
436(3)
7.13.2 The LLL Lattice Reduction Algorithm
439(9)
7.13.3 Using LLL to Solve apprCVP
448(1)
7.13.4 Generalizations of LLL
449(1)
7.14 Applications of LLL to Cryptanalysis
450(21)
7.14.1 Congruential Cryptosystems
451(1)
7.14.2 Applying LLL to Knapsacks
451(1)
7.14.3 Applying LLL to GGH
452(1)
7.14.4 Applying LLL to NTRU
453(1)
Exercises
454(17)
8 Additional Topics in Cryptography
471(32)
8.1 Hash Functions
472(2)
8.2 Random Numbers and Pseudorandom Number
474(3)
8.3 Zero-Knowledge Proofs
477(3)
8.4 Secret Sharing Schemes
480(1)
8.5 Identification Schemes
481(1)
8.6 Padding Schemes and the Random Oracle Model
482(3)
8.7 Building Protocols from Cryptographic Primitives
485(2)
8.8 Blind Digital Signatures, Digital Cash, and Bitcoin
487(3)
8.9 Homomorphic Encryption
490(4)
8.10 Hyperelliptic Curve Cryptography
494(3)
8.11 Quantum Computing
497(2)
8.12 Modern Symmetric Cryptosystems: DES and AES
499(4)
List of Notation 503(4)
References 507(10)
Index 517
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 University 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 since 1988. He served as the Chair of the Brown Mathematics department from 20012004. 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 than 130 publications and has had more than 20 doctoral students.