An introduction to mathematical cryptography:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
Springer
2014
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverzeichnis Seite 507 - 516 |
| Beschreibung: | XVII, 538 S. |
| ISBN: | 9781493917105 9781493939381 |
Internformat
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| 245 | 1 | 0 | |a An introduction to mathematical cryptography |c Jeffrey Hoffstein ; Jill Pipher ; Joseph H. Silverman |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York, NY [u.a.] |b Springer |c 2014 | |
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| 490 | 0 | |a Undergraduate texts in mathematics | |
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Datensatz im Suchindex
| _version_ | 1804152666023002112 |
|---|---|
| adam_text | Titel: An introduction to mathematical cryptography
Autor: Hoffstein, Jeffrey
Jahr: 2014
Contents
Preface v
Introduction xiii
1 An Introduction to Cryptography 1
1.1 Simple Substitution Ciphers........................................1
1.1.1 Cryptanalysis of Simple Substitution Ciphers............4
1.2 Divisibility and Greatest Common Divisors......................10
1.3 Modular Arithmetic ................................................19
1.3.1 Modular Arithmetic and Shift Ciphers....................23
1.3.2 The Fast Powering Algorithm..............................24
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..............................37
1.7.1 Symmetric Ciphers..........................................37
1.7.2 Encoding Schemes..........................................39
1.7.3 Symmetric Encryption of Encoded Blocks................40
1.7.4 Examples of Symmetric Ciphers ..........................41
1.7.5 Random Bit Sequences and Symmetric Ciphers..........44
1.7.6 Asymmetric Ciphers Make a First Appearance..........46
Exercises ..................................................................47
2 Discrete Logarithms and Diffie—Hellman 61
2.1 The Birth of Public Key Cryptography ..........................61
2.2 The Discrete Logarithm Problem..................................64
2.3 Difhe-Hellman Key Exchange......................................67
2.4 The Elgamal Public Key Cryptosystem ..........................70
2.5 An Overview of the Theory of Groups............................74
2.6 How Hard Is the Discrete Logarithm Problem?..................77
2.7 A Collision Algorithm for the DLP ..............................81
2.8 The Chinese Remainder Theorem - - - -............83
2.8.1 Solving Congruences with Composite Moduli............86
2.9 The Pohlig-Hellinan Algorithm ............................88
2.10 Rings, Quotients, Polynomials, and Finite Fields................94
2.10.1 An Overview of the Theory of Rings........... 95
2.10.2 Divisibility and Quotient Rings............................90
2.10.3 Polynomial Rings and the Euclidean Algorithm.....98
2.10.4 Polynomial Ring Quotients and Finite Fields......102
Exercises .................................107
Integer Factorization and RS A 117
3.1 Euler s Formula and Roots Modulo pq..............117
3.2 The RSA Public Key Cryptosystem ...............123
3.3 Implementation and Security Issues................120
3.4 Primality Testing .........................128
3.4.1 The Distribution of the Set of Primes..........133
3.4.2 Primality Proofs Versus Probabilistic Tests.......130
3.5 Pollard s p — 1 Factorization Algorithm .............137
3.0 Factorization via Difference of Squares .............141
3.7 Smooth Numbers and Sieves....................150
3.7.1 Smooth Numbers......................150
3.7.2 The Quadratic Sieve....................155
3.7.3 The Number Field Sieve..................102
3.8 The Index Calculus and Discrete Logarithms..........100
3.9 Quadratic Residues and Quadratic Reciprocity.........109
3.10 Probabilistic Encryption......................177
Exercises .................................180
Digital Signatures 193
4.1 What Is a Digital Signature?...................193
4.2 RSA Digital Signatures......................196
4.3 Elgamal Digital Signatures and DSA...............198
Exercises .................................203
Combinatorics, Probability, and Information Theory 207
5.1 Basic Principles of Counting ...................208
5.1.1 Permutations....................................210
5.1.2 Combinations..................................211
5.1.3 The Binomial Theorem........................213
5.2 The Vigenère Cipher..............................214
5.2.1 Cryptanalysis of the Vigenère Cipher: Theory......218
5.2.2 Cryptanalysis of the Vigenère Cipher: Practice.....223
5.3 Probability Theory .............................228
5.3.1 Basic Concepts of Probability Theory..........228
5.3.2 Bayes s Formula......................233
5.3.3 Monte Carlo Algorithms..................236
5.3.4 Random Variables.....................238
5.3.5 Expected Value.......................244
5.4 Collision Algorithms and Meet-in-the-Middle Attacks......246
5.4.1 The Birthday Paradox...................246
5.4.2 A Collision Theorem....................247
5.4.3 A Discrete Logarithm Collision Algorithm........250
5.5 Pollard s p Method.........................253
5.5.1 Abstract Formulation of Pollard s p Method.......254
5.5.2 Discrete Logarithms via Pollard s p Method.......259
5.6 Information Theory ........................263
5.6.1 Perfect Secrecy.......................263
5.6.2 Entropy...........................269
5.6.3 Redundancy and the Entropy
of Natural Language....................275
5.6.4 The Algebra of Secrecy Systems .............277
5.7 Complexity Theory and V Versus NV..............278
Exercises .................................282
6 Elliptic Curves and Cryptography 299
6.1 Elliptic Curves...........................299
6.2 Elliptic Curves over Finite Fields.................306
6.3 The Elliptic Curve Discrete Logarithm Problem.........310
6.3.1 The Double-and-Add Algorithm .............312
6.3.2 How Hard Is the ECDLP?.................315
6.4 Elliptic Curve Cryptography ...................316
6.4.1 Elliptic Diffie-Hellman Key Exchange..........316
6.4.2 Elliptic Elgainal Public Key Cryptosystem .......319
6.4.3 Elliptic Curve Signatures .................321
6.5 The Evolution of Public Key Cryptography...........321
6.6 Lenstra s Elliptic Curve Factorization Algorithm........324
6.7 Elliptic Curves over F2 and over F2fe...............329
6.8 Bilinear Pairings on Elliptic Curves................336
6.8.1 Points of Finite Order on Elliptic Curves ........337
6.8.2 Rational Functions and Divisors on Elliptic Curves . . . 338
6.8.3 The Weil Pairing......................340
6.8.4 An Efficient Algorithm to Compute the Weil Pairing . . 343
6.8.5 The Tate Pairing......................346
6.9 The Weil Pairing over Fields of Prime Power Order.......347
6.9.1 Embedding Degree and the MOV Algorithm ......347
6.9.2 Distortion Maps and a Modified Weil Pairing......350
6.9.3 A Distortion Map on y2 = x3 + x.............352
6 10 Applications of the Weil Pairing .................356
6.10.1 Tripartite Diffie-Hellman Key Exchange.........356
6 10.2 ID-Based Public Key Cryptosystems...........358
Exercises .......................361
373
Lattices and Cryptography
7.1 A Congruential Public Key Cryptosysteni............373
7.2 Subset-Sum Problems and Knapsack Cryptosystems......377
7.3 A Brief Review of Vector Spaces .................384
7.4 Lattices: Basic Definitions and Properties............388
7.5 Short Vectors in Lattices......................395
7.5.1 The Shortest and the Closest Vector Problems.....395
7.5.2 Beruhte s Theorem and Minkowski s Theorem .....396
7.5.3 The Gaussian Heuristic..................400
7.6 Babai s Algorithm.........................403
7.7 Cryptosystems Based on Hard Lattice Problems ........407
7.8 The GGH Public Key Cryptosysteni...............409
7.9 Convolution Polynomial Rings ..................412
7.10 The NTRU Public Key Cryptosysteni ..............416
7.10.1 NTRUEncrypt.......................417
7.10.2 Mathematical Problems for NTRUEncrypt........422
7.11 NTRUEncrypt as a Lattice Cryptosysteni............425
7.11.1 The NTRU Lattice.....................425
7.11.2 Quantifying the Security of an NTRU Lattice......427
7.12 Lattice-Based Digital Signature Schemes.............128
7.12.1 The GGH Digital Signature Scheme...........428
7.12.2 Transcript Analysis.....................430
7.12.3 Rejection Sampling.....................431
7.12.4 Rejection Sampling Applied to an Abstract Signature
Scheme...........................433
7.12.5 The NTRU Modular Lattice Signature Scheme.....434
7.13 Lattice Reduction Algorithms ..................436
7.13.1 Gaussian Lattice Reduction in Dimension 2.......436
7.13.2 The LLL Lattice Reduction Algorithm..........439
7.13.3 Using LLL to Solve apprCVP...............448
7.13.4 Generalizations of LLL...................449
7.14 Applications of LLL to Cryptanalysis...............450
7.14.1 Congruential Cryptosystems............ . 451
7.14.2 Applying LLL to Knapsacks................451
7.14.3 Applying LLL to GGH...................452
7.14.4 Applying LLL to NTRU.................453
Exercises ................
8 Additional Topics in Cryptography 471
8.1 Hash Functions...........................472
8.2 Random Numbers and Pseudorandom Number.........474
8.3 Zero-Knowledge Proofs.......................477
8.4 Secret Sharing Schemes ......................480
8.5 Identification Schemes.......................481
8.6 Padding Schemes and the Random Oracle Model........482
8.7 Building Protocols from Cryptographic Primitives .......485
8.8 Blind Digital Signatures, Digital Cash, and Bitcoin.......487
8.9 Homomorphic Encryption.....................490
8.10 Hyperelliptic Curve Cryptography ................494
8.11 Quantum Computing........................497
8.12 Modern Symmetric Cryptosystems: DES and AES.......499
List of Notation 503
References 507
Index 517
|
| any_adam_object | 1 |
| author | Hoffstein, Jeffrey 1953- Pipher, Jill 1955- Silverman, Joseph H. 1955- |
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| discipline | Informatik Mathematik Wirtschaftswissenschaften |
| edition | 2. ed. |
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| spelling | Hoffstein, Jeffrey 1953- Verfasser (DE-588)1029472769 aut An introduction to mathematical cryptography Jeffrey Hoffstein ; Jill Pipher ; Joseph H. Silverman 2. ed. New York, NY [u.a.] Springer 2014 XVII, 538 S. txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics Literaturverzeichnis Seite 507 - 516 Kryptologie (DE-588)4033329-2 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Kryptologie (DE-588)4033329-2 s DE-604 Pipher, Jill 1955- Verfasser (DE-588)1029472998 aut Silverman, Joseph H. 1955- Verfasser (DE-588)118906933 aut Erscheint auch als Online-Ausgabe 978-1-4939-1711-2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027605308&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Hoffstein, Jeffrey 1953- Pipher, Jill 1955- Silverman, Joseph H. 1955- An introduction to mathematical cryptography Kryptologie (DE-588)4033329-2 gnd |
| subject_GND | (DE-588)4033329-2 (DE-588)4123623-3 |
| title | An introduction to mathematical cryptography |
| title_auth | An introduction to mathematical cryptography |
| title_exact_search | An introduction to mathematical cryptography |
| title_full | An introduction to mathematical cryptography Jeffrey Hoffstein ; Jill Pipher ; Joseph H. Silverman |
| title_fullStr | An introduction to mathematical cryptography Jeffrey Hoffstein ; Jill Pipher ; Joseph H. Silverman |
| title_full_unstemmed | An introduction to mathematical cryptography Jeffrey Hoffstein ; Jill Pipher ; Joseph H. Silverman |
| title_short | An introduction to mathematical cryptography |
| title_sort | an introduction to mathematical cryptography |
| topic | Kryptologie (DE-588)4033329-2 gnd |
| topic_facet | Kryptologie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027605308&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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