Cryptanalysis of RSA and its variants:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fl. [u.a.]
Chapman & Hall / CRC
2010
|
| Series: | Chapman & Hall/CRC cryptography and network security
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVIII, 268 S. Ill., graph. Darst. |
| ISBN: | 9781420075182 |
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| 245 | 1 | 0 | |a Cryptanalysis of RSA and its variants |c M. Jason Hinek |
| 264 | 1 | |a Boca Raton, Fl. [u.a.] |b Chapman & Hall / CRC |c 2010 | |
| 300 | |a XVIII, 268 S. |b Ill., graph. Darst. | ||
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| 490 | 0 | |a Chapman & Hall/CRC cryptography and network security | |
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| 650 | 4 | |a Public key cryptography | |
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Record in the Search Index
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| adam_text | Titel: Cryptanalysis of RSA and its variants
Autor: Hinek, M. Jason
Jahr: 2010
Contents
List of Figures xiii
List of Tables xv
Preface xvii
I Preliminaries 1
1 The RSA Cryptosystem 3
1.1 Public-Key Cryptography.................... 3
1.2 The RSA Cryptosystem..................... 4
1.3 The Security of RSA ...................... 8
1.3.1 Integer Factorization................... 8
1.3.2 Breaking RSA ...................... 9
1.3.3 Cryptanalysis of RSA.................. 10
1.3.4 The Homomorphic Property of RSA.......... 11
1.3.5 Semantic Security .................... 12
1.4 Efficiency of RSA ........................ 12
1.4.1 Prime Generation .................... 13
1.4.2 Modular Exponentiation................. 13
1.5 RSA Signature Scheme ..................... 14
1.6 Variants of RSA ......................... 14
1.7 Additional Notes......................... 16
2 Some Notation, Mathematics and Techniques 17
2.1 Some Notation.......................... 17
2.2 Some Mathematics Results ................... 19
2.3 Integer Factorization ...................... 21
2.4 Continued Fractions....................... 22
2.5 Lattices.............................. 23
2.5.1 Definitions and Basic Facts............... 24
2.5.2 LLL-Reduced Bases................... 25
2.5.3 The Shortest Vector Problem.............. 27
2.6 Solving Linear Equations .................... 28
2.6.1 Modular Linear Equations................ 31
2.7 Coppersmith s Methods..................... 33
vii
Vlll
2.7.1 Small Solutions of Modular Polynomials........ 34
2.7.1.1 Known Results for Modular Polynomials . . 39
2.7.2 Small Solutions of Integer Polynomials......... 40
2.7.2.1 Known Results for Integer Polynomials ... 42
2.7.3 Computing and Improving the Bounds......... 43
2.7.4 Assumptions for Coppersmith s Methods ....... 45
2.8 On Attacks and Proofs ..................... 46
2.9 Additional Notes......................... 47
II Cryptanalysis of RSA 49
3 Some Early Attacks 51
3.1 Common Modulus Attack.................... 51
3.2 Hastad s Broadcast Attack ................... 52
3.2.1 Common Plaintext Attack................ 53
3.2.2 Related Plaintext Attack ................ 53
3.3 Cycling Attacks ......................... 54
3.4 Additional Notes......................... 55
4 Small Public Exponent Attacks 57
4.1 Stereotyped Message Attack .................. 57
4.2 Related Message Attacks .................... 59
4.3 Random Padding Attack .................... 61
4.4 Leaking Information....................... 61
4.5 Additional Notes......................... 63
5 Small Private Exponent Attacks 65
5.1 Wiener s Continued Fraction Attack.............. 66
5.1.1 Extending Wiener s Attack............... 69
5.1.2 Wiener s Attack with Lattices.............. 70
5.1.2.1 Heuristic Approach.............. 71
5.1.2.2 Provable Approach .............. 72
5.2 Boneh and Durfee s Lattice Attacks .............. 74
5.2.1 Lattice Attack...................... 74
5.2.2 Sub-Lattice Attack.................... 79
5.2.3 Blomer and May s Attack................ 83
5.3 Effectiveness of the Attacks................... 87
5.4 Additional Notes......................... 89
6 Partial Key Exposure Attacks 91
6.1 Factoring with a Hint ...................... 91
6.2 Partially Known Private Exponent: MSBs .......... 93
6.2.1 Arbitrary Exponents................... 94
6.2.2 Full Sized Public Exponent............... 98
6.2.3 Full Sized Private Exponent............... 99
6.2.4 Effectiveness in Practice................. 105
IX
6.3 Partially Known Private Exponent: LSBs........... 106
6.3.1 Arbitrary Exponents................... 106
6.3.2 Full Sized Public Exponent............... 108
6.3.3 Full Sized Private Exponent............... 108
6.3.4 Practical Effectiveness.................. 112
6.4 Partially Known Primes..................... 113
6.4.1 Small Private Exponent Attacks............ 114
6.4.2 Partial Key Exposure Attacks.............. 115
6.4.3 Exhaustive Search Attacks ............... 116
6.5 Key Reconstruction with Random Errors ........... 117
6.6 Additional Notes......................... 118
7 More Small Private Exponent Attacks 121
7.1 Common Modulus Attack.................... 121
7.1.1 Practical Effectiveness.................. 128
7.2 Common Private Exponent Attack............... 129
7.2.1 Practical Effectiveness.................. 132
7.3 Additional Notes......................... 134
III Cryptanalysis of Variants of RSA 137
8 CRT-RSA 139
8.1 CRT-RSA ............................ 139
8.1.1 Variations of CRT-RSA................. 140
8.1.2 Efficiency of CRT-RSA ................. 141
8.1.3 Breaking CRT-RSA................... 142
8.2 Small CRT-Exponent Attacks ................. 143
8.3 Partial Key Exposure Attacks ................. 148
8.4 Key Reconstruction with Random Errors ........... 151
8.5 Additional Notes......................... 152
9 Multi-Prime RSA 155
9.1 Multi-Prime RSA ........................ 155
9.1.1 Efficiency of Multi-Prime RSA............. 157
9.1.2 Breaking Multi-Prime RSA............... 158
9.2 Factoring the Modulus ..................... 158
9.2.1 Factoring with a Hint.................. 159
9.3 Small Private Exponent Attacks ................ 161
9.4 Partial Key Exposure Attacks ................. 163
9.4.1 Partial Private Exponent: MSBs............ 164
9.4.1.1 Arbitrary Exponents.............. 164
9.4.1.2 Full Sized Public Exponent.......... 166
9.4.1.3 Full Sized Private Exponent ......... 167
9.4.1.4 Effectiveness in Practice............ 168
9.4.2 Partial Private Exponent: LSBs............. 169
9.4.2.1 Arbitrary Exponents.............. 170
9.4.2.2 Full Sized Public Exponent.......... 171
9.4.2.3 Full Sized Private Exponent ......... 171
9.4.2.4 Effectiveness in Practice............ 172
9.4.3 Known Primes...................... 172
9.4.3.1 Small Private Exponent Attack........ 172
9.4.3.2 Partially Known Private Exponent Attack . 174
9.4.4 Partial Primes ...................... 176
9.5 Common Modulus Attacks................... 176
9.5.1 Howgrave-Graham and Seifert s Attack ........ 176
9.5.1.1 Practical Effectiveness............. 178
9.5.2 Guo s Attack....................... 178
9.5.2.1 Practical Effectiveness............. 180
9.6 CRT Attacks........................... 181
9.6.1 Small CRT-Exponent Attack.............. 181
9.6.2 Partial CRT-Exponent Attack ............. 182
9.7 Additional Notes......................... 184
10 Multi-Power RSA 187
10.1 Takagi s Scheme ......................... 187
10.1.1 Efficiency of Takagi s Scheme.............. 188
10.1.2 Breaking Takagi s Scheme................ 190
10.2 Factoring the Modulus ..................... 191
10.2.1 Factoring with a Hint.................. 192
10.3 Small Private Exponent Attacks ................ 192
10.4 Partial Key Exposure Attacks ................. 194
10.5 Common Modulus Attack.................... 195
10.6 Multi-Exponent RSA...................... 196
10.6.1 Small Private Exponent Attacks............ 197
10.6.2 Partial Key Exposure Attacks.............. 198
10.7 Additional Notes......................... 200
11 Common Prime RSA 201
11.1 Common Prime RSA ...................... 201
11.1.1 Efficiency of Common Prime RSA........... 202
11.1.2 Breaking Common Prime RSA............. 203
11.2 Factoring the Modulus ..................... 203
11.2.1 Known a and b...................... 204
11.2.2 Known g ......................... 205
11.2.3 Factoring JV - 1 (or Computing g)........... 207
11.3 Small Private Exponent Attacks ................ 208
11.3.1 Wiener s Attack ..................... 209
11.3.2 Lattice-Based Attacks.................. 210
11.3.3 Known g Attacks..................... 214
11.3.4 Summary of Attacks................... 216
11.4 Small CRT-Exponent Attacks ................. 218
XI
11.5 Additional Notes......................... 219
12 Dual RSA 221
12.1 Dual RSA ............................ 221
12.2 Small Public Exponent ..................... 222
12.3 Small Private Exponent..................... 225
12.4 Dual CRT-RSA ......................... 226
12.5 Efficiency and Comparison ................... 228
12.5.1 Key Generation...................... 228
12.5.2 Space Requirements................... 229
12.6 Additional Notes......................... 230
A Distribution of g = gcd(p - 1, q - 1) 233
B Geometrically Progressive Matrices 235
C Some Algorithms 239
C.I Dual RSA ............................ 239
Further Reading 241
Bibliography 243
Index 265
List of Figures
2.1 Sylvester matrix.......................... 20
5.1 Boneh and Durfee s basis matrix................. 77
6.1 Partial key exposure attacks with known MSBs (theory). ... 94
6.2 Partial key exposure attacks with known MSBs (practice). . . 105
6.3 Partial key exposure attacks with known LSBs (theory). . . . 107
6.4 Partial key exposure attacks with known LSBs (practice). , . 112
7.1 Common private exponent attack with 2 r 5 (practice). . 133
7.2 Common private exponent attack with 10 r 35 (practice). 133
7.3 Common private exponent attack with different modulus sizes
(practice).............................. 134
8.1 Small CRT-exponent attacks (theory).............. 147
9.1 Partial key exposure attacks with known MSBs (theory) ... 165
9.2 Partial key exposure attacks with known MSBs (practice). . . 169
9.3 Partial key exposure attacks with known LSBs (theory). ... 170
9.4 Partial key exposure attacks with known LSBs (practice). . . 173
11.1 Small private exponent bounds for common prime RSA with
infinite modulus size (theory)................... 217
11.2 Small private exponent bounds for common prime RSA with
finite modulus size (theory).................... 217
12.1 Small public and CRT-exponent attacks on Dual CRT-RSA
(theory)............................... 226
XIll
List of Tables
5.1 Effectiveness of small private exponent attacks ........ 88
6.1 Effectiveness of exhaustive search attacks........... 117
7.1 Effectiveness of common modulus attack with two instances . 128
7.2 Effectiveness of common modulus attack with three instances 128
9.1 Maximum number of safe primes for multi-prime RSA .... 159
9.2 Fraction of bits required for factoring with a hint....... 161
9.3 Small private exponent bounds for multi-prime RSA..... 164
9.4 Howgrave-Graham and Seifert s common modulus attack in
practice.............................. 179
9.5 Guo s common modulus attack in practice........... 181
10.1 Number of modular exponentiations for decryption algorithms 190
10.2 Maximum number of safe primes for multi-power RSA .... 191
10.3 Small private exponent bounds for Takagi s scheme...... 193
10.4 Effectiveness of small private exponent attacks for Takagi s
scheme............................... 194
10.5 Common modulus bounds for Takagi s scheme......... 196
10.6 Effectiveness of common modulus attack for Takagi s scheme 196
10.7 Small private exponent bounds for multi-power RSA..... 198
11.1 Safe sizes for common factors.................. 208
12.1 Space requirements for standard decryption and small public
or private exponent........................ 230
12.2 Space requirements for CRT decryption with small CRT-
exponents ............................. 230
A.I Distribution of 2 g 20.................... 233
A.2 Distribution of 20 g 100................... 234
|
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| spelling | Hinek, M. Jason Verfasser (DE-588)138915865 aut Cryptanalysis of RSA and its variants M. Jason Hinek Boca Raton, Fl. [u.a.] Chapman & Hall / CRC 2010 XVIII, 268 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC cryptography and network security Includes bibliographical references and index Mathematik Telecommunication Security measures Mathematics Public key cryptography Computer security Kryptoanalyse (DE-588)4830502-9 gnd rswk-swf RSA-Verschlüsselung (DE-588)4738133-4 gnd rswk-swf RSA-Verschlüsselung (DE-588)4738133-4 s Kryptoanalyse (DE-588)4830502-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018608975&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hinek, M. Jason Cryptanalysis of RSA and its variants Mathematik Telecommunication Security measures Mathematics Public key cryptography Computer security Kryptoanalyse (DE-588)4830502-9 gnd RSA-Verschlüsselung (DE-588)4738133-4 gnd |
| subject_GND | (DE-588)4830502-9 (DE-588)4738133-4 |
| title | Cryptanalysis of RSA and its variants |
| title_auth | Cryptanalysis of RSA and its variants |
| title_exact_search | Cryptanalysis of RSA and its variants |
| title_full | Cryptanalysis of RSA and its variants M. Jason Hinek |
| title_fullStr | Cryptanalysis of RSA and its variants M. Jason Hinek |
| title_full_unstemmed | Cryptanalysis of RSA and its variants M. Jason Hinek |
| title_short | Cryptanalysis of RSA and its variants |
| title_sort | cryptanalysis of rsa and its variants |
| topic | Mathematik Telecommunication Security measures Mathematics Public key cryptography Computer security Kryptoanalyse (DE-588)4830502-9 gnd RSA-Verschlüsselung (DE-588)4738133-4 gnd |
| topic_facet | Mathematik Telecommunication Security measures Mathematics Public key cryptography Computer security Kryptoanalyse RSA-Verschlüsselung |
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