Algebraic curves in cryptography:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton ; London ; New York
CRC Press, Taylor & Francis Group
[2018]
|
| Series: | Discrete mathematics and its applications
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | First issued in paperback 2018 |
| Physical Description: | xvii, 321 Seiten Illustrationen |
| ISBN: | 9781420079463 9781138381414 |
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MARC
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Record in the Search Index
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| adam_text | Contents Preface xiii List of Figures xv List of Tables xvii 1 Introduction to Algebraic Curves 1.1 Plane Curves................. 1.2 Algebraic Curves and Their Function Fields ........................ 1.3 Smooth Curves ......... . ................................. 1.4 Riemann-Roch Theorem .................... 1.5 Rational Points and Zeta Functions ....................................... 1 1 6 8 11 15 2 Introduction to Error-Correcting Codes 2.1 Introduction .................................................................... 2.2 Linear Codes....................... 2.3 Bounds ........................ ... ............... .. .·■.................... 2.4 Algebraic Geometry Codes .................................................... 2.5 Asymptotic Behavior of Codes . ... .............. 21 21 21 29 35 38 3 Elliptic Curves and Their Applications to Cryptography , 3.1 Basic Introduction............ .. ..................................................... 3.1.1 The Divisor Class Group of an Elliptic Curve . . ... 3.1.2 The Group Law on Elliptic Curves.............................. 3.2 Maps between Elliptic Curves ................................................ 3.2.1 Morphisms of Elliptic Curves . ..................... 3.2.2 The Frobenius and Multiplication Morphisms............ 3.3 The Group £(Fq) and Its Torsion Subgroups ......... 3.3.1 The Torsion Group £[m] . .......................................... .. 3.3.2 The Group 5(Fq) .................................... 3.3.3 Supersingular Elliptic Curves .............. 3.4 Computational Considerations on Elliptic Curves ...... 3.4.1
Finding Multiples of a Point . . ............... ................. 3.4.2 Computing the Miller Functions ............ 3.4.3 Finding the Order of £(Fg)..................... . ............... 3.4.4 The Discrete Logarithm Problem on an Elliptic Curve 43 44 49 51 54 54 60 62 62 64 67 68 68 70 72 73 ix
x Contents 3.5 Pairings on an Elliptic Curve.................................................. 3.5.1 The Weil Pairing . . . ................................................. 3.5.2 The Tate-Lichtenbaum Pairing.............. 3.5.3 Embedding Degrees..................................................... 3.5.4 Modified Weil Pairing on Supersingular Elliptic Curves 3.5.5 Pairings and the Discrete Logarithm Problem............ 3.6 Elliptic Curve Cryptography ................................................... 3.6.1 The Elliptic Curve Factorization Method.................. 3.6.2 Discrete Logarithm-Based Elliptic Curve Schemes . . 3.6.3 Pairing-Based Schemes................................................... 4 Secret Sharing Schemes 4.1 The Shamir Threshold Scheme................................................ 4.2 Other Threshold Schemes......................................................... 4.2.1 The Karnin-Greene-Hellman (n, n)-Threshold Scheme 4.2.2 The Blakley Threshold Scheme . . . ......................... 4.2.3 The Asmuth-Bloom Threshold Scheme ........................ 4.3 General Secret Sharing Schemes ................ 4.3.1 Secret Sharing Schemes from Cumulative Arrays ... 4.3.2 The Benaloh-Leichter Secret Sharing Scheme............ 4.4 Information Rate ...... .... .............. 4.4.1 Entropy ................................. .....՛............ .. 4.4.2 Perfect Secret Sharing Schemes ............. 4.5 Quasi-Perfect Secret Sharing Schemes............ ... ............... .. 4.6 Linear Secret Sharing Schemes ................................................ 4.7
Multiplicative Linear Secret Sharing Schemes........................ 4.8 Secret Sharing from Error-Correcting Codes . . . . . . . . . 4.9 Secret Sharing from Algebraic Geometry Codes .................. 5 75 75 78 80 81 82 83 83 85 87 91 91 94 94 95 96 97 98 101 103 104 107 112 115 121 127 131 Authentication Codes 137 5.1 Authentication Codes ........................ .. ................................... 138 5.2 Bounds for A-Godes................. ............................................... 141 5.2.1 Information-Theoretic Bounds for A-Codes............... 141 5.2.2 Combinatorial Bounds for A-Codes . . . ..................... 142 5.3 A-Codes and Error-Correcting Codes . . . ........................... 146 5.3.1 From A-Codes to Error-Correcting Codes.................. 146 5.3.2 From Linear Error-Correcting Codes to A-Codes ... 148 5.4 Universal Hash Families and A-Codes .................................... 151 5.4.1 є-AU Hash Families..................................................... 154 5.4.2 e-ASU Hash Families . ............................................ 156 5.5 A-Codes from Algebraic Curves ............................................. 160 5.6 Linear Authentication Codes . .................. .. .................. .. . 163 5.6.1 Interpreting a Linear A-Code as a Family of Subspaces 165 5.6.2 Bounds for Linear A-Codes............... 167 5.6.3 Constructions of Linear A-Codes ................................. 170
Contents 6 Frameproof Codes 6.1 Introduction . ... . ................ .................. ... . . . ................ 6.2 Constructions of Frameproof Codes without Algebraic Geome try ................................. 6.2.1 Constructions of Frameproof Codes............................... 6.2.2 Two Characterizations of Binary Frameproof Codes . 6.2.3 Combinatorial Constructions of Binary Frameproof Codes .................................. ............... 6.3 Asymptotic Bounds and Constructions from Algebraic Geom etry ............................ 6.4 Improvements to the Asymptotic Bound ........... xi 173 173 174 175 178 180 182 190 7 Key Distribution Schemes 199 7.1 Key Predistribution..................................... ... ......................... 199 7.2 Key Predistribution Schemes with Optimal Information Rates 201 7.2.1 Bounds for KPSs ............................... 201 7.2.2 The Blom Scheme .................... 203 7.2.3 The Blundo et al. Scheme ............................................... 204 7.2.4 The Fiat-Naor Scheme........................... 204 7.3 Linear Key Predistribution Schemes........................................ 205 7.4 Key Predistribution Schemes from Algebraic Geometry . . . 209 7.5 Key Predistribution Schemes from Cover-Free Families ... 211 7.5.1 Bounds for Cover-Free Families . .................................. 215 7.5.2 Constructions from Error-Correcting Codes................ 218 7.5.3 Constructions from Designs . . . . ............................... 220 7.5.4 Constructions from Perfect Hash Families . ................ 222 7.6 Perfect
Hash Families and Algebraic Geometry ...................... 224 8 Broadcast Encryption and Multicast Security 8.1 One-Time Broadcast Encryption ............................................... 8.1.1 Two Trivial Constructions............ ................................. 8.1.2 The Blundo-Frota Mattos-Stinson Scheme............ ... . 8.1.3 KIO Construction........................... 8.1.4 The Fiat-Naor OTBES .................. 8.2 Multicast Re-Keying Schemes ................................................. 8.2.1 Re-Keying Schemes from Cover-Free Families ..... 8.2.2 Re-Keying Schemes from Secret Sharing.................. . 8.2.3 Logical Key Hierarchy Schemes ...................................... 8.2.3.1 Key Initialization Scheme from Resilient LKH 8.2.3.2 The OR Broadcast Scheme . . . ............... 8.2.3.3 The ANDBroadcast Scheme.......................... 8.3 Re-Keying Schemes with Dynamic Group Controllers .... 8.3.1 The OR Re-Keying Scheme with Dynamic Controller 8.3.2 The AND Re-Keying Scheme with Dynamic Controller 8.4 Some Applications from Algebraic Geometry ......... 231 232 234 235 237 241 244 245 246 248 249 251 251 252 253 256 257
Contents xii 8.4.1 8.4.2 8.4.3 8.4.4 9 OTBESs over Constant Size Fields......................... Improving the Fiat-Naor OTBES.................................. Improving the Blacklisting Scheme............................... Construction of տ-Resilient LKHs ............................... Sequences 9.1 Introduction . . . ................ .................. ... ............................... 9.2 Linear Feedback Shift Register Sequences............................... 9.3 Constructions of Almost Perfect Sequences . . ...................... 9.3.1 Linear Complexity........................... 9.3.2 Constructions of (¿-Perfect Sequences from Algebraic Curves ................................................................................. 9.3.3 Examples of d-Perfect Sequences ............ 9.4 Constructions of Multisequences ............................... 9.5 Sequences with Low Correlation and Large Linear Complexity 9.5.1 Construction Using a Projective Line ............................ 9.5.2 Construction Using Elliptic Curves............................... 257 258 258 259 261 261 262 264 265 267 270 278 286 293 297 Bibliography 301 Index 315
|
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| author | Ling, San 1964- Wang, Huaxiong Xing, Chaoping 1963- |
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| illustrated | Illustrated |
| indexdate | 2024-07-10T08:36:50Z |
| institution | BVB |
| isbn | 9781420079463 9781138381414 |
| language | English |
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| physical | xvii, 321 Seiten Illustrationen |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | CRC Press, Taylor & Francis Group |
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| series2 | Discrete mathematics and its applications |
| spelling | Ling, San 1964- Verfasser (DE-588)138740755 aut Algebraic curves in cryptography San Ling, Huaxiong Wang, Chaoping Xing Boca Raton ; London ; New York CRC Press, Taylor & Francis Group [2018] xvii, 321 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Discrete mathematics and its applications First issued in paperback 2018 Kryptologie (DE-588)4033329-2 gnd rswk-swf Elliptische Kurve (DE-588)4014487-2 gnd rswk-swf Algebraische Kurve (DE-588)4001165-3 gnd rswk-swf Algebraische Kurve (DE-588)4001165-3 s Kryptologie (DE-588)4033329-2 s DE-604 Elliptische Kurve (DE-588)4014487-2 s Wang, Huaxiong Verfasser aut Xing, Chaoping 1963- Verfasser (DE-588)138740844 aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031537520&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ling, San 1964- Wang, Huaxiong Xing, Chaoping 1963- Algebraic curves in cryptography Kryptologie (DE-588)4033329-2 gnd Elliptische Kurve (DE-588)4014487-2 gnd Algebraische Kurve (DE-588)4001165-3 gnd |
| subject_GND | (DE-588)4033329-2 (DE-588)4014487-2 (DE-588)4001165-3 |
| title | Algebraic curves in cryptography |
| title_auth | Algebraic curves in cryptography |
| title_exact_search | Algebraic curves in cryptography |
| title_full | Algebraic curves in cryptography San Ling, Huaxiong Wang, Chaoping Xing |
| title_fullStr | Algebraic curves in cryptography San Ling, Huaxiong Wang, Chaoping Xing |
| title_full_unstemmed | Algebraic curves in cryptography San Ling, Huaxiong Wang, Chaoping Xing |
| title_short | Algebraic curves in cryptography |
| title_sort | algebraic curves in cryptography |
| topic | Kryptologie (DE-588)4033329-2 gnd Elliptische Kurve (DE-588)4014487-2 gnd Algebraische Kurve (DE-588)4001165-3 gnd |
| topic_facet | Kryptologie Elliptische Kurve Algebraische Kurve |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031537520&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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