About the Author |
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ix | |
Preface |
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xi | |
Acknowledgments |
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xiii | |
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3 | (32) |
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1.1 What is Number Theory? |
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3 | (6) |
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1.2 What is Computation Theory? |
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9 | (6) |
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1.3 What is Computational Number Theory? |
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15 | (14) |
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1.4 What is Modern Cryptography? |
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29 | (3) |
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1.5 Bibliographic Notes and Further Reading |
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32 | (3) |
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32 | (3) |
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35 | (124) |
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2.1 Basic Algebraic Structures |
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35 | (11) |
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46 | (29) |
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75 | (14) |
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89 | (42) |
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131 | (10) |
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141 | (13) |
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2.7 Bibliographic Notes and Further Reading |
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154 | (5) |
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155 | (4) |
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Part II Computational Number Theory |
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159 | (32) |
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159 | (9) |
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168 | (5) |
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173 | (5) |
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178 | (9) |
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3.5 Bibliographic Notes and Further Reading |
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187 | (4) |
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188 | (3) |
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191 | (44) |
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191 | (3) |
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4.2 Trial Divisions Factoring |
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194 | (4) |
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198 | (7) |
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4.4 Elliptic Curve Method |
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205 | (4) |
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4.5 Continued Fraction Method |
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209 | (5) |
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214 | (5) |
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219 | (12) |
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4.8 Bibliographic Notes and Further Reading |
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231 | (4) |
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232 | (3) |
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235 | (30) |
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235 | (2) |
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5.2 Baby-Step Giant-Step Method |
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237 | (3) |
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5.3 Pohlig-Hellman Method |
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240 | (6) |
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246 | (5) |
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5.5 Elliptic Curve Discrete Logarithms |
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251 | (9) |
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5.6 Bibliographic Notes and Further Reading |
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260 | (5) |
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261 | (4) |
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Part III Modern Cryptography |
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6 Secret-Key Cryptography |
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265 | (28) |
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6.1 Cryptography and Cryptanalysis |
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265 | (12) |
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6.2 Classic Secret-Key Cryptography |
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277 | (8) |
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6.3 Modern Secret-Key Cryptography |
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285 | (6) |
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6.4 Bibliographic Notes and Further Reading |
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291 | (2) |
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291 | (2) |
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7 Integer Factorization Based Cryptography |
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293 | (44) |
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293 | (9) |
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302 | (17) |
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319 | (7) |
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7.4 Residuosity Based Cryptography |
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326 | (5) |
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331 | (4) |
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7.6 Bibliographic Notes and Further Reading |
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335 | (2) |
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335 | (2) |
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8 Discrete Logarithm Based Cryptography |
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337 | (16) |
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8.1 Diffie-Hellman-Merkle Key-Exchange Protocol |
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337 | (5) |
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342 | (2) |
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8.3 Massey-Omura Cryptography |
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344 | (4) |
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8.4 DLP-Based Digital Signatures |
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348 | (3) |
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8.5 Bibliographic Notes and Further Reading |
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351 | (2) |
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351 | (2) |
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9 Elliptic Curve Discrete Logarithm Based Cryptography |
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353 | (26) |
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353 | (3) |
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9.2 Elliptic Curve Diffie-Hellman-Merkle Key Exchange Scheme |
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356 | (4) |
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9.3 Elliptic Curve Massey-Omura Cryptography |
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360 | (5) |
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9.4 Elliptic Curve ElGamal Cryptography |
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365 | (5) |
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9.5 Elliptic Curve RSA Cryptosystem |
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370 | (1) |
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9.6 Menezes-Vanstone Elliptic Curve Cryptography |
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371 | (2) |
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373 | (1) |
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9.8 Bibliographic Notes and Further Reading |
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374 | (5) |
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375 | (4) |
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Part IV Quantum Resistant Cryptography |
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10 Quantum Computational Number Theory |
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379 | (22) |
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10.1 Quantum Algorithms for Order Finding |
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379 | (6) |
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10.2 Quantum Algorithms for Integer Factorization |
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385 | (5) |
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10.3 Quantum Algorithms for Discrete Logarithms |
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390 | (3) |
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10.4 Quantum Algorithms for Elliptic Curve Discrete Logarithms |
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393 | (4) |
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10.5 Bibliographic Notes and Further Reading |
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397 | (4) |
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397 | (4) |
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11 Quantum Resistant Cryptography |
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401 | (12) |
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11.1 Coding-Based Cryptography |
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401 | (2) |
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11.2 Lattice-Based Cryptography |
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403 | (1) |
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11.3 Quantum Cryptography |
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404 | (2) |
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11.4 DNA Biological Cryptography |
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406 | (3) |
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11.5 Bibliographic Notes and Further Reading |
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409 | (4) |
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410 | (3) |
Index |
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413 | |