Mathematics of public key cryptography:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2012
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 579 - 602 |
| Physical Description: | xiv, 615 Seiten 25 cm |
| ISBN: | 9781107013926 1107013925 |
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Titel: Mathematics of public key cryptography
Autor: Galbraith, Steven D.
Jahr: 2012
Contents
Preface page xiii
Acknowledgements xiv
1 Introduction 1
1.1 Public key cryptography 2
1.2 The textbook RSA cryptosystem 2
1.3 Formal definition of public key cryptography 4
PART I BACKGROUND 11
2 Basic algorithmic number theory 13
2.1 Algorithms and complexity 13
2.2 Integer operations 21
2.3 Euclid's algorithm 24
2.4 Computing Legendre and Jacobi symbols 27
2.5 Modular arithmetic 29
2.6 Chinese remainder theorem 31
2.7 Linear algebra 32
2.8 Modular exponentiation 33
2.9 Square roots modulo p 36
2.10 Polynomial arithmetic 38
2.11 Arithmetic in finite fields 39
2.12 Factoring polynomials over finite fields 40
2.13 Hensel lifting 43
2.14 Algorithms in finite fields 43
2.15 Computing orders of elements and primitive roots 47
2.16 Fast evaluation of polynomials at multiple points 51
2.17 Pseudorandom generation 53
2.18 Summary 53
3 Hash functions and MACs 54
3.1 Security properties of hash functions 54
3.2 Birthday attack 55
vi Contents
3.3 Message authentication codes 56
3.4 Constructions of hash functions 56
3.5 Number-theoretic hash functions 57
3.6 Full domain hash 57
3.7 Random oracle model 58
PARTII ALGEBRAIC GROUPS 59
4 Preliminary remarks on algebraic groups 61
4.1 Informal definition of an algebraic group 61
4.2 Examples of algebraic groups 62
4.3 Algebraic group quotients 63
4.4 Algebraic groups over rings 64
5 Varieties 66
5.1 Affine algebraic sets 66
5.2 Projective algebraic sets 69
5.3 Irreducibility 74
5.4 Function fields 76
5.5 Rational maps and morphisms 79
5.6 Dimension 83
5.7 Weil restriction of scalare 84
6 Tori, LUC and XTR 86
6.1 Cyclotomic subgroups of finite fields 86
6.2 Algebraic tori 88
6.3 The group Gq,2 89
6.4 The group Gqfi 94
6.5 Further remarks 99
6.6 Algebraic tori over rings 99
7 Curves and divisor class groups 101
7.1 Non-singular varieties 101
7.2 Weierstrass equations 105
7.3 Uniformisers on curves 106
7.4 Valuation at a point on a curve 108
7.5 Valuations and points on curves 110
7.6 Divisors 111
7.7 Principal divisors 112
7.8 Divisor class group 114
7.9 Elliptic curves 116
8 Rational maps on curves and divisors 121
8.1 Rational maps of curves and the degree 121
8.2 Extensions of valuations 123
Contents vii
8.3 Maps on divisor classes 126
8.4 Riemann-Roch spaces 129
8.5 Derivations and differentials 130
8.6 Genus zero curves 136
8.7 Riemann-Roch theorem and Hurwitz genus formula 137
9 Elliptic curves 138
9.1 Group law 138
9.2 Morphisms between elliptic curves 140
9.3 Isomorphisms of elliptic curves 142
9.4 Automorphisms 143
9.5 Twists 144
9.6 Isogenies 146
9.7 The invariant differential 153
9.8 Multiplication by « and division polynomials 155
9.9 Endomorphism structure 156
9.10 Frobeniusmap 158
9.11 Supersingular elliptic curves 164
9.12 Alternative models for elliptic curves 168
9.13 Statistical properties of elliptic curves over finite fields 175
9.14 Elliptic curves over rings 177
10 Hyperelliptic curves 178
10.1 Non-singular models for hyperelliptic curves 179
10.2 Isomorphisms, automorphisms and twists 186
10.3 Effective affine divisors on hyperelliptic curves 188
10.4 Addition in the divisor class group 196
10.5 Jacobians, Abelian varieties and isogenics 204
10.6 Elements of order n 206
10.7 Hyperelliptic curves over finite fields 206
10.8 Supersingular curves 209
PART HI EXPONENTIATION, FACTORING AND DISCRETE
LOGARITHMS 213
11 Basic algorithms for algebraic groups 215
11.1 Efficient exponentiation using signed exponents 215
11.2 Multi-exponentiation 219
11.3 Efficient exponentiation in specific algebraic groups 221
11.4 Sampling from algebraic groups 231
11.5 Determining group structure and computing generators for elliptic
curves 235
11.6 Testing subgroup membership 236
viii Contents
12 Primality testing and integer factorisation using algebraic groups 238
12.1 Primality testing 238
12.2 Generating random primes 240
12.3 The p - 1 factoring method 242
12.4 Elliptic curve method 244
12.5 Pollard-Strassen method 245
13 Basic discrete logarithm algorithms 246
13.1 Exhaustive search 247
13.2 The Pohlig-Hellman method 247
13.3 Baby-step-giant-step (BSGS) method 250
13.4 Lower bound on complexity of generic algorithms for the DLP 253
13.5 Generalised discrete logarithm problems 256
13.6 Low Hamming weight DLP 258
13.7 Low Hamming weight product exponents 260
14 Factoring and discrete logarithms using pseudorandom walks 262
14.1 Birthday paradox 262
14.2 The Pollard rho method 264
14.3 Distributed Pollard rho 273
14.4 Speeding up the rho algorithm using equivalence classes 276
14.5 The kangaroo method 280
14.6 Distributed kangaroo algorithm 287
14.7 The Gaudry-Schost algorithm 292
14.8 Parallel collision search in other contexts 296
14.9 Pollard rho factoring method 297
15 Factoring and discrete logarithms in subexponential time 301
15.1 Smooth integers 301
15.2 Factoring using random squares 303
15.3 Elliptic curve method revisited 310
15.4 The number field sieve 312
15.5 Index calculus in finite fields 313
15.6 Discrete logarithms on hyperelliptic curves 324
15.7 Weil descent 328
15.8 Discrete logarithms on elliptic curves over extension fields 329
15.9 Further results 332
PART IV LATTICES 335
16 Lattices 337
16.1 Basic notions on lattices 338
16.2 The Hermite and Minkowski bounds 343
16.3 Computational problems in lattices 345
Contents ix
17 Lattice basis reduction 347
17.1 Lattice basis reduction in two dimensions 347
17.2 LLL-reduced lattice bases 352
17.3 The Gram-Schmidt algorithm 356
17.4 The LLL algorithm 358
17.5 Complexity of LLL 362
17.6 Variants of the LLL algorithm 365
18 Algorithms for the closest and shortest vector problems 366
18.1 Babai's nearest plane method 366
18.2 Babai's rounding technique 371
18.3 The embedding technique 373
18.4 Enumerating all short vectors 375
18.5 Korkine-Zolotarev bases 379
19 Coppersmith's method and related applications 380
19.1 Coppersmith's method for modular univariate polynomials 380
19.2 Multivariate modular polynomial equations 387
19.3 Bivariate integer polynomials 387
19.4 Some applications of Coppersmith's method 390
19.5 Simultaneous Diophantine approximation 397
19.6 Approximate integer greatest common divisors 398
19.7 Learning with errors 400
19.8 Further applications of lattice reduction 402
PART V CRYPTOGRAPHY RELATED TO DISCRETE LOGARITHMS 403
20 The Diffie-Hellman problem and cryptographic applications 405
20.1 The discrete logarithm assumption 405
20.2 Key exchange 405
20.3 Textbook Elgamal encryption 408
20.4 Security of textbook Elgamal encryption 410
20.5 Security of Diffie-Hellman key exchange 414
20.6 Efficiency considerations for discrete logarithm cryptography 416
21 The Diffie-Hellman problem 418
21.1 Variants of the Diffie-Hellman problem 418
21.2 Lower bound on the complexity of CDH for generic
algorithms 422
21.3 Random self-reducibility and self-correction of CDH 423
21.4 The den Boer and Maurer reductions 426
21.5 Algorithms for static Diffie-Hellman 435
21.6 Hard bits of discrete logarithms 439
21.7 Bit security of Diffie-Hellman 443
x Contents
22 Digital signatures based on discrete logarithms 452
22.1 Schnorr signatures 452
22.2 Other public key signature schemes 459
22.3 Lattice attacks on signatures 466
22.4 Other signature functionalities 467
23 Public key encryption based on discrete logarithms 469
23.1 CCA secure Elgamal encryption 469
23.2 Cramer-Shoup encryption 474
23.3 Other encryption functionalities 478
PART VI CRYPTOGRAPHY RELATED TO INTEGER
FACTORISATION 483
24 The RSA and Rabin cryptosystems 485
24.1 The textbook RSA cryptosystem 485
24.2 The textbook Rabin cryptosystem 491
24.3 Homomorphic encryption 498
24.4 Algebraic attacks on textbook RSA and Rabin 499
24.5 Attacks on RSA parameters 504
24.6 Digital signatures based on RSA and Rabin 507
24.7 Public key encryption based on RSA and Rabin 511
PART VII ADVANCED TOPICS IN ELLIPTIC AND
HYPERELLIPTIC CURVES 513
25 Isogenies of elliptic curves 515
25.1 Isogenies and kernels 515
25.2 Isogenies from y-invariants 523
25.3 Isogeny graphs of elliptic curves over finite fields 529
25.4 The structure of the ordinary isogeny graph 535
25.5 Constructing isogenies between elliptic curves 540
25.6 Relating the discrete logarithm problem on isogenous curves 543
26 Pairings on elliptic curves 545
26.1 Weil reciprocity 545
26.2 The Weil pairing 546
26.3 The Tate-Lichtenbaum pairing 548
26.4 Reduction of ECDLP to finite fields 557
26.5 Computational problems 559
26.6 Pairing-friendly elliptic curves 561
Appendix A Background mathematics 564
A.l Basic notation 564
A.2 Groups 564
Contents xi
A.3 Rings 565
A.4 Modules 565
A.5 Polynomials 566
A.6 Field extensions 567
A.7 Galois theory 569
A.8 Finite fields 570
A.9 Ideals 571
A. 10 Vector spaces and linear algebra 572
A. 11 Hermite normal form 575
A. 12 Orders in quadratic fields 575
A. 13 Binary strings 576
A. 14 Probability and combinatorics 576
References 579
Author index 603
Subject index 608 |
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| discipline | Informatik Mathematik |
| edition | 1. publ. |
| format | Book |
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| spelling | Galbraith, Steven D. 1968- Verfasser (DE-588)121893928 aut Mathematics of public key cryptography Steven D. Galbraith, University of Auckland Cambridge Cambridge University Press 2012 xiv, 615 Seiten 25 cm txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 579 - 602 Mathematik (DE-588)4037944-9 gnd rswk-swf Kryptologie (DE-588)4033329-2 gnd rswk-swf Public-Key-Kryptosystem (DE-588)4209133-0 gnd rswk-swf Kryptologie (DE-588)4033329-2 s Public-Key-Kryptosystem (DE-588)4209133-0 s Mathematik (DE-588)4037944-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024975105&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Galbraith, Steven D. 1968- Mathematics of public key cryptography Mathematik (DE-588)4037944-9 gnd Kryptologie (DE-588)4033329-2 gnd Public-Key-Kryptosystem (DE-588)4209133-0 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4033329-2 (DE-588)4209133-0 |
| title | Mathematics of public key cryptography |
| title_auth | Mathematics of public key cryptography |
| title_exact_search | Mathematics of public key cryptography |
| title_full | Mathematics of public key cryptography Steven D. Galbraith, University of Auckland |
| title_fullStr | Mathematics of public key cryptography Steven D. Galbraith, University of Auckland |
| title_full_unstemmed | Mathematics of public key cryptography Steven D. Galbraith, University of Auckland |
| title_short | Mathematics of public key cryptography |
| title_sort | mathematics of public key cryptography |
| topic | Mathematik (DE-588)4037944-9 gnd Kryptologie (DE-588)4033329-2 gnd Public-Key-Kryptosystem (DE-588)4209133-0 gnd |
| topic_facet | Mathematik Kryptologie Public-Key-Kryptosystem |
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