Handbook of elliptic and hyperelliptic curve cryptography
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| 245 | 1 | 0 | |a Handbook of elliptic and hyperelliptic curve cryptography |c Henri Cohen ...[u.a.] |
| 264 | 1 | |a Boca Raton |b Chapman & Hall/CRC |c 2006 | |
| 300 | |a XXXIV, 808 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Discrete mathematics and its applications | |
| 650 | 4 | |a Automates mathématiques, Théorie des - Guides, manuels, etc | |
| 650 | 4 | |a Courbes elliptiques - Guides, manuels, etc | |
| 650 | 4 | |a Cryptographie - Mathématiques - Guides, manuels, etc | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Curves, Elliptic |v Handbooks, manuals, etc | |
| 650 | 4 | |a Cryptography |x Mathematics |v Handbooks, manuals, etc | |
| 650 | 4 | |a Machine theory |v Handbooks, manuals, etc | |
| 650 | 0 | 7 | |a Elliptische Kurve |0 (DE-588)4014487-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Kryptologie |0 (DE-588)4033329-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Kryptologie |0 (DE-588)4033329-2 |D s |
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Table of Contents
List of Algorithms.xxiii
Preface.xxix
1 Introduction to Public-Key Cryptography.1
1.1 Cryptography.2
1.2 Complexity.2
1.3 Public-key cryptography.5
1.4 Factorization and primality.6
1.4.1 Primality.6
1.4.2 Complexity of factoring.6
1.4.3 RSA.7
1.5 Discrete logarithm systems.8
1.5.1 Generic discrete logarithm systems.8
1.5.2 Discrete logarithm systems with bilinear structure . 9
1.6 Protocols.9
1.6.1 Diffie-Hellman key exchange.10
1.6.2 Asymmetric Diffie-Hellman and EIGamal encryption . . . .10
1.6.3 Signature scheme of EIGamal-type.12
1.6.4 Tripartite key exchange.13
1.7 Other problems.14
I Mathematical Background
2 Algebraic Background.19
2.1 Elementary algebraic structures.19
2.1.1 Groups.19
2.1.2 Rings.21
2.1.3 Fields.23
2.1.4 Vector spaces.24
2.2 Introduction to number theory.24
2.2.1 Extension of fields.25
2.2.2 Algebraic closure.27
2.2.3 Galois theory.27
2.2.4 Number fields.29
2.3 Finite fields.31
2.3.1 First properties.31
2.3.2 Algebraic extensions of a finite field.32
2.3.3 Finite field representations.33
2.3.4 Finite field characters.35
XI
XII
Table of Contents
3 Background on p-adic Numbers.39
3.1 Definition of Qp and first properties.39
3.2 Complete discrete valuation rings and fields.41
3.2.1 First properties.41
3.2.2 Lifting a solution of a polynomial equation.42
3.3 The field Qp and its extensions.43
3.3.1 Unramified extensions.43
3.3.2 Totally ramified extensions.43
3.3.3 Multiplicative system of representatives.44
3.3.4 Witt vectors.44
4 Background on Curves and Jacobians.45
4.1 Algebraic varieties.45
4.1.1 Affine and projective varieties.46
4.2 Function fields.51
4.2.1 Morphisms of affine varieties.52
4.2.2 Rational maps of affine varieties.53
4.2.3 Regular functions.54
4.2.4 Generalization to projective varieties.55
4.3 Abelian varieties.55
4.3.1 Algebraic groups.55
4.3.2 Birational group laws.56
4.3.3 Homomorphisms of abelian varieties.57
4.3.4 Isomorphisms and isogenies.58
4.3.5 Points of finite order and Tate modules.60
4.3.6 Background on ¿-adic representations.61
4.3.7 Complex multiplication.63
4.4 Arithmetic of curves.64
4.4.1 Local rings and smoothness.64
4.4.2 Genus and Riemann-Roch theorem.66
4.4.3 Divisor class group.76
4.4.4 The Jacobian variety of curves.77
4.4.5 Jacobian variety of elliptic curves and group law.79
4.4.6 Ideal class group.81
4.4.7 Class groups of hyperelliptic curves.83
5 Varieties over Special Fields.87
5.1 Varieties over the field of complex numbers.87
5.1.1 Analytic varieties.87
5.1.2 Curves over C.89
5.1.3 Complex tori and abelian varieties.92
5.1.4 Isogenies of abelian varieties over C.94
5.1.5 Elliptic curves over C.95
5.1.6 Hyperelliptic curves over C.100
5.2 Varieties over finite fields.108
5.2.1 The Frobenius morphism.109
5.2.2 The characteristic polynomial of the Frobenius endomorphism . .109
5.2.3 The theorem of Hasse-Weil for Jacobians.110
5.2.4 Tate’s isogeny theorem.112
Table of Contents
xiii
6 Background on Pairings.115
6.1 General duality results.115
6.2 The Tate pairing.116
6.3 Pairings over local fields. . . 117
6.3.1 The local Tate pairing.118
6.3.2 The Lichtenbaum pairing on Jacobian varieties.119
6.4 An explicit pairing.122
6.4.1 The Tate-Lichtenbaum pairing.122
6.4.2 Size of the embedding degree.123
7 Background on Weil Descent.125
7.1 Affine Weil descent.125
7.2 The projective Weil descent.127
7.3 Descent by Galois theory.128
7.4 Zariski closed subsets inside of the Weil descent.129
7.4.1 Hyperplane sections.129
7.4.2 Trace zero varieties.130
7.4.3 Covers of curves.131
7.4.4 The GHS approach.131
8 Cohomological Background on Point Counting.133
8.1 General principle.133
8.1.1 Zeta function and the Weil conjectures.134
8.1.2 Cohomology and Lefschetz fixed point formula.135
8.2 Overview of T-adic methods.137
8.3 Overview of p-adic methods.138
8.3.1 Serre-Tate canonical lift.138
8.3.2 Monsky-Washnitzer cohomology.139
II Elementary Arithmetic
9 Exponentiation.145
9.1 Generic methods.146
9.1.1 Binary methods.146
9.1.2 Left-to-right 2fe-ary algorithm.148
9.1.3 Sliding window method.149
9.1.4 Signed-digit recoding.150
9.1.5 Multi-exponentiation.154
9.2 Fixed exponent.157
9.2.1 Introduction to addition chains.157
9.2.2 Short addition chains search.160
9.2.3 Exponentiation using addition chains.163
9.3 Fixed base point.164
9.3.1 Yao’s method.165
9.3.2 Euclidean method.166
9.3.3 Fixed-base comb method.166
x/V Table of Contents
10 Integer Arithmetic.169
10.1 Multiprecision integers.170
10.1.1 Introduction.170
10.1.2 Internal representation.171
10.1.3 Elementary operations.172
10.2 Addition and subtraction.172
10.3 Multiplication.174
10.3.1 Schoolbook multiplication.174
10.3.2 Karatsuba multiplication.176
10.3.3 Squaring.177
10.4 Modular reduction.178
10.4.1 Barrett method.178
10.4.2 Montgomery reduction.180
10.4.3 Special moduli.182
10.4.4 Reduction modulo several primes.184
10.5 Division.184
10.5.1 Schoolbook division.185
10.5.2 Recursive division.187
10.5.3 Exact division.189
10.6 Greatest common divisor.190
10.6.1 Euclid extended gcd.191
10.6.2 Lehmer extended gcd.192
10.6.3 Binary extended gcd.194
10.6.4 Chinese remainder theorem.196
10.7 Square root.197
10.7.1 Integer square root.197
10.7.2 Perfect square detection.198
11 Finite Field Arithmetic.201
11.1 Prime fields of odd characteristic.201
11.1.1 Representations and reductions.202
11.1.2 Multiplication.202
11.1.3 Inversion and division.205
11.1.4 Exponentiation.209
11.1.5 Squares and square roots.210
11.2 Finite fields of characteristic 2.213
11.2.1 Representation.213
11.2.2 Multiplication.218
11.2.3 Squaring.221
11.2.4 Inversion and division.222
11.2.5 Exponentiation.225
11.2.6 Square roots and quadratic equations.228
11.3 Optimal extension fields.229
11.3.1 Introduction.229
11.3.2 Multiplication.231
11.3.3 Exponentiation.231
11.3.4 Inversion.233
11.3.5 Squares and square roots.234
11.3.6 Specific improvements for degrees 3 and 5.235
Table of Contents
xv
12 Arithmetic of p-adic Numbers.239
12.1 Representation.239
12.1.1 Introduction.239
12.1.2 Computing the Teichmuller modulus.240
12.2 Modular arithmetic.244
12.2.1 Modular multiplication.244
12.2.2 Fast division with remainder.244
12.3 Newton lifting.246
12.3.1 Inverse.247
12.3.2 Inverse square root.248
12.3.3 Square root.249
12.4 Hensel lifting.249
12.5 Frobenius substitution.250
12.5.1 Sparse modulus.251
12.5.2 Teichmuller modulus.252
12.5.3 Gaussian normal basis.252
12.6 Artin-Schreier equations.252
12.6.1 Lercier-Lubicz algorithm.253
12.6.2 Harley’s algorithm.254
12.7 Generalized Newton lifting.256
12.8 Applications.257
12.8.1 Teichmuller lift.257
12.8.2 Logarithm.258
12.8.3 Exponential.259
12.8.4 Trace.260
12.8.5 Norm.261
III Arithmetic of Curves
13 Arithmetic of Elliptic Curves.267
13.1 Summary of background on elliptic curves.268
13.1.1 First properties and group law.268
13.1.2 Scalar multiplication.271
13.1.3 Rational points.272
13.1.4 Torsion points.273
13.1.5 Isomorphisms.273
13.1.6 Isogenies.277
13.1.7 Endomorphisms.277
13.1.8 Cardinality.278
13.2 Arithmetic of elliptic curves defined over Fp.280
13.2.1 Choice of the coordinates.280
13.2.2 Mixed coordinates.283
13.2.3 Montgomery scalar multiplication.285
13.2.4 Parallel implementations.288
13.2.5 Compression of points.288
13.3 Arithmetic of elliptic curves defined over F2 (.289
13.3.1 Choice of the coordinates.291
13.3.2 Faster doublings in affine coordinates.295
XVI
Table of Contents
13.3.3 Mixed coordinates.296
13.3.4 Montgomery scalar multiplication.298
13.3.5 Point halving and applications.299
13.3.6 Parallel implementation.302
13.3.7 Compression of points.302
14 Arithmetic of HyperelHptic Curves.303
14.1 Summary of background on hyperelliptic curves.304
14.1.1 Group law for hyperelliptic curves.304
14.1.2 Divisor class group and ideal class group.306
14.1.3 Isomorphisms and isogenies.308
14.1.4 Torsion elements.309
14.1.5 Endomorphisms.310
14.1.6 Cardinality . . .310
14.2 Compression techniques.311
14.2.1 Compression in odd characteristic.311
14.2.2 Compression in even characteristic.313
14.3 Arithmetic on genus 2 curves over arbitrary characteristic . . . . .313
14.3.1 Different cases.314
14.3.2 Addition and doubling in affine coordinates.316
14.4 Arithmetic on genus 2 curves in odd characteristic.320
14.4.1 Projective coordinates.321
14.4.2 New coordinates in odd characteristic.323
14.4.3 Different sets of coordinates in odd characteristic.325
14.4.4 Montgomery arithmetic for genus 2 curves in odd characteristic . . 328
14.5 Arithmetic on genus 2 curves in even characteristic.334
14.5.1 Classification of genus 2 curves in even characteristic. . . . 334
14.5.2 Explicit formulas in even characteristic in affine coordinates . . . 336
14.5.3 Inversion-free systems for even characteristic when h i ^ 0. . . 341
14.5.4 Projective coordinates.341
14.5.5 Inversion-free systems for even characteristic when /i2 = 0. . . 345
14.6 Arithmetic on genus 3 curves.348
14.6.1 Addition in most common case.348
14.6.2 Doubling in most common case.349
14.6.3 Doubling on genus 3 curves for even characteristic when h(x) = 1 351
14.7 Other curves and comparison.352
15 Arithmetic of Special Curves.355
15.1 Koblitz curves.355
15.1.1 Elliptic binary Koblitz curves.356
15.1.2 Generalized Koblitz curves.367
15.1.3 Alternative setup.375
15.2 Scalar multiplication using endomorphisms.376
15.2.1 GLV method. 377
15.2.2 Generalizations.380
15.2.3 Combination of GLV and Koblitz curve strategies.381
15.2.4 Curves with endomorphisms for identity-based parameters. . . 382
15.3 Trace zero varieties.383
15.3.1 Background on trace zero varieties.384
15.3.2 Arithmetic in G.385
Table of Contents_xvii
16 Implementation of Pairings.389
16.1 The basic algorithm.389
16.1.1 The setting.390
16.1.2 Preparation.391
16.1.3 The pairing computation algorithm.391
16.1.4 The case of nontrivial embedding degree A;.393
16.1.5 Comparison with the Weil pairing.395
16.2 Elliptic curves.396
16.2.1 The basic step.396
16.2.2 The representation.396
16.2.3 The pairing algorithm.397
16.2.4 Example.397
16.3 Hyperelliptic curves of genus 2.398
16.3.1 The basic step.399
16.3.2 Representation for k 2.399
16.4 Improving the pairing algorithm.400
16.4.1 Elimination of divisions.400
16.4.2 Choice of the representation.400
16.4.3 Precomputations.400
16.5 Specific improvements for elliptic curves.400
16.5.1 Systems of coordinates.401
16.5.2 Subfield computations.401
16.5.3 Even embedding degree.402
16.5.4 Example.403
IV Point Counting
17 Point Counting on Elliptic and Hyperelliptic Curves.407
17.1 Elementary methods.407
17.1.1 Enumeration.407
17.1.2 Subfield curves.409
17.1.3 Square root algorithms.410
17.1.4 Cartier-Manin operator.411
17.2 Overview of ¿-adic methods.413
17.2.1 Schoof’s algorithm.413
17.2.2 Schoof-Elkies-Atkin’s algorithm.414
17.2.3 Modular polynomials.416
17.2.4 Computing separable isogenies in finite fields of large characteristic . 419
17.2.5 Complete SEA algorithm.421
17.3 Overview of p-adic methods.422
17.3.1 Satoh’s algorithm.423
17.3.2 Arithmetic-Geometric-Mean algorithm.434
17.3.3 Kedlaya’s algorithm.449
xv/// Table of Contents
18 Complex Multiplication.455
18.1 CM for elliptic curves.456
18.1.1 Summary of background.456
18.1.2 Outline of the algorithm.456
18.1.3 Computation of class polynomials.457
18.1.4 Computation of norms.458
18.1.5 The algorithm.459
18.1.6 Experimental results.459
18.2 CM for curves of genus 2.460
18.2.1 Summary of background.462
18.2.2 Outline of the algorithm.462
18.2.3 CM-types and period matrices.463
18.2.4 Computation of the class polynomials.465
18.2.5 Finding a curve.467
18.2.6 The algorithm.469
18.3 CM for larger genera.470
18.3.1 Strategy and difficulties in the general case.470
18.3.2 Hyperelliptic curves with automorphisms.471
18.3.3 The case of genus 3.472
V Computation of Discrete Logarithms
19 Generic Algorithms for Computing Discrete Logarithms.477
19.1 Introduction.478
19.2 Brute force.479
19.3 Chinese remaindering.479
19.4 Baby-step giant-step.480
19.4.1 Adaptive giant-step width.481
19.4.2 Search in intervals and parallelization.482
19.4.3 Congruence classes.483
19.5 Pollard’s rho method.483
19.5.1 Cycle detection.484
19.5.2 Application to DL.488
19.5.3 More on random walks.489
19.5.4 Parallelization.489
19.5.5 Automorphisms of the group.490
19.6 Pollard’s kangaroo method.491
19.6.1 The lambda method.492
19.6.2 Parallelization.493
19.6.3 Automorphisms of the group.494
20 Index Calculus.495
20.1 Introduction.495
20.2 Arithmetical formations.496
20.2.1 Examples of formations.497
20.3 The algorithm.498
20.3.1 On the relation search.499
20.3.2 Parallelization of the relation search.500
Table of Contents_xix
20.3.3 On the linear algebra.500
20.3.4 Filtering.503
20.3.5 Automorphisms of the group.505
20.4 An important example: finite fields.506
20.5 Large primes.507
20.5.1 One large prime.507
20.5.2 Two large primes.508
20.5.3 More large primes.509
21 Index Calculus for Hyperelliptic Curves.511
21.1 General algorithm.511
21.1.1 Hyperelliptic involution.512
21.1.2 Adleman-DeMarrais-Huang.512
21.1.3 Enge-Gaudry.516
21.2 Curves of small genus.516
21.2.1 Gaudry’s algorithm.517
21.2.2 Refined factor base.517
21.2.3 Harvesting.518
21.3 Large prime methods.519
21.3.1 Single large prime.520
21.3.2 Double large primes.521
22 Transfer of Discrete Logarithms.529
22.1 Transfer of discrete logarithms to Fg-vector spaces.529
22.2 Transfer of discrete logarithms by pairings.530
22.3 Transfer of discrete logarithms by Weil descent.530
22.3.1 Summary of background.531
22.3.2 The GHS algorithm.531
22.3.3 Odd characteristic.536
22.3.4 Transfer via covers.538
22.3.5 Index calculus method via hyperplane sections.541
VI Applications
23 Algebraic Realizations of DL Systems.547
23.1 Candidates for secure DL systems.547
23.1.1 Groups with numeration and the DLP.548
23.1.2 Ideal class groups and divisor class groups.548
23.1.3 Examples: elliptic and hyperelliptic curves.551
23.1.4 Conclusion.553
23.2 Security of systems based on Pic£.554
23.2.1 Security under index calculus attacks.554
23.2.2 Transfers by Galois theory.555
23.3 Efficient systems.557
23.3.1 Choice of the finite field.558
23.3.2 Choice of genus and curve equation.560
23.3.3 Special choices of curves and scalar multiplication.563
23.4 Construction of systems.564
XX
Table of Contents
23.4.1 Heuristics of class group orders.564
23.4.2 Finding groups of suitable size.565
23.5 Protocols.569
23.5.1 System parameters.569
23.5.2 Protocols on Pic^.570
23.6 Summary.571
24 Pairing-Based Cryptography.573
24.1 Protocols.573
24.1.1 Multiparty key exchange.574
24.1.2 Identity-based cryptography.576
24.1.3 Short signatures.578
24.2 Realization.579
24.2.1 Supersingular elliptic curves.580
24.2.2 Supersingular hyperelliptic curves.584
24.2.3 Ordinary curves with small embedding degree.586
24.2.4 Performance.589
24.2.5 Hash functions on the Jacobian.590
25 Compositeness and Primality Testing - Factoring.591
25.1 Compositeness tests.592
25.1.1 Trial division.592
25.1.2 Fermat tests.593
25.1.3 Rabin-Millertest.594
25.1.4 Lucas pseudoprime tests.595
25.1.5 BPSW tests.596
25.2 Primality tests.596
25.2.1 Introduction.596
25.2.2 Atkin-Morain ECPP test.597
25.2.3 APRCL Jacobi sum test.599
25.2.4 Theoretical considerations and the AKS test.600
25.3 Factoring.601
25.3.1 Pollard’s rho method.601
25.3.2 Pollard’s p - 1 method.603
25.3.3 Factoring with elliptic curves.604
25.3.4 Fermat-Morrison-Brillhart approach.607
VII Realization of Discrete Logarithm Systems
26 Fast Arithmetic in Hardware.617
26.1 Design of cryptographic coprocessors.618
26.1.1 Design criterions.618
26.2 Complement representations of signed numbers.620
26.3 The operation XY + Z.622
26.3.1 Multiplication using left shifts.623
26.3.2 Multiplication using right shifts.624
26.4 Reducing the number of partial products.625
26.4.1 Booth or signed digit encoding.625
Table of Contents_ xxi
26.4.2 Advanced recoding techniques.626
26.5 Accumulation of partial products.627
26.5.1 Full adders.627
26.5.2 Faster carry propagation.628
26.5.3 Analysis of carry propagation.631
26.5.4 Multi-operand operations.633
26.6 Modular reduction in hardware.638
26.7 Finite fields of characteristic 2.641
26.7.1 Polynomial basis.642
26.7.2 Normal basis.643
26.8 Unified multipliers.644
26.9 Modular inversion in hardware.645
27 Smart Cards.647
27.1 History.647
27.2 Smart card properties.648
27.2.1 Physical properties.648
27.2.2 Electrical properties.650
27.2.3 Memory.651
27.2.4 Environment and software.656
27.3 Smart card interfaces.659
27.3.1 Transmission protocols.659
27.3.2 Physical interfaces.663
27.4 Types of smart cards.664
27.4.1 Memory only cards (synchronous cards).664
27.4.2 Microprocessor cards (asynchronous cards).665
28 Practical Attacks on Smart Cards.669
28.1 Introduction.669
28.2 Invasive attacks.670
28.2.1 Gaining access to the chip.670
28.2.2 Reconstitution of the layers.670
28.2.3 Reading the memories.671
28.2.4 Probing.671
28.2.5 FIB and test engineers scheme flaws.672
28.3 Non-invasive attacks.673
28.3.1 Timing attacks.673
28.3.2 Power consumption analysis.675
28.3.3 Electromagnetic radiation attacks.682
28.3.4 Differential fault analysis (DFA) and fault injection attacks . . . 683
29 Mathematical Countermeasures against Side-Channel Attacks 687
29.1 Countermeasures against simple SCA.688
29.1.1 Dummy arithmetic instructions.689
29.1.2 Unified addition formulas.694
29.1.3 Montgomery arithmetic.696
29.2 Countermeasures against differential SCA.697
29.2.1 Implementation of DSCA.698
29.2.2 Scalar randomization.699
29.2.3 Randomization of group elements.700
XXII
Table of Contents
29.2.4 Randomization of the curve equation.700
29.3 Countermeasures against Goubin type attacks.703
29.4 Countermeasures against higher order differential SCA.704
29.5 Countermeasures against timing attacks.705
29.6 Countermeasures against fault attacks.705
29.6.1 Countermeasures against simple fault analysis.706
29.6.2 Countermeasures against differential fault analysis.706
29.6.3 Conclusion on fault induction.708
29.7 Countermeasures for special curves.709
29.7.1 Countermeasures against SSCA on Koblitz curves . 709
29.7.2 Countermeasures against DSCA on Koblitz curves.711
29.7.3 Countermeasures for GLV curves.713
30 Random Numbers - Generation and Testing.715
30.1 Definition of a random sequence.715
30.2 Random number generators.717
30.2.1 History.717
30.2.2 Properties of random number generators.718
30.2.3 Types of random number generators.718
30.2.4 Popular random number generators.720
30.3 Testing of random number generators.722
30.4 Testing a device.722
30.5 Statistical (empirical) tests.723
30.6 Some examples of statistical models on S".725
30.7 Hypothesis testings and random sequences.726
30.8 Empirical test examples for binary sequences.727
30.8.1 Random walk.727
30.8.2 Runs.728
30.8.3 Autocorrelation.728
30.9 Pseudorandom number generators.729
30.9.1 Relevant measures.730
30.9.2 Pseudorandom number generators from curves.732
30.9.3 Other applications.735
References.737
Notation Index.777
General Index.785 |
| adam_txt |
Table of Contents
List of Algorithms.xxiii
Preface.xxix
1 Introduction to Public-Key Cryptography.1
1.1 Cryptography.2
1.2 Complexity.2
1.3 Public-key cryptography.5
1.4 Factorization and primality.6
1.4.1 Primality.6
1.4.2 Complexity of factoring.6
1.4.3 RSA.7
1.5 Discrete logarithm systems.8
1.5.1 Generic discrete logarithm systems.8
1.5.2 Discrete logarithm systems with bilinear structure . 9
1.6 Protocols.9
1.6.1 Diffie-Hellman key exchange.10
1.6.2 Asymmetric Diffie-Hellman and EIGamal encryption . . . .10
1.6.3 Signature scheme of EIGamal-type.12
1.6.4 Tripartite key exchange.13
1.7 Other problems.14
I Mathematical Background
2 Algebraic Background.19
2.1 Elementary algebraic structures.19
2.1.1 Groups.19
2.1.2 Rings.21
2.1.3 Fields.23
2.1.4 Vector spaces.24
2.2 Introduction to number theory.24
2.2.1 Extension of fields.25
2.2.2 Algebraic closure.27
2.2.3 Galois theory.27
2.2.4 Number fields.29
2.3 Finite fields.31
2.3.1 First properties.31
2.3.2 Algebraic extensions of a finite field.32
2.3.3 Finite field representations.33
2.3.4 Finite field characters.35
XI
XII
Table of Contents
3 Background on p-adic Numbers.39
3.1 Definition of Qp and first properties.39
3.2 Complete discrete valuation rings and fields.41
3.2.1 First properties.41
3.2.2 Lifting a solution of a polynomial equation.42
3.3 The field Qp and its extensions.43
3.3.1 Unramified extensions.43
3.3.2 Totally ramified extensions.43
3.3.3 Multiplicative system of representatives.44
3.3.4 Witt vectors.44
4 Background on Curves and Jacobians.45
4.1 Algebraic varieties.45
4.1.1 Affine and projective varieties.46
4.2 Function fields.51
4.2.1 Morphisms of affine varieties.52
4.2.2 Rational maps of affine varieties.53
4.2.3 Regular functions.54
4.2.4 Generalization to projective varieties.55
4.3 Abelian varieties.55
4.3.1 Algebraic groups.55
4.3.2 Birational group laws.56
4.3.3 Homomorphisms of abelian varieties.57
4.3.4 Isomorphisms and isogenies.58
4.3.5 Points of finite order and Tate modules.60
4.3.6 Background on ¿-adic representations.61
4.3.7 Complex multiplication.63
4.4 Arithmetic of curves.64
4.4.1 Local rings and smoothness.64
4.4.2 Genus and Riemann-Roch theorem.66
4.4.3 Divisor class group.76
4.4.4 The Jacobian variety of curves.77
4.4.5 Jacobian variety of elliptic curves and group law.79
4.4.6 Ideal class group.81
4.4.7 Class groups of hyperelliptic curves.83
5 Varieties over Special Fields.87
5.1 Varieties over the field of complex numbers.87
5.1.1 Analytic varieties.87
5.1.2 Curves over C.89
5.1.3 Complex tori and abelian varieties.92
5.1.4 Isogenies of abelian varieties over C.94
5.1.5 Elliptic curves over C.95
5.1.6 Hyperelliptic curves over C.100
5.2 Varieties over finite fields.108
5.2.1 The Frobenius morphism.109
5.2.2 The characteristic polynomial of the Frobenius endomorphism . .109
5.2.3 The theorem of Hasse-Weil for Jacobians.110
5.2.4 Tate’s isogeny theorem.112
Table of Contents
xiii
6 Background on Pairings.115
6.1 General duality results.115
6.2 The Tate pairing.116
6.3 Pairings over local fields. . . 117
6.3.1 The local Tate pairing.118
6.3.2 The Lichtenbaum pairing on Jacobian varieties.119
6.4 An explicit pairing.122
6.4.1 The Tate-Lichtenbaum pairing.122
6.4.2 Size of the embedding degree.123
7 Background on Weil Descent.125
7.1 Affine Weil descent.125
7.2 The projective Weil descent.127
7.3 Descent by Galois theory.128
7.4 Zariski closed subsets inside of the Weil descent.129
7.4.1 Hyperplane sections.129
7.4.2 Trace zero varieties.130
7.4.3 Covers of curves.131
7.4.4 The GHS approach.131
8 Cohomological Background on Point Counting.133
8.1 General principle.133
8.1.1 Zeta function and the Weil conjectures.134
8.1.2 Cohomology and Lefschetz fixed point formula.135
8.2 Overview of T-adic methods.137
8.3 Overview of p-adic methods.138
8.3.1 Serre-Tate canonical lift.138
8.3.2 Monsky-Washnitzer cohomology.139
II Elementary Arithmetic
9 Exponentiation.145
9.1 Generic methods.146
9.1.1 Binary methods.146
9.1.2 Left-to-right 2fe-ary algorithm.148
9.1.3 Sliding window method.149
9.1.4 Signed-digit recoding.150
9.1.5 Multi-exponentiation.154
9.2 Fixed exponent.157
9.2.1 Introduction to addition chains.157
9.2.2 Short addition chains search.160
9.2.3 Exponentiation using addition chains.163
9.3 Fixed base point.164
9.3.1 Yao’s method.165
9.3.2 Euclidean method.166
9.3.3 Fixed-base comb method.166
x/V Table of Contents
10 Integer Arithmetic.169
10.1 Multiprecision integers.170
10.1.1 Introduction.170
10.1.2 Internal representation.171
10.1.3 Elementary operations.172
10.2 Addition and subtraction.172
10.3 Multiplication.174
10.3.1 Schoolbook multiplication.174
10.3.2 Karatsuba multiplication.176
10.3.3 Squaring.177
10.4 Modular reduction.178
10.4.1 Barrett method.178
10.4.2 Montgomery reduction.180
10.4.3 Special moduli.182
10.4.4 Reduction modulo several primes.184
10.5 Division.184
10.5.1 Schoolbook division.185
10.5.2 Recursive division.187
10.5.3 Exact division.189
10.6 Greatest common divisor.190
10.6.1 Euclid extended gcd.191
10.6.2 Lehmer extended gcd.192
10.6.3 Binary extended gcd.194
10.6.4 Chinese remainder theorem.196
10.7 Square root.197
10.7.1 Integer square root.197
10.7.2 Perfect square detection.198
11 Finite Field Arithmetic.201
11.1 Prime fields of odd characteristic.201
11.1.1 Representations and reductions.202
11.1.2 Multiplication.202
11.1.3 Inversion and division.205
11.1.4 Exponentiation.209
11.1.5 Squares and square roots.210
11.2 Finite fields of characteristic 2.213
11.2.1 Representation.213
11.2.2 Multiplication.218
11.2.3 Squaring.221
11.2.4 Inversion and division.222
11.2.5 Exponentiation.225
11.2.6 Square roots and quadratic equations.228
11.3 Optimal extension fields.229
11.3.1 Introduction.229
11.3.2 Multiplication.231
11.3.3 Exponentiation.231
11.3.4 Inversion.233
11.3.5 Squares and square roots.234
11.3.6 Specific improvements for degrees 3 and 5.235
Table of Contents
xv
12 Arithmetic of p-adic Numbers.239
12.1 Representation.239
12.1.1 Introduction.239
12.1.2 Computing the Teichmuller modulus.240
12.2 Modular arithmetic.244
12.2.1 Modular multiplication.244
12.2.2 Fast division with remainder.244
12.3 Newton lifting.246
12.3.1 Inverse.247
12.3.2 Inverse square root.248
12.3.3 Square root.249
12.4 Hensel lifting.249
12.5 Frobenius substitution.250
12.5.1 Sparse modulus.251
12.5.2 Teichmuller modulus.252
12.5.3 Gaussian normal basis.252
12.6 Artin-Schreier equations.252
12.6.1 Lercier-Lubicz algorithm.253
12.6.2 Harley’s algorithm.254
12.7 Generalized Newton lifting.256
12.8 Applications.257
12.8.1 Teichmuller lift.257
12.8.2 Logarithm.258
12.8.3 Exponential.259
12.8.4 Trace.260
12.8.5 Norm.261
III Arithmetic of Curves
13 Arithmetic of Elliptic Curves.267
13.1 Summary of background on elliptic curves.268
13.1.1 First properties and group law.268
13.1.2 Scalar multiplication.271
13.1.3 Rational points.272
13.1.4 Torsion points.273
13.1.5 Isomorphisms.273
13.1.6 Isogenies.277
13.1.7 Endomorphisms.277
13.1.8 Cardinality.278
13.2 Arithmetic of elliptic curves defined over Fp.280
13.2.1 Choice of the coordinates.280
13.2.2 Mixed coordinates.283
13.2.3 Montgomery scalar multiplication.285
13.2.4 Parallel implementations.288
13.2.5 Compression of points.288
13.3 Arithmetic of elliptic curves defined over F2 (.289
13.3.1 Choice of the coordinates.291
13.3.2 Faster doublings in affine coordinates.295
XVI
Table of Contents
13.3.3 Mixed coordinates.296
13.3.4 Montgomery scalar multiplication.298
13.3.5 Point halving and applications.299
13.3.6 Parallel implementation.302
13.3.7 Compression of points.302
14 Arithmetic of HyperelHptic Curves.303
14.1 Summary of background on hyperelliptic curves.304
14.1.1 Group law for hyperelliptic curves.304
14.1.2 Divisor class group and ideal class group.306
14.1.3 Isomorphisms and isogenies.308
14.1.4 Torsion elements.309
14.1.5 Endomorphisms.310
14.1.6 Cardinality . . .310
14.2 Compression techniques.311
14.2.1 Compression in odd characteristic.311
14.2.2 Compression in even characteristic.313
14.3 Arithmetic on genus 2 curves over arbitrary characteristic . . . . .313
14.3.1 Different cases.314
14.3.2 Addition and doubling in affine coordinates.316
14.4 Arithmetic on genus 2 curves in odd characteristic.320
14.4.1 Projective coordinates.321
14.4.2 New coordinates in odd characteristic.323
14.4.3 Different sets of coordinates in odd characteristic.325
14.4.4 Montgomery arithmetic for genus 2 curves in odd characteristic . . 328
14.5 Arithmetic on genus 2 curves in even characteristic.334
14.5.1 Classification of genus 2 curves in even characteristic. . . . 334
14.5.2 Explicit formulas in even characteristic in affine coordinates . . . 336
14.5.3 Inversion-free systems for even characteristic when h i ^ 0. . . 341
14.5.4 Projective coordinates.341
14.5.5 Inversion-free systems for even characteristic when /i2 = 0. . . 345
14.6 Arithmetic on genus 3 curves.348
14.6.1 Addition in most common case.348
14.6.2 Doubling in most common case.349
14.6.3 Doubling on genus 3 curves for even characteristic when h(x) = 1 351
14.7 Other curves and comparison.352
15 Arithmetic of Special Curves.355
15.1 Koblitz curves.355
15.1.1 Elliptic binary Koblitz curves.356
15.1.2 Generalized Koblitz curves.367
15.1.3 Alternative setup.375
15.2 Scalar multiplication using endomorphisms.376
15.2.1 GLV method. 377
15.2.2 Generalizations.380
15.2.3 Combination of GLV and Koblitz curve strategies.381
15.2.4 Curves with endomorphisms for identity-based parameters. . . 382
15.3 Trace zero varieties.383
15.3.1 Background on trace zero varieties.384
15.3.2 Arithmetic in G.385
Table of Contents_xvii
16 Implementation of Pairings.389
16.1 The basic algorithm.389
16.1.1 The setting.390
16.1.2 Preparation.391
16.1.3 The pairing computation algorithm.391
16.1.4 The case of nontrivial embedding degree A;.393
16.1.5 Comparison with the Weil pairing.395
16.2 Elliptic curves.396
16.2.1 The basic step.396
16.2.2 The representation.396
16.2.3 The pairing algorithm.397
16.2.4 Example.397
16.3 Hyperelliptic curves of genus 2.398
16.3.1 The basic step.399
16.3.2 Representation for k 2.399
16.4 Improving the pairing algorithm.400
16.4.1 Elimination of divisions.400
16.4.2 Choice of the representation.400
16.4.3 Precomputations.400
16.5 Specific improvements for elliptic curves.400
16.5.1 Systems of coordinates.401
16.5.2 Subfield computations.401
16.5.3 Even embedding degree.402
16.5.4 Example.403
IV Point Counting
17 Point Counting on Elliptic and Hyperelliptic Curves.407
17.1 Elementary methods.407
17.1.1 Enumeration.407
17.1.2 Subfield curves.409
17.1.3 Square root algorithms.410
17.1.4 Cartier-Manin operator.411
17.2 Overview of ¿-adic methods.413
17.2.1 Schoof’s algorithm.413
17.2.2 Schoof-Elkies-Atkin’s algorithm.414
17.2.3 Modular polynomials.416
17.2.4 Computing separable isogenies in finite fields of large characteristic . 419
17.2.5 Complete SEA algorithm.421
17.3 Overview of p-adic methods.422
17.3.1 Satoh’s algorithm.423
17.3.2 Arithmetic-Geometric-Mean algorithm.434
17.3.3 Kedlaya’s algorithm.449
xv/// Table of Contents
18 Complex Multiplication.455
18.1 CM for elliptic curves.456
18.1.1 Summary of background.456
18.1.2 Outline of the algorithm.456
18.1.3 Computation of class polynomials.457
18.1.4 Computation of norms.458
18.1.5 The algorithm.459
18.1.6 Experimental results.459
18.2 CM for curves of genus 2.460
18.2.1 Summary of background.462
18.2.2 Outline of the algorithm.462
18.2.3 CM-types and period matrices.463
18.2.4 Computation of the class polynomials.465
18.2.5 Finding a curve.467
18.2.6 The algorithm.469
18.3 CM for larger genera.470
18.3.1 Strategy and difficulties in the general case.470
18.3.2 Hyperelliptic curves with automorphisms.471
18.3.3 The case of genus 3.472
V Computation of Discrete Logarithms
19 Generic Algorithms for Computing Discrete Logarithms.477
19.1 Introduction.478
19.2 Brute force.479
19.3 Chinese remaindering.479
19.4 Baby-step giant-step.480
19.4.1 Adaptive giant-step width.481
19.4.2 Search in intervals and parallelization.482
19.4.3 Congruence classes.483
19.5 Pollard’s rho method.483
19.5.1 Cycle detection.484
19.5.2 Application to DL.488
19.5.3 More on random walks.489
19.5.4 Parallelization.489
19.5.5 Automorphisms of the group.490
19.6 Pollard’s kangaroo method.491
19.6.1 The lambda method.492
19.6.2 Parallelization.493
19.6.3 Automorphisms of the group.494
20 Index Calculus.495
20.1 Introduction.495
20.2 Arithmetical formations.496
20.2.1 Examples of formations.497
20.3 The algorithm.498
20.3.1 On the relation search.499
20.3.2 Parallelization of the relation search.500
Table of Contents_xix
20.3.3 On the linear algebra.500
20.3.4 Filtering.503
20.3.5 Automorphisms of the group.505
20.4 An important example: finite fields.506
20.5 Large primes.507
20.5.1 One large prime.507
20.5.2 Two large primes.508
20.5.3 More large primes.509
21 Index Calculus for Hyperelliptic Curves.511
21.1 General algorithm.511
21.1.1 Hyperelliptic involution.512
21.1.2 Adleman-DeMarrais-Huang.512
21.1.3 Enge-Gaudry.516
21.2 Curves of small genus.516
21.2.1 Gaudry’s algorithm.517
21.2.2 Refined factor base.517
21.2.3 Harvesting.518
21.3 Large prime methods.519
21.3.1 Single large prime.520
21.3.2 Double large primes.521
22 Transfer of Discrete Logarithms.529
22.1 Transfer of discrete logarithms to Fg-vector spaces.529
22.2 Transfer of discrete logarithms by pairings.530
22.3 Transfer of discrete logarithms by Weil descent.530
22.3.1 Summary of background.531
22.3.2 The GHS algorithm.531
22.3.3 Odd characteristic.536
22.3.4 Transfer via covers.538
22.3.5 Index calculus method via hyperplane sections.541
VI Applications
23 Algebraic Realizations of DL Systems.547
23.1 Candidates for secure DL systems.547
23.1.1 Groups with numeration and the DLP.548
23.1.2 Ideal class groups and divisor class groups.548
23.1.3 Examples: elliptic and hyperelliptic curves.551
23.1.4 Conclusion.553
23.2 Security of systems based on Pic£.554
23.2.1 Security under index calculus attacks.554
23.2.2 Transfers by Galois theory.555
23.3 Efficient systems.557
23.3.1 Choice of the finite field.558
23.3.2 Choice of genus and curve equation.560
23.3.3 Special choices of curves and scalar multiplication.563
23.4 Construction of systems.564
XX
Table of Contents
23.4.1 Heuristics of class group orders.564
23.4.2 Finding groups of suitable size.565
23.5 Protocols.569
23.5.1 System parameters.569
23.5.2 Protocols on Pic^.570
23.6 Summary.571
24 Pairing-Based Cryptography.573
24.1 Protocols.573
24.1.1 Multiparty key exchange.574
24.1.2 Identity-based cryptography.576
24.1.3 Short signatures.578
24.2 Realization.579
24.2.1 Supersingular elliptic curves.580
24.2.2 Supersingular hyperelliptic curves.584
24.2.3 Ordinary curves with small embedding degree.586
24.2.4 Performance.589
24.2.5 Hash functions on the Jacobian.590
25 Compositeness and Primality Testing - Factoring.591
25.1 Compositeness tests.592
25.1.1 Trial division.592
25.1.2 Fermat tests.593
25.1.3 Rabin-Millertest.594
25.1.4 Lucas pseudoprime tests.595
25.1.5 BPSW tests.596
25.2 Primality tests.596
25.2.1 Introduction.596
25.2.2 Atkin-Morain ECPP test.597
25.2.3 APRCL Jacobi sum test.599
25.2.4 Theoretical considerations and the AKS test.600
25.3 Factoring.601
25.3.1 Pollard’s rho method.601
25.3.2 Pollard’s p - 1 method.603
25.3.3 Factoring with elliptic curves.604
25.3.4 Fermat-Morrison-Brillhart approach.607
VII Realization of Discrete Logarithm Systems
26 Fast Arithmetic in Hardware.617
26.1 Design of cryptographic coprocessors.618
26.1.1 Design criterions.618
26.2 Complement representations of signed numbers.620
26.3 The operation XY + Z.622
26.3.1 Multiplication using left shifts.623
26.3.2 Multiplication using right shifts.624
26.4 Reducing the number of partial products.625
26.4.1 Booth or signed digit encoding.625
Table of Contents_ xxi
26.4.2 Advanced recoding techniques.626
26.5 Accumulation of partial products.627
26.5.1 Full adders.627
26.5.2 Faster carry propagation.628
26.5.3 Analysis of carry propagation.631
26.5.4 Multi-operand operations.633
26.6 Modular reduction in hardware.638
26.7 Finite fields of characteristic 2.641
26.7.1 Polynomial basis.642
26.7.2 Normal basis.643
26.8 Unified multipliers.644
26.9 Modular inversion in hardware.645
27 Smart Cards.647
27.1 History.647
27.2 Smart card properties.648
27.2.1 Physical properties.648
27.2.2 Electrical properties.650
27.2.3 Memory.651
27.2.4 Environment and software.656
27.3 Smart card interfaces.659
27.3.1 Transmission protocols.659
27.3.2 Physical interfaces.663
27.4 Types of smart cards.664
27.4.1 Memory only cards (synchronous cards).664
27.4.2 Microprocessor cards (asynchronous cards).665
28 Practical Attacks on Smart Cards.669
28.1 Introduction.669
28.2 Invasive attacks.670
28.2.1 Gaining access to the chip.670
28.2.2 Reconstitution of the layers.670
28.2.3 Reading the memories.671
28.2.4 Probing.671
28.2.5 FIB and test engineers scheme flaws.672
28.3 Non-invasive attacks.673
28.3.1 Timing attacks.673
28.3.2 Power consumption analysis.675
28.3.3 Electromagnetic radiation attacks.682
28.3.4 Differential fault analysis (DFA) and fault injection attacks . . . 683
29 Mathematical Countermeasures against Side-Channel Attacks 687
29.1 Countermeasures against simple SCA.688
29.1.1 Dummy arithmetic instructions.689
29.1.2 Unified addition formulas.694
29.1.3 Montgomery arithmetic.696
29.2 Countermeasures against differential SCA.697
29.2.1 Implementation of DSCA.698
29.2.2 Scalar randomization.699
29.2.3 Randomization of group elements.700
XXII
Table of Contents
29.2.4 Randomization of the curve equation.700
29.3 Countermeasures against Goubin type attacks.703
29.4 Countermeasures against higher order differential SCA.704
29.5 Countermeasures against timing attacks.705
29.6 Countermeasures against fault attacks.705
29.6.1 Countermeasures against simple fault analysis.706
29.6.2 Countermeasures against differential fault analysis.706
29.6.3 Conclusion on fault induction.708
29.7 Countermeasures for special curves.709
29.7.1 Countermeasures against SSCA on Koblitz curves . 709
29.7.2 Countermeasures against DSCA on Koblitz curves.711
29.7.3 Countermeasures for GLV curves.713
30 Random Numbers - Generation and Testing.715
30.1 Definition of a random sequence.715
30.2 Random number generators.717
30.2.1 History.717
30.2.2 Properties of random number generators.718
30.2.3 Types of random number generators.718
30.2.4 Popular random number generators.720
30.3 Testing of random number generators.722
30.4 Testing a device.722
30.5 Statistical (empirical) tests.723
30.6 Some examples of statistical models on S".725
30.7 Hypothesis testings and random sequences.726
30.8 Empirical test examples for binary sequences.727
30.8.1 Random walk.727
30.8.2 Runs.728
30.8.3 Autocorrelation.728
30.9 Pseudorandom number generators.729
30.9.1 Relevant measures.730
30.9.2 Pseudorandom number generators from curves.732
30.9.3 Other applications.735
References.737
Notation Index.777
General Index.785 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author_GND | (DE-588)1018621717 |
| building | Verbundindex |
| bvnumber | BV021273577 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 170 ST 276 |
| classification_tum | MAT 145f DAT 465f |
| ctrlnum | (OCoLC)58546549 (DE-599)BVBBV021273577 |
| dewey-full | 516.3/52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
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| dewey-tens | 510 - Mathematics |
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| discipline_str_mv | Informatik Mathematik |
| format | Book |
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| id | DE-604.BV021273577 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T13:45:16Z |
| indexdate | 2024-11-25T17:26:05Z |
| institution | BVB |
| isbn | 9781584885184 1584885181 |
| language | English |
| lccn | 2005041841 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014594665 |
| oclc_num | 58546549 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-20 DE-703 DE-19 DE-BY-UBM DE-634 DE-83 DE-739 DE-706 |
| owner_facet | DE-91G DE-BY-TUM DE-20 DE-703 DE-19 DE-BY-UBM DE-634 DE-83 DE-739 DE-706 |
| physical | XXXIV, 808 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Chapman & Hall/CRC |
| record_format | marc |
| series2 | Discrete mathematics and its applications |
| spellingShingle | Handbook of elliptic and hyperelliptic curve cryptography Automates mathématiques, Théorie des - Guides, manuels, etc Courbes elliptiques - Guides, manuels, etc Cryptographie - Mathématiques - Guides, manuels, etc Mathematik Curves, Elliptic Handbooks, manuals, etc Cryptography Mathematics Handbooks, manuals, etc Machine theory Handbooks, manuals, etc Elliptische Kurve (DE-588)4014487-2 gnd Kryptologie (DE-588)4033329-2 gnd |
| subject_GND | (DE-588)4014487-2 (DE-588)4033329-2 |
| title | Handbook of elliptic and hyperelliptic curve cryptography |
| title_auth | Handbook of elliptic and hyperelliptic curve cryptography |
| title_exact_search | Handbook of elliptic and hyperelliptic curve cryptography |
| title_exact_search_txtP | Handbook of elliptic and hyperelliptic curve cryptography |
| title_full | Handbook of elliptic and hyperelliptic curve cryptography Henri Cohen ...[u.a.] |
| title_fullStr | Handbook of elliptic and hyperelliptic curve cryptography Henri Cohen ...[u.a.] |
| title_full_unstemmed | Handbook of elliptic and hyperelliptic curve cryptography Henri Cohen ...[u.a.] |
| title_short | Handbook of elliptic and hyperelliptic curve cryptography |
| title_sort | handbook of elliptic and hyperelliptic curve cryptography |
| topic | Automates mathématiques, Théorie des - Guides, manuels, etc Courbes elliptiques - Guides, manuels, etc Cryptographie - Mathématiques - Guides, manuels, etc Mathematik Curves, Elliptic Handbooks, manuals, etc Cryptography Mathematics Handbooks, manuals, etc Machine theory Handbooks, manuals, etc Elliptische Kurve (DE-588)4014487-2 gnd Kryptologie (DE-588)4033329-2 gnd |
| topic_facet | Automates mathématiques, Théorie des - Guides, manuels, etc Courbes elliptiques - Guides, manuels, etc Cryptographie - Mathématiques - Guides, manuels, etc Mathematik Curves, Elliptic Handbooks, manuals, etc Cryptography Mathematics Handbooks, manuals, etc Machine theory Handbooks, manuals, etc Elliptische Kurve Kryptologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014594665&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT cohenhenri handbookofellipticandhyperellipticcurvecryptography |