Group-based cryptography:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2008
|
| Series: | Advanced Courses in Mathematics - CRM Barcelona
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 183 S. graph. Darst. |
| ISBN: | 9783764388263 3764388269 |
Staff View
MARC
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| 100 | 1 | |a Myasnikov, Alexei |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Group-based cryptography |c Alexei Myasnikov ; Vladimir Shpilrain ; Alexander Ushakov |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2008 | |
| 300 | |a XV, 183 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Advanced Courses in Mathematics - CRM Barcelona | |
| 650 | 4 | |a Algorithms | |
| 650 | 4 | |a Combinatorial group theory | |
| 650 | 4 | |a Cryptography | |
| 650 | 0 | 7 | |a Public-Key-Kryptosystem |0 (DE-588)4209133-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Berechnungskomplexität |0 (DE-588)4134751-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Kombinatorische Gruppentheorie |0 (DE-588)4219556-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Public-Key-Kryptosystem |0 (DE-588)4209133-0 |D s |
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| 689 | 0 | 2 | |a Berechnungskomplexität |0 (DE-588)4134751-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Shpilrain, Vladimir |e Verfasser |4 aut | |
| 700 | 1 | |a Ušakov, Aleksandr A. |d 1946- |e Verfasser |0 (DE-588)118900269 |4 aut | |
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Record in the Search Index
| _version_ | 1804137672365572096 |
|---|---|
| adam_text | Contents
Preface
xi
Introduction
xiii
I Background on Groups, Complexity, and Cryptography
1
1
Background on Public Key Cryptography
3
1.1
From key establishment to encryption
................ 4
1.2
The Diffie-Hellman key establishment
................ 5
1.3
The ElGamal cryptosystem
...................... 6
1.4
Authentication
............................. 7
2
Background on Combinatorial Group Theory
9
2.1
Basic definitions and notation
..................... 9
2.2
Presentations of groups by generators and
relatore
......... 11
2.3
Algorithmic problems of group theory
................ 11
2.3.1
The word problem
....................... 11
2.3.2
The conjugacy problem
.................... 12
2.3.3
The decomposition and factorization problems
....... 12
2.3.4
The membership problem
................... 13
2.3.
δ
The isomorphism problem
................... 14
2.4
Nielsen s and Schreier s methods
................... 14
2.5
Tietze s method
............................. 16
2.6
Normal forms
.............................. 17
3
Background on Computational Complexity
19
3.1
Algorithms
............................... 19
3.1.1
Deterministic Turing machines
................ 19
3.1.2
Non-deterministic Turing machines
.............. 20
3.1.3
Probabilistic Turing machines
................. 21
3.2
Computational problems
........................ 21
3.2.1
Decision and search computational problems
........ 21
Contents
3.2.2
Size functions
.......................... 23
3.2.3
Stratification
..........................
25
3.2.4
Reductions and complete problems
.............. 26
3.2.5
Many-one reductions
...................... 27
3.2.6
Turing reductions
........................ 27
3.3
The worst case complexity
....................... 28
3.3.1
Complexity classes
....................... 28
3.3.2
Class NP
............................ 29
3.3.3
Polynomial-time many-one reductions and class NP
.... 30
3.3.4
NP-complete problems
.................... 31
3.3.5
Deficiency of the worst case complexity
........... 33
II Non-commutative Cryptography
35
4
Canonical Non-commutative Cryptography
37
4.1
Protocols based on the conjugacy search problem
.......... 37
4.2
Protocols based on the decomposition problem
........... 39
4.2.1
Twisted protocol
....................... 40
4.2.2
Hiding one of the subgroups
.................. 41
4.2.3
Using the triple decomposition problem
........... 42
4.3
A protocol based on the factorization search problem
........ 43
4.4
Stickel s key exchange protocol
.................... 43
4.4.1
Linear algebra attack
..................... 45
4.5
The Anshel-Anshel-Goldfeld protocol
................. 47
4.6
Authentication protocols based on the conjugacy problem
..... 49
4.6.1
A Diffie-Hellman-like scheme
................. 49
4.6.2
A Fiat-Shamir-like scheme
................... 50
4.6.3
An authentication scheme based on the twisted conjugacy
problem
............................. 51
4.7
Relations between different problems
................. 52
5
Platform Groups
55
5.1
Braid groups
.............................. 55
5.1.1
A group of braids and its presentation
............ 56
5.1.2
Dehomoy handle free form
.................. 59
5.1.3
Garside normal form
...................... 60
5.2
Thompson s group
........................... 61
5.3
Groups of matrices
........................... 65
5.4
Small cancellation groups
....................... 67
5.4.1
Dehn s algorithm
........................ 67
5.5
Solvable groups
............................. 68
5.5.1
Normal forms in free metabelian groups
........... 68
5.6 Artin
groups
.............................. 71
Contents ix
5.7
Grigorchuk s group
........................... 72
6
Using Decision Problems in Public Key Cryptography
77
6.1
The Shpilrain-Zapata scheme
..................... 78
6.1.1
The protocol
.......................... 78
6.1.2
Pool of group presentations
.................. 81
6.1.3
Tietze transformations: elementary isomorphisms
...... 82
6.1.4
Generating random elements in finitely presented groups
. . 84
6.1.5
Isomorphism attack
...................... 87
6.1.6
Quotient attack
......................... 88
6.2
Public key encryption and encryption emulation attacks
...... 89
III Generic Complexity and Cryptanalysis
95
7
Distributional Problems and the Average Case Complexity
99
7.1
Distributional computational problems
................ 99
7.1.1
Distributions and computational problems
.......... 99
7.1.2
Stratified problems with ensembles of distributions
..... 101
7.1.3
Randomized many-one reductions
.............. 102
7.2
Average case complexity
........................ 103
7.2.1
Polynomial on average functions
............... 103
7.2.2
Average case behavior of functions
.............. 109
7.2.3
Average case complexity of algorithms
............ 109
7.2.4
Average case vs worst case
.................. 110
7.2.5
Average case behavior as a trade-off
.............
Ill
7.2.6
Deficiency of average case complexity
............ 114
8
Generic Case Complexity
117
8.1
Generic Complexity
.......................... 117
8.1.1
Generic sets
........................... 117
8.1.2
Asymptotic density
....................... 118
8.1.3
Convergence rates
....................... 120
8.1.4
Generic complexity of algorithms and algorithmic problems
121
8.1.5
Deficiency of the generic complexity
............. 122
8.2
Generic- versus average case complexity
............... 123
8.2.1
Comparing generic and average case complexities
...... 123
8.2.2
When average polynomial time implies generic polynomial
time
............................... 124
8.2.3
When generically easy implies easy on average
....... 125
x
Contents
9
Generic Complexity of NP-complete Problems
129
9.1
The linear generic time complexity of subset sum problem
..... 129
9.2
A practical algorithm for subset sum problem
............ 131
9.3
3-Satisfiability
.............................. 131
IV Asymptotically Dominant Properties and Cryptanalysis
135
10
Asymptotically Dominant Properties
139
10.1
A brief description
........................... 139
10.2
Random subgroups and generating tuples
.............. 141
10.3
Asymptotic properties of subgroups
.................. 142
10.4
Groups with generic free basis property
............... 143
10.5
Quasi-isometrically embedded subgroups
............... 145
11
Length-Based and Quotient Attacks
149
11.1
Anshel-Anshel-Goldfeld scheme
.................... 149
11.1.1
Description of the Anshel-Anshel-Goldfeld scheme
..... 149
11.1.2
Security assumptions of the AAG scheme
.......... 150
11.2
Length-based attacks
.......................... 152
11.2.1
A general description
...................... 152
11.2.2
LBA
in free groups
....................... 155
11.2.3
LBA
in groups from TBexv
.................. 156
11.3
Computing the geodesic length in a subgroup
............ 157
11.3.1
Related algorithmic problems
................. 158
11.3.2
Geodesic length in braid groups
................ 159
11.4
Quotient attacks
............................ 161
11.4.1
Membership problems in free groups
............. 162
11.4.2
Conjugacy problems in free groups
.............. 164
11.4.3
The MSP and SCSP* problems in groups with good quo¬
tients
.............................. 167
Bibliography
169
Abbreviations and Notation
179
Index
181
|
| adam_txt |
Contents
Preface
xi
Introduction
xiii
I Background on Groups, Complexity, and Cryptography
1
1
Background on Public Key Cryptography
3
1.1
From key establishment to encryption
. 4
1.2
The Diffie-Hellman key establishment
. 5
1.3
The ElGamal cryptosystem
. 6
1.4
Authentication
. 7
2
Background on Combinatorial Group Theory
9
2.1
Basic definitions and notation
. 9
2.2
Presentations of groups by generators and
relatore
. 11
2.3
Algorithmic problems of group theory
. 11
2.3.1
The word problem
. 11
2.3.2
The conjugacy problem
. 12
2.3.3
The decomposition and factorization problems
. 12
2.3.4
The membership problem
. 13
2.3.
δ
The isomorphism problem
. 14
2.4
Nielsen's and Schreier's methods
. 14
2.5
Tietze's method
. 16
2.6
Normal forms
. 17
3
Background on Computational Complexity
19
3.1
Algorithms
. 19
3.1.1
Deterministic Turing machines
. 19
3.1.2
Non-deterministic Turing machines
. 20
3.1.3
Probabilistic Turing machines
. 21
3.2
Computational problems
. 21
3.2.1
Decision and search computational problems
. 21
Contents
3.2.2
Size functions
. 23
3.2.3
Stratification
.
25
3.2.4
Reductions and complete problems
. 26
3.2.5
Many-one reductions
. 27
3.2.6
Turing reductions
. 27
3.3
The worst case complexity
. 28
3.3.1
Complexity classes
. 28
3.3.2
Class NP
. 29
3.3.3
Polynomial-time many-one reductions and class NP
. 30
3.3.4
NP-complete problems
. 31
3.3.5
Deficiency of the worst case complexity
. 33
II Non-commutative Cryptography
35
4
Canonical Non-commutative Cryptography
37
4.1
Protocols based on the conjugacy search problem
. 37
4.2
Protocols based on the decomposition problem
. 39
4.2.1
"Twisted" protocol
. 40
4.2.2
Hiding one of the subgroups
. 41
4.2.3
Using the triple decomposition problem
. 42
4.3
A protocol based on the factorization search problem
. 43
4.4
Stickel's key exchange protocol
. 43
4.4.1
Linear algebra attack
. 45
4.5
The Anshel-Anshel-Goldfeld protocol
. 47
4.6
Authentication protocols based on the conjugacy problem
. 49
4.6.1
A Diffie-Hellman-like scheme
. 49
4.6.2
A Fiat-Shamir-like scheme
. 50
4.6.3
An authentication scheme based on the twisted conjugacy
problem
. 51
4.7
Relations between different problems
. 52
5
Platform Groups
55
5.1
Braid groups
. 55
5.1.1
A group of braids and its presentation
. 56
5.1.2
Dehomoy handle free form
. 59
5.1.3
Garside normal form
. 60
5.2
Thompson's group
. 61
5.3
Groups of matrices
. 65
5.4
Small cancellation groups
. 67
5.4.1
Dehn's algorithm
. 67
5.5
Solvable groups
. 68
5.5.1
Normal forms in free metabelian groups
. 68
5.6 Artin
groups
. 71
Contents ix
5.7
Grigorchuk's group
. 72
6
Using Decision Problems in Public Key Cryptography
77
6.1
The Shpilrain-Zapata scheme
. 78
6.1.1
The protocol
. 78
6.1.2
Pool of group presentations
. 81
6.1.3
Tietze transformations: elementary isomorphisms
. 82
6.1.4
Generating random elements in finitely presented groups
. . 84
6.1.5
Isomorphism attack
. 87
6.1.6
Quotient attack
. 88
6.2
Public key encryption and encryption emulation attacks
. 89
III Generic Complexity and Cryptanalysis
95
7
Distributional Problems and the Average Case Complexity
99
7.1
Distributional computational problems
. 99
7.1.1
Distributions and computational problems
. 99
7.1.2
Stratified problems with ensembles of distributions
. 101
7.1.3
Randomized many-one reductions
. 102
7.2
Average case complexity
. 103
7.2.1
Polynomial on average functions
. 103
7.2.2
Average case behavior of functions
. 109
7.2.3
Average case complexity of algorithms
. 109
7.2.4
Average case vs worst case
. 110
7.2.5
Average case behavior as a trade-off
.
Ill
7.2.6
Deficiency of average case complexity
. 114
8
Generic Case Complexity
117
8.1
Generic Complexity
. 117
8.1.1
Generic sets
. 117
8.1.2
Asymptotic density
. 118
8.1.3
Convergence rates
. 120
8.1.4
Generic complexity of algorithms and algorithmic problems
121
8.1.5
Deficiency of the generic complexity
. 122
8.2
Generic- versus average case complexity
. 123
8.2.1
Comparing generic and average case complexities
. 123
8.2.2
When average polynomial time implies generic polynomial
time
. 124
8.2.3
When generically easy implies easy on average
. 125
x
Contents
9
Generic Complexity of NP-complete Problems
129
9.1
The linear generic time complexity of subset sum problem
. 129
9.2
A practical algorithm for subset sum problem
. 131
9.3
3-Satisfiability
. 131
IV Asymptotically Dominant Properties and Cryptanalysis
135
10
Asymptotically Dominant Properties
139
10.1
A brief description
. 139
10.2
Random subgroups and generating tuples
. 141
10.3
Asymptotic properties of subgroups
. 142
10.4
Groups with generic free basis property
. 143
10.5
Quasi-isometrically embedded subgroups
. 145
11
Length-Based and Quotient Attacks
149
11.1
Anshel-Anshel-Goldfeld scheme
. 149
11.1.1
Description of the Anshel-Anshel-Goldfeld scheme
. 149
11.1.2
Security assumptions of the AAG scheme
. 150
11.2
Length-based attacks
. 152
11.2.1
A general description
. 152
11.2.2
LBA
in free groups
. 155
11.2.3
LBA
in groups from TBexv
. 156
11.3
Computing the geodesic length in a subgroup
. 157
11.3.1
Related algorithmic problems
. 158
11.3.2
Geodesic length in braid groups
. 159
11.4
Quotient attacks
. 161
11.4.1
Membership problems in free groups
. 162
11.4.2
Conjugacy problems in free groups
. 164
11.4.3
The MSP and SCSP* problems in groups with "good" quo¬
tients
. 167
Bibliography
169
Abbreviations and Notation
179
Index
181 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Myasnikov, Alexei Shpilrain, Vladimir Ušakov, Aleksandr A. 1946- |
| author_GND | (DE-588)118900269 |
| author_facet | Myasnikov, Alexei Shpilrain, Vladimir Ušakov, Aleksandr A. 1946- |
| author_role | aut aut aut |
| author_sort | Myasnikov, Alexei |
| author_variant | a m am v s vs a a u aa aau |
| building | Verbundindex |
| bvnumber | BV023329597 |
| callnumber-first | Q - Science |
| callnumber-label | QA182 |
| callnumber-raw | QA182.5 |
| callnumber-search | QA182.5 |
| callnumber-sort | QA 3182.5 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 170 SK 260 ST 134 |
| classification_tum | MAT 050f DAT 517f MAT 209f DAT 465f |
| ctrlnum | (OCoLC)226281300 (DE-599)DNB988023997 |
| dewey-full | 512/.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.2 |
| dewey-search | 512/.2 |
| dewey-sort | 3512 12 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
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| id | DE-604.BV023329597 |
| illustrated | Illustrated |
| index_date | 2024-07-02T20:57:06Z |
| indexdate | 2024-07-09T21:16:01Z |
| institution | BVB |
| isbn | 9783764388263 3764388269 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016513549 |
| oclc_num | 226281300 |
| open_access_boolean | |
| owner | DE-703 DE-91G DE-BY-TUM DE-29T DE-355 DE-BY-UBR DE-83 DE-11 DE-739 DE-20 |
| owner_facet | DE-703 DE-91G DE-BY-TUM DE-29T DE-355 DE-BY-UBR DE-83 DE-11 DE-739 DE-20 |
| physical | XV, 183 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Advanced Courses in Mathematics - CRM Barcelona |
| spelling | Myasnikov, Alexei Verfasser aut Group-based cryptography Alexei Myasnikov ; Vladimir Shpilrain ; Alexander Ushakov Basel [u.a.] Birkhäuser 2008 XV, 183 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Advanced Courses in Mathematics - CRM Barcelona Algorithms Combinatorial group theory Cryptography Public-Key-Kryptosystem (DE-588)4209133-0 gnd rswk-swf Berechnungskomplexität (DE-588)4134751-1 gnd rswk-swf Kombinatorische Gruppentheorie (DE-588)4219556-1 gnd rswk-swf Public-Key-Kryptosystem (DE-588)4209133-0 s Kombinatorische Gruppentheorie (DE-588)4219556-1 s Berechnungskomplexität (DE-588)4134751-1 s DE-604 Shpilrain, Vladimir Verfasser aut Ušakov, Aleksandr A. 1946- Verfasser (DE-588)118900269 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016513549&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Myasnikov, Alexei Shpilrain, Vladimir Ušakov, Aleksandr A. 1946- Group-based cryptography Algorithms Combinatorial group theory Cryptography Public-Key-Kryptosystem (DE-588)4209133-0 gnd Berechnungskomplexität (DE-588)4134751-1 gnd Kombinatorische Gruppentheorie (DE-588)4219556-1 gnd |
| subject_GND | (DE-588)4209133-0 (DE-588)4134751-1 (DE-588)4219556-1 |
| title | Group-based cryptography |
| title_auth | Group-based cryptography |
| title_exact_search | Group-based cryptography |
| title_exact_search_txtP | Group-based cryptography |
| title_full | Group-based cryptography Alexei Myasnikov ; Vladimir Shpilrain ; Alexander Ushakov |
| title_fullStr | Group-based cryptography Alexei Myasnikov ; Vladimir Shpilrain ; Alexander Ushakov |
| title_full_unstemmed | Group-based cryptography Alexei Myasnikov ; Vladimir Shpilrain ; Alexander Ushakov |
| title_short | Group-based cryptography |
| title_sort | group based cryptography |
| topic | Algorithms Combinatorial group theory Cryptography Public-Key-Kryptosystem (DE-588)4209133-0 gnd Berechnungskomplexität (DE-588)4134751-1 gnd Kombinatorische Gruppentheorie (DE-588)4219556-1 gnd |
| topic_facet | Algorithms Combinatorial group theory Cryptography Public-Key-Kryptosystem Berechnungskomplexität Kombinatorische Gruppentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016513549&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT myasnikovalexei groupbasedcryptography AT shpilrainvladimir groupbasedcryptography AT usakovaleksandra groupbasedcryptography |