Modern cryptography and elliptic curves: a beginner's guide
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2017]
|
| Series: | Student mathematical library
83 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite 245-247 |
| Physical Description: | xii, 250 Seiten Diagramme |
| ISBN: | 9781470435820 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface
Introduction
vii
ix
Chapter
1.1.
1.2.
1.3.
1. Three Motivating Problems
Fermat’s Last Theorem
The Congruent Number Problem
Cryptography
Chapter 2. Back to the Beginning
§2.1. The Unit Circle: Real vs. Rational Points
§2.2. Parametrizing the Rational Points on the Unit Circle
§2.3. Finding all Pythagorean Triples
§2.4. Looking for Underlying Structure: Geometry vs.
Algebra
2.5. More about Points on Curves
2.6. Gathering Some Insight about Plane Curves
2.7. Additional Exercises
Chapter 3. Some Elementary Number Theory
§3.1. The Integers
§3.2. Some Basic Properties of the Integers
1
3
5
6
9
10
12
16
27
34
38
43
45
46
47
111
IV
Contents
3.3. Euclid’s Algorithm
3.4. A First Pass at Modular Arithmetic
3.5. Elementary Cryptography: Caesar Cipher
3.6. Affine Ciphers and Linear Congruences
3.7. Systems of Congruences
Chapter
§4.1.
§4.2.
§4.3.
§4.4.
§4.5.
4. A Second View of Modular Arithmetic: Z^ and Un
Groups and Rings
Fractions and the Notion of an Equivalence Relation
Modular Arithmetic
A Few More Comments on the Euler Totient Function
An Application to Factoring
Chapter
5.1.
5.2.
5.3.
5.4.
Chapter
§6.1.
§6.2.
§6.3.
§6.4.
§6.5.
§6.6.
§6.7.
§6.8.
Chapter
§7.1.
§7.2.
5. Public-Key Cryptography and RSA
A Brief Overview of Cryptographic Systems
RSA
Hash Functions
Breaking Cryptosystems and Practical RSA Security
Considerations
6. A Little More Algebra
Towards a Classification of Groups
Cayley Tables
A Couple of Non-abelian Groups
Cyclic Groups and Direct Products
Fundamental Theorem of Finite Abelian Groups
Primitive Roots
Diffie-Hellman Key Exchange
ElGamal Encryption
7. Curves in Affine and Projective Space
Affine and Projective Space
Curves in the Affine and Projective Plane
52
56
63
66
70
73
73
77
79
93
95
101
102
107
114
123
127
128
128
131
134
138
141
143
144
147
147
153
Contents
V
§7.3. Rational Points on Curves
§7.4. The Group Law for Points on an Elliptic Curve
§7.5. A Formula for the Group Law on an Elliptic Curve
§7.6. The Number of Points on an Elliptic Curve
Chapter 8. Applications of Elliptic Curves
§8.1. Elliptic Curves and Factoring
§8.2. Elliptic Curves and Cryptography
§8.3. Remarks on a Post-Quantum Cryptographic World
Appendix A. Deeper Results and Concluding Thoughts
§A.l. The Congruent Number Problem and Tunnell’s
Solution
§A.2. A Digression on Functions of a Complex Variable
§A.3. Return to the Birch and Swinnerton-Dyer Conjecture
§A.4. Elliptic Curves over C
Appendix B. Answers to Selected Exercises
§B.l. Chapter 2
§B.2. Chapter 3
§B.3. Chapter 4
§B.4. Chapter 5
§B.5. Chapter 6
§B.6. Chapter 7
Bibliography
156
159
179
185
189
190
196
198
203
203
209
211
212
219
219
231
233
236
238
241
245
Index
249
|
| any_adam_object | 1 |
| author | Shemanske, Thomas R. 1952- |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 170 SK 180 |
| ctrlnum | (OCoLC)995169919 (DE-599)BVBBV044341683 |
| dewey-full | 516.3/52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/52 |
| dewey-search | 516.3/52 |
| dewey-sort | 3516.3 252 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| spelling | Shemanske, Thomas R. 1952- Verfasser (DE-588)114125591X aut Modern cryptography and elliptic curves a beginner's guide Thomas R. Shemanske Providence, Rhode Island American Mathematical Society [2017] © 2017 xii, 250 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Student mathematical library volume 83 Literaturverzeichnis Seite 245-247 Number theory ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Computer science ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Number theory ... Elementary number theory ... Elementary number theory / msc Algebraic geometry ... Arithmetic problems. Diophantine geometry ... Applications to coding theory and cryptography / msc Number theory ... Finite fields and commutative rings (number-theoretic aspects) ... Algebraic coding theory; cryptography / msc Computer science ... Theory of data ... Data encryption / msc Number theory ... Computational number theory ... Factorization / msc Information and communication, circuits ... Communication, information ... Cryptography / msc Number theory ... Arithmetic algebraic geometry (Diophantine geometry) ... Elliptic curves over global fields / msc Quantum theory ... Axiomatics, foundations, philosophy ... Quantum computation / msc Curves, Elliptic Textbooks Geometry, Algebraic Textbooks Cryptography Textbooks Number theory ... Instructional exposition (textbooks, tutorial papers, etc.) Computer science ... Instructional exposition (textbooks, tutorial papers, etc.) Number theory ... Elementary number theory ... Elementary number theory Algebraic geometry ... Arithmetic problems. Diophantine geometry ... Applications to coding theory and cryptography Number theory ... Finite fields and commutative rings (number-theoretic aspects) ... Algebraic coding theory; cryptography Computer science ... Theory of data ... Data encryption Number theory ... Computational number theory ... Factorization Information and communication, circuits ... Communication, information ... Cryptography Number theory ... Arithmetic algebraic geometry (Diophantine geometry) ... Elliptic curves over global fields Quantum theory ... Axiomatics, foundations, philosophy ... Quantum computation Computersicherheit (DE-588)4274324-2 gnd rswk-swf Public-Key-Kryptosystem (DE-588)4209133-0 gnd rswk-swf Kryptologie (DE-588)4033329-2 gnd rswk-swf Computersicherheit (DE-588)4274324-2 s Kryptologie (DE-588)4033329-2 s Public-Key-Kryptosystem (DE-588)4209133-0 s DE-604 Student mathematical library 83 (DE-604)BV013184751 83 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=029744707&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4274324-2 (DE-588)4209133-0 (DE-588)4033329-2 |
| title | Modern cryptography and elliptic curves a beginner's guide |
| title_auth | Modern cryptography and elliptic curves a beginner's guide |
| title_exact_search | Modern cryptography and elliptic curves a beginner's guide |
| title_full | Modern cryptography and elliptic curves a beginner's guide Thomas R. Shemanske |
| title_fullStr | Modern cryptography and elliptic curves a beginner's guide Thomas R. Shemanske |
| title_full_unstemmed | Modern cryptography and elliptic curves a beginner's guide Thomas R. Shemanske |
| title_short | Modern cryptography and elliptic curves |
| title_sort | modern cryptography and elliptic curves a beginner s guide |
| title_sub | a beginner's guide |
| topic | Number theory ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Computer science ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Number theory ... Elementary number theory ... Elementary number theory / msc Algebraic geometry ... Arithmetic problems. Diophantine geometry ... Applications to coding theory and cryptography / msc Number theory ... Finite fields and commutative rings (number-theoretic aspects) ... Algebraic coding theory; cryptography / msc Computer science ... Theory of data ... Data encryption / msc Number theory ... Computational number theory ... Factorization / msc Information and communication, circuits ... Communication, information ... Cryptography / msc Number theory ... Arithmetic algebraic geometry (Diophantine geometry) ... Elliptic curves over global fields / msc Quantum theory ... Axiomatics, foundations, philosophy ... Quantum computation / msc Curves, Elliptic Textbooks Geometry, Algebraic Textbooks Cryptography Textbooks Number theory ... Instructional exposition (textbooks, tutorial papers, etc.) Computer science ... Instructional exposition (textbooks, tutorial papers, etc.) Number theory ... Elementary number theory ... Elementary number theory Algebraic geometry ... Arithmetic problems. Diophantine geometry ... Applications to coding theory and cryptography Number theory ... Finite fields and commutative rings (number-theoretic aspects) ... Algebraic coding theory; cryptography Computer science ... Theory of data ... Data encryption Number theory ... Computational number theory ... Factorization Information and communication, circuits ... Communication, information ... Cryptography Number theory ... Arithmetic algebraic geometry (Diophantine geometry) ... Elliptic curves over global fields Quantum theory ... Axiomatics, foundations, philosophy ... Quantum computation Computersicherheit (DE-588)4274324-2 gnd Public-Key-Kryptosystem (DE-588)4209133-0 gnd Kryptologie (DE-588)4033329-2 gnd |
| topic_facet | Number theory ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Computer science ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Number theory ... Elementary number theory ... Elementary number theory / msc Algebraic geometry ... Arithmetic problems. Diophantine geometry ... Applications to coding theory and cryptography / msc Number theory ... Finite fields and commutative rings (number-theoretic aspects) ... Algebraic coding theory; cryptography / msc Computer science ... Theory of data ... Data encryption / msc Number theory ... Computational number theory ... Factorization / msc Information and communication, circuits ... Communication, information ... Cryptography / msc Number theory ... Arithmetic algebraic geometry (Diophantine geometry) ... Elliptic curves over global fields / msc Quantum theory ... Axiomatics, foundations, philosophy ... Quantum computation / msc Curves, Elliptic Textbooks Geometry, Algebraic Textbooks Cryptography Textbooks Number theory ... Instructional exposition (textbooks, tutorial papers, etc.) Computer science ... Instructional exposition (textbooks, tutorial papers, etc.) Number theory ... Elementary number theory ... Elementary number theory Algebraic geometry ... Arithmetic problems. Diophantine geometry ... Applications to coding theory and cryptography Number theory ... Finite fields and commutative rings (number-theoretic aspects) ... Algebraic coding theory; cryptography Computer science ... Theory of data ... Data encryption Number theory ... Computational number theory ... Factorization Information and communication, circuits ... Communication, information ... Cryptography Number theory ... Arithmetic algebraic geometry (Diophantine geometry) ... Elliptic curves over global fields Quantum theory ... Axiomatics, foundations, philosophy ... Quantum computation Computersicherheit Public-Key-Kryptosystem Kryptologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029744707&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013184751 |
| work_keys_str_mv | AT shemanskethomasr moderncryptographyandellipticcurvesabeginnersguide |