Number theory for computing:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2002
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 435 S. Ill., graph. Darst. |
| ISBN: | 3540430725 9783642077104 9783540430728 |
Staff View
MARC
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| 100 | 1 | |a Yan, Song Y. |d 1954- |e Verfasser |0 (DE-588)121556034 |4 aut | |
| 245 | 1 | 0 | |a Number theory for computing |c Song Y. Yan |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2002 | |
| 300 | |a XXII, 435 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 7 | |a Teoria dos números |2 larpcal | |
| 650 | 4 | |a Informatik | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Number theory | |
| 650 | 4 | |a Computer science -- Mathematics | |
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Record in the Search Index
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Table of Contents
1. Elementary Number Theory 1
1.1 Introduction 1
1.1.1 What is Number Theory? 1
1.1.2 Applications of Number Theory 13
1.1.3 Algebraic Preliminaries 14
1.2 Theory of Divisibility 21
1.2.1 Basic Concepts and Properties of Divisibility 21
1.2.2 Fundamental Theorem of Arithmetic 27
1.2.3 Mersenne Primes and Fermat Numbers 33
1.2.4 Euclid's Algorithm 40
1.2.5 Continued Fractions 44
1.3 Diophantine Equations 52
1.3.1 Basic Concepts of Diophantine Equations 52
1.3.2 Linear Diophantine Equations 54
1.3.3 Pell's Equations 57
1.4 Arithmetic Functions 63
1.4.1 Multiplicative Functions 63
1.4.2 Functions r(n), a{n) and s(n) 66
1.4.3 Perfect, Amicable and Sociable Numbers 71
1.4.4 Functions 4 (n), X(n) and fi(n) 79
1.5 Distribution of Prime Numbers 85
1.5.1 Prime Distribution Function n(x) 85
1.5.2 Approximations of n(x) by x/ lnx 87
1.5.3 Approximations of n(x) by Li(x) 94
1.5.4 The Riemann ([ Function £(•?) 95
1.5.5 The nth Prime 104
1.5.6 Distribution of Twin Primes 106
1.5.7 The Arithmetic Progression of Primes 110
1.6 Theory of Congruences Ill
1.6.1 Basic Concepts and Properties of Congruences Ill
1.6.2 Modular Arithmetic 118
1.6.3 Linear Congruences 123
1.6.4 The Chinese Remainder Theorem 130
1.6.5 High Order Congruences 133
xiv Table of Contents
1.6.6 Legendre and Jacobi Symbols 139
1.6.7 Orders and Primitive Roots 150
1.6.8 Indices and kth Power Residues 155
1.7 Arithmetic of Elliptic Curves 160
1.7.1 Basic Concepts of Elliptic Curves 160
1.7.2 Geometric Composition Laws of Elliptic Curves 163
1.7.3 Algebraic Computation Laws for Elliptic Curves 164
1.7.4 Group Laws on Elliptic Curves 168
1.7.5 Number of Points on Elliptic Curves 169
1.8 Bibliographic Notes and Further Reading 171
2. Computational/Algorithmic Number Theory 173
2.1 Introduction 173
2.1.1 What is Computational/Algorithmic Number Theory? . 174
2.1.2 Effective Computability 177
2.1.3 Computational Complexity 181
2.1.4 Complexity of Number Theoretic Algorithms 188
2.1.5 Fast Modular Exponentiations 194
2.1.6 Fast Group Operations on Elliptic Curves 198
2.2 Algorithms for Primality Testing 202
2.2.1 Deterministic and Rigorous Primality Tests 202
2.2.2 Fermat's Pseudoprimality Test 206
2.2.3 Strong Pseudoprimality Test 208
2.2.4 Lucas Pseudoprimality Test 215
2.2.5 Elliptic Curve Test 222
2.2.6 Historical Notes on Primality Testing 225
2.3 Algorithms for Integer Factorization 228
2.3.1 Complexity of Integer Factorization 228
2.3.2 Trial Division and Fermat Method 232
2.3.3 Legendre's Congruence 234
2.3.4 Continued FRACtion Method (CFRAC) 237
2.3.5 Quadratic and Number Field Sieves (QS/NFS) 240
2.3.6 Polland's ilrho" and "p 1" Methods 244
2.3.7 Lenstra's Elliptic Curve Method (ECM) 251
2.4 Algorithms for Discrete Logarithms 254
2.4.1 Shanks' Baby Step Giant Step Algorithm 255
2.4.2 Silver Pohlig Hellman Algorithm 258
2.4.3 Index Calculus for Discrete Logarithms 262
2.4.4 Algorithms for Elliptic Curve Discrete Logarithms 266
2.4.5 Algorithm for Root Finding Problem 270
2.5 Quantum Number Theoretic Algorithms 273
2.5.1 Quantum Information and Computation 273
2.5.2 Quantum Computability and Complexity 278
2.5.3 Quantum Algorithm for Integer Factorization 279
2.5.4 Quantum Algorithms for Discrete Logarithms 285
Table of Contents xv
2.6 Miscellaneous Algorithms in Number Theory 287
2.6.1 Algorithms for Computing n(x) 287
2.6.2 Algorithms for Generating Amicable Pairs 292
2.6.3 Algorithms for Verifying Goldbach's Conjecture 295
2.6.4 Algorithm for Finding Odd Perfect Numbers 299
2.7 Bibliographic Notes and Further Reading 300
3. Applied Number Theory in Computing/ Cryptography . . . 303
3.1 Why Applied Number Theory? 303
3.2 Computer Systems Design 305
3.2.1 Representing Numbers in Residue Number Systems 305
3.2.2 Fast Computations in Residue Number Systems 308
3.2.3 Residue Computers 312
3.2.4 Complementary Arithmetic 315
3.2.5 Hash Functions 317
3.2.6 Error Detection and Correction Methods 321
3.2.7 Random Number Generation 326
3.3 Cryptography and Information Security 332
3.3.1 Introduction 332
3.3.2 Secret Key Cryptography 333
3.3.3 Data/Advanced Encryption Standard (DES/AES) 344
3.3.4 Public Key Cryptography 348
3.3.5 Discrete Logarithm Based Cryptosystems 354
3.3.6 RSA Public Key Cryptosystem 358
3.3.7 Quadratic Residuosity Cryptosystems 373
3.3.8 Elliptic Curve Public Key Cryptosystems 379
3.3.9 Digital Signatures 385
3.3.10 Digital Signature Standard (DSS) 392
3.3.11 Database Security 395
3.3.12 Secret Sharing 399
3.3.13 Internet/Web Security and Electronic Commerce 403
3.3.14 Steganography 409
3.3.15 Quantum Cryptography 410
3.4 Bibliographic Notes and Further Reading 411
Bibliography 415
Index 429 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-search | 512/.7 512/.7 21 |
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| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| edition | 2. ed. |
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| indexdate | 2024-09-25T12:02:49Z |
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| isbn | 3540430725 9783642077104 9783540430728 |
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| spelling | Yan, Song Y. 1954- Verfasser (DE-588)121556034 aut Number theory for computing Song Y. Yan 2. ed. Berlin [u.a.] Springer 2002 XXII, 435 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Teoria dos números larpcal Informatik Mathematik Number theory Computer science -- Mathematics Elementare Zahlentheorie (DE-588)4294368-1 gnd rswk-swf Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Algorithmische Zahlentheorie (DE-588)4314054-3 gnd rswk-swf Elementare Zahlentheorie (DE-588)4294368-1 s DE-604 Algorithmische Zahlentheorie (DE-588)4314054-3 s Zahlentheorie (DE-588)4067277-3 s Datenverarbeitung (DE-588)4011152-0 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009656962&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Yan, Song Y. 1954- Number theory for computing Teoria dos números larpcal Informatik Mathematik Number theory Computer science -- Mathematics Elementare Zahlentheorie (DE-588)4294368-1 gnd Datenverarbeitung (DE-588)4011152-0 gnd Zahlentheorie (DE-588)4067277-3 gnd Algorithmische Zahlentheorie (DE-588)4314054-3 gnd |
| subject_GND | (DE-588)4294368-1 (DE-588)4011152-0 (DE-588)4067277-3 (DE-588)4314054-3 |
| title | Number theory for computing |
| title_auth | Number theory for computing |
| title_exact_search | Number theory for computing |
| title_full | Number theory for computing Song Y. Yan |
| title_fullStr | Number theory for computing Song Y. Yan |
| title_full_unstemmed | Number theory for computing Song Y. Yan |
| title_short | Number theory for computing |
| title_sort | number theory for computing |
| topic | Teoria dos números larpcal Informatik Mathematik Number theory Computer science -- Mathematics Elementare Zahlentheorie (DE-588)4294368-1 gnd Datenverarbeitung (DE-588)4011152-0 gnd Zahlentheorie (DE-588)4067277-3 gnd Algorithmische Zahlentheorie (DE-588)4314054-3 gnd |
| topic_facet | Teoria dos números Informatik Mathematik Number theory Computer science -- Mathematics Elementare Zahlentheorie Datenverarbeitung Zahlentheorie Algorithmische Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009656962&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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