Number theory in science and communication: with applications in cryptography, physics, digital information, computing, and self-similarity
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | Undetermined |
| Veröffentlicht: |
Berlin u.a.
Springer
1986
|
| Ausgabe: | 2. enl. ed. |
| Schriftenreihe: | Springer series in information sciences
7 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIX, 374 S. Ill., graph. Darst. |
| ISBN: | 3540158006 0387158006 |
Internformat
MARC
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| 245 | 1 | 0 | |a Number theory in science and communication |b with applications in cryptography, physics, digital information, computing, and self-similarity |c M. R. Schroeder |
| 250 | |a 2. enl. ed. | ||
| 264 | 1 | |a Berlin u.a. |b Springer |c 1986 | |
| 300 | |a XIX, 374 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer series in information sciences |v 7 | |
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Datensatz im Suchindex
| _version_ | 1804119820625510400 |
|---|---|
| adam_text | Contents
Part I A Few Fundamentals
1. Introduction 1
1.1 Fibonacci, Continued Fractions and the Golden Ratio 4
1.2 Fermat, Primes and Cyclotomy 7
1.3 Euler, Totients and Cryptography 9
1.4 Gauss, Congruences and Diffraction 10
1.5 Galois, Fields and Codes 12
2. The Natural Numbers 17
2.1 The Fundamental Theorem 17
2.2 The Least Common Multiple 18
2.3 Planetary Gears 19
2.4 The Greatest Common Divisor 20
2.5 Human Pitch Perception 22
2.6 Octaves, Temperament, Kilos and Decibels 22
2.7 Coprimes 25
2.8 Euclid s Algorithm 25
3. Primes 26
3.1 How Many Primes are There? 26
3.2 The Sieve of Eratosthenes 27
3.3 A Chinese Theorem in Error 29
3.4 A Formula for Primes 29
3.5 Mersenne Primes 30
3.6 Repunits 34
3.7 Perfect Numbers 35
3.8 Fermat Primes 37
3.9 Gauss and the Impossible Heptagon 38
4. The Prime Distribution 40
4.1 A Probabilistic Argument 40
4.2 The Prime Counting Function 7r(x) 43
4.3 David Hilbert and Large Nuclei 47
4.4 Coprime Probabilities 48
XIV Contents
4.5 Twin Primes 51
4.6 Primeless Expanses 53
4.7 Square Free and Coprime Integers 54
Part II Some Simple Applications
5. Fractions: Continued, Egyptian and Farey 55
5.1 A Neglected Subject 55
5.2 Relations with Measure Theory 60
5.3 Periodic Continued Fractions 60
5.4 Electrical Networks and Squared Squares 64
5.5 Fibonacci Numbers and the Golden Ratio 65
5.6 Fibonacci, Rabbits and Computers 70
5.7 Fibonacci and Divisibility 72
5.8 Generalized Fibonacci and Lucas Numbers 73
5.9 Egyptian Fractions, Inheritance and Some Unsolved Problems 76
5.10 Farey Fractions 77
5.11 Fibonacci and the Problem of Bank Deposits 80
5.12 Error Free Computing 81
Part III Congruences and the Like
6. Linear Congruences 87
6.1 Residues 87
6.2 Some Simple Fields 90
6.3 Powers and Congruences 92
7. Diophantine Equations 95
7.1 Relation with Congruences 95
7.2 A Gaussian Trick 96
7.3 Nonlinear Diophantine Equations 98
7.4 Triangular Numbers 100
7.5 Pythagorean Numbers 102
7.6 Exponential Diophantine Equations 103
7.7 Fermat s Last Theorem 104
7.8 The Demise of a Conjecture by Euler 105
7.9 A Nonlinear Diophantine Equation in Physics and the
Geometry of Numbers 106
7.10 Normal Mode Degeneracy in Room Acoustics
(A Number Theoretic Application) 1°8
7.11 Waring s Problem 109
Contents XV
8. The Theorems of Fennat, Wilson and Euler Ill
8.1 Fermat sTheorem Ill
8.2 Wilson s Theorem 112
8.3 Euler s Theorem 113
8.4 The Impossible Star of David 115
8.5 Dirichlet and Linear Progression 116
Part IV Cryptography and Divisors
9. Euler Trap Doors and Public Key Encryption 118
9.1 A Numerical Trap Door 118
9.2 Digital Encryption 119
9.3 Public Key Encryption 121
9.4 A Simple Example 123
9.5 Repeated Encryption 123
9.6 Summary and Encryption Requirements 125
10. The Divisor Functions 127
10.1 The Number of Divisors 127
10.2 The Average of the Divisor Function 130
10.3 The Geometric Mean of the Divisors 131
10.4 The Summatory Function of the Divisor Function 131
10.5 The Generalized Divisor Functions 132
10.6 The Average Value of Euler s Function 133
11. The Prime Divisor Functions 135
11.1 The Number of Different Prime Divisors 135
11.2 The Distribution of co(n) 138
11.3 The Number of Prime Divisors 141
11.4 The Harmonic Mean of Q{n) 144
11.5 Medians and Percentiles of Q(n) 146
11.6 Implications for Public Key Encryption 147
12. Certified Signatures 149
12.1 A Story of Creative Financing 149
12.2 Certified Signature for Public Key Encryption 149
13. Primitive Roots 151
13.1 Orders 151
13.2 Periods of Decimal and Binary Fractions 154
13.3 A Primitive Proof of Wilson s Theorem 157
13.4 The Index A Number Theoretic Logarithm 158
13.5 Solution of Exponential Congruences 159
13.6 What is the Order Tm of an Integers Modulo a Prime pi .. 161
13.7 Index Encryption 162
XVI Contents
13.8 A Fourier Property of Primitive Roots and Concert Hall
Acoustics 163
13.9 More Spacious Sounding Sound 164
13.10 A Negative Property of the Fermat Primes 167
14. Knapsack Encryption 168
14.1 An Easy Knapsack 168
14.2 AHardKnapsack 169
Part V Residues and Diffraction
15. Quadratic Residues 172
15.1 Quadratic Congruences 172
15.2 Euler s Criterion 173
15.3 The Legendre Symbol 175
15.4 A Fourier Property of Legendre Sequences 176
15.5 Gauss Sums 177
15.6 Pretty Diffraction 179
15.7 Quadratic Reciprocity 179
15.8 A Fourier Property of Quadratic Residue Sequences 180
15.9 Spread Spectrum Communication 183
15.10 Generalized Legendre Sequences Obtained Through
Complexification of the Euler Criterion 183
Part VI Chinese and Other Fast Algorithms
16. The Chinese Remainder Theorem and Simultaneous Congruences . 186
16.1 Simultaneous Congruences 186
16.2 The Sino Representation: A Chinese Number System 187
16.3 Applications of the Sino Representation 189
16.4 Discrete Fourier Transformation in Sino 190
16.5 A Sino Optical Fourier Transformer 191
16.6 Generalized Sino Representation 192
16.7 Fast Prime Length Fourier Transform 194
17. Fast Transformations and Kronecker Products 196
17.1 A Fast Hadamard Transform 196
17.2 The Basic Principle of the Fast Fourier Transforms 199
18. Quadratic Congruences 201
18.1 Application of the Chinese Remainder Theorem (CRT) 201
Contents XVII
Part VII Pseudoprimes, Mobius Transform, and Partitions
19. Pseudoprimes, Poker and Remote Coin Tossing 203
19.1 Pulling Roots to Ferret Out Composites 203
19.2 Factors from a Square Root 205
19.3 Coin Tossing by Telephone 206
19.4 Absolute and Strong Pseudoprimes 209
19.5 Fermat and Strong Pseudoprimes 211
19.6 Deterministic Primality Testing 212
19.7 A Very Simple Factoring Algorithm 213
20. The Mobius Function and the Mobius Transform 215
20.1 The Mobius Transform and Its Inverse 215
20.2 Proof of the Inversion Formula 217
20.3 Second Inversion Formula 218
20.4 Third Inversion Formula 219
20.5 Fourth Inversion Formula 219
20.6 Riemann s Hypothesis and the Disproof of the Mertens
Conjecture 219
20.7 Dirichlet Series and the Mobius Function 220
21. Generating Functions and Partitions 223
21.1 Generating Functions 223
21.2 Partitions of Integers 225
21.3 Generating Functions of Partitions 226
21.4 Restricted Partitions 227
Part VIII Cyclotomy and Polynomials
22. Cyclotomic Polynomials 232
22.1 How to Divide a Circle into Equal Parts 232
22.2 Gauss s Great Insight 235
22.3 Factoring in Different Fields 240
22.4 Cyclotomy in the Complex Plane 240
22.5 How to Divide a Circle with Compass and Straightedge 242
22.5.1 Rational Factors of zN 1 243
22.6 An Alternative Rational Factorization 244
22.7 Relation Between Rational Factors and Complex Roots 245
22.8 How to Calculate with Cyclotomic Polynomials 247
23. Linear Systems and Polynomials 249
23.1 Impulse Responses 249
23.2 Time Discrete Systems and the z Transform 250
XVIII Contents
23.3 Discrete Convolution 251
23.4 Cyclotomic Polynomials and zTransform 251
24. Polynomial Theory 253
24.1 Some Basic Facts of Polynomial Life 253
24.2 Polynomial Residues 254
24.3 Chinese Remainders for Polynomials 256
24.4 Euclid s Algorithm for Polynomials 257
Part IX Galois Fields and More Applications
25. GaloisFields 259
25.1 Prime Order 259
25.2 Prime Power Order 259
25.3 Generation of GF(24) 262
25.4 How Many Primitive Elements? 263
25.5 Recursive Relations 264
25.6 How to Calculate in GF(pm) 266
25.7 Zech Logarithm, Doppler Radar and Optimum Ambiguity
Functions 267
25.8 A Unique Phase Array Based on the Zech Logarithm 271
25.9 Spread Spectrum Communication and Zech Logarithms ... 272
26. Spectral Properties of Galois Sequences 274
26.1 Circular Correlation 274
26.2 Application to Error Correcting Codes
and Speech Recognition 277
26.3 Application to Precision Measurements 278
26.4 Conceit Hall Measurements 279
26.5 The Fourth Effect of General Relativity 280
26.6 Toward Better Concert Hall Acoustics 281
26.7 Higher Dimensional Diffusors 287
26.8 Active Array Applications 287
27. Random Number Generators 289
27.1 Pseudorandom Galois Sequences 290
27.2 Randomness from Congruences 291
27.3 Continuous Distributions 292
27.4 Four Ways to Generate a Gaussian Variable 293
27.5 Pseudorandom Sequences in Cryptography 295
28. Waveforms and Radiation Patterns 296
28.1 Special Phases 297
28.2 The Rudin Shapiro Polynomials 299
28.3 Gauss Sums and Peak Factors 300
Contents XIX
28.4 Galois Sequences and the Smallest Peak Factors 302
28.5 Minimum Redundancy Antennas 305
29. Number Theory, Randomness and Art 307
29.1 Number Theory and Graphic Design 307
29.2 The Primes of Gauss and Eisenstein 309
29.3 Galois Fields and Impossible Necklaces 310
Part X Self Similarity, Fractals and Art
30. Self Similarity, Fractals, Deterministic Chaos and a New State
of Matter 315
30.1 Fibonacci, Noble Numbers and a New State of Matter 319
30.2 Cantor Sets, Fractals and a Musical Paradox 324
30.3 The Twin Dragon: a Fractal from a Complex Number
System 330
30.4 Statistical Fractals 331
30.5 Some Crazy Mappings 333
30.6 The Logistic Parabola and Strange Attractors 337
30.7 Conclusion 340
Appendix 341
A. A Calculator Program for Exponentiation
and Residue Reduction 341
B. A Calculator Program for Calculating Fibonacci
and Lucas Numbers 345
C. A Calculator Program for Decomposing an Integer
According to the Fibonacci Number System 346
Glossary of Symbols 349
References 353
Name Index 363
Subject Index 367
|
| any_adam_object | 1 |
| author | Schroeder, Manfred R. 1926-2009 |
| author_GND | (DE-588)120787318 |
| author_facet | Schroeder, Manfred R. 1926-2009 |
| author_role | aut |
| author_sort | Schroeder, Manfred R. 1926-2009 |
| author_variant | m r s mr mrs |
| building | Verbundindex |
| bvnumber | BV005610452 |
| classification_rvk | SK 180 |
| ctrlnum | (DE-599)BVBBV005610452 |
| discipline | Mathematik |
| edition | 2. enl. ed. |
| format | Book |
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| id | DE-604.BV005610452 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:32:16Z |
| institution | BVB |
| isbn | 3540158006 0387158006 |
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| publishDate | 1986 |
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| series2 | Springer series in information sciences |
| spelling | Schroeder, Manfred R. 1926-2009 Verfasser (DE-588)120787318 aut Number theory in science and communication with applications in cryptography, physics, digital information, computing, and self-similarity M. R. Schroeder 2. enl. ed. Berlin u.a. Springer 1986 XIX, 374 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in information sciences 7 Anwendung (DE-588)4196864-5 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 Anwendung (DE-588)4196864-5 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003513282&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schroeder, Manfred R. 1926-2009 Number theory in science and communication with applications in cryptography, physics, digital information, computing, and self-similarity Anwendung (DE-588)4196864-5 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4196864-5 (DE-588)4067277-3 |
| title | Number theory in science and communication with applications in cryptography, physics, digital information, computing, and self-similarity |
| title_auth | Number theory in science and communication with applications in cryptography, physics, digital information, computing, and self-similarity |
| title_exact_search | Number theory in science and communication with applications in cryptography, physics, digital information, computing, and self-similarity |
| title_full | Number theory in science and communication with applications in cryptography, physics, digital information, computing, and self-similarity M. R. Schroeder |
| title_fullStr | Number theory in science and communication with applications in cryptography, physics, digital information, computing, and self-similarity M. R. Schroeder |
| title_full_unstemmed | Number theory in science and communication with applications in cryptography, physics, digital information, computing, and self-similarity M. R. Schroeder |
| title_short | Number theory in science and communication |
| title_sort | number theory in science and communication with applications in cryptography physics digital information computing and self similarity |
| title_sub | with applications in cryptography, physics, digital information, computing, and self-similarity |
| topic | Anwendung (DE-588)4196864-5 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Anwendung Zahlentheorie |
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