Number theory in science and communication: with applications in cryptography, physics, digital information, computing, and self-similarity
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2005
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| Ausgabe: | 4. ed. |
| Schriftenreihe: | Springer series in information sciences
7 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XXII, 363 S. Ill., graph. Darst. |
| ISBN: | 9783540265962 |
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 |
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Datensatz im Suchindex
| _version_ | 1804135015391428608 |
|---|---|
| adam_text | M. R. SCHROEDER
NUMBER THEORY
IN SCIENCE
AND COMMUNICATION
WITH APPLICATIONS IN CRYPTOGRAPHY,
PHYSICS, DIGITAL INFORMATION, COMPUTING,
AND SELF-SIMILARITY
FOURTH EDITION
WITH 99 FIGURES
4Y SPRINGER
CONTENTS
PAR
T I. A FEW FUNDAMENTAL
S
1. INTRODUCTIO
N
1
THE FAMILY OF NUMBERS 4
1.1 FIBONACCI, CONTINUED FRACTIONS AND THE GOLDEN RATIO 7
1.2 FERMAT, PRIMES AND CYCLOTOMY 9
1.3 EULER, TOTIENTS AND CRYPTOGRAPHY 11
1.4 GAUSS, CONGRUENCES AND DIFFRACTION 13
1.5 GALOIS, FIELDS AND CODES 14
2. TH
E NATURA
L NUMBER
S
19
2.1 THE FUNDAMENTAL THEOREM 19
2.2 TH
E LEAST COMMON MULTIPLE 20
2.3 PLANETARY GEARS 21
2.4 THE GREATEST COMMON DIVISOR 21
2.5 HUMAN PITCH PERCEPTION 23
2.6 OCTAVES, TEMPERAMENT, KILOS AND DECIBELS 24
2.7 COPRIMES 26
2.8 EUCLID S ALGORITHM 26
2.9 THE DECIMAL SYSTEM DECIMATED 27
3
. PRIME
S
28
3.1 HOW MANY PRIMES ARE THERE? 28
3.2 THE SIEVE OF ERATOSTHENES 29
3.3 A CHINESE THEOREM IN ERROR 30
3.4 A FORMULA FOR PRIMES 31
3.5 MERSENNE PRIMES 32
3.6 REPUNITS 36
3.7 PERFECT NUMBERS 37
3.8 FERMAT PRIMES 38
3.9 GAUSS AND THE IMPOSSIBLE HEPTAGON 39
4. TH
E PRIM
E DISTRIBUTIO
N
41
4.1 A PROBABILISTIC ARGUMENT 41
4.2 THE PRIME-COUNTING FUNCTION
N(X)
43
XX CONTENTS
4.3 DAVID HILBERT AND LARGE NUCLEI 47
4.4 COPRIME PROBABILITIES 48
4.5 PRIMES IN PROGRESSIONS 51
4.6 PRIMCLESS EXPANSES 53
4.7 SQUAREFREE AND COPRIME INTEGERS 54
4.8 TWIN PRIMES 54
4.9 PRIME TRIPLETS 56
4.10 PRIME QUADRUPLETS AND QUINTUPLETS 57
4.11 PRIMES AT ANY DISTANCE 58
4.12 SPACING DISTRIBUTION BETWEEN ADJACENT PRIMES 61
4.13 GOLDBACH S CONJECTURE 61
4.14 SUM OF THREE PRIMES 63
PART II. SOM
E SIMPL
E APPLICATION
S
5. FRACTIONS
: CONTINUED
, EGYPTIA
N AN
D FAREY
65
5.1 A NEGLECTED SUBJECT 65
5.2 RELATIONS WITH MEASURE THEORY 69
5.3 PERIODIC CONTINUED FRACTIONS 70
5.4 ELECTRICAL NETWORKS AND SQUARED SQUARES 73
5.5 FIBONACCI NUMBERS AND THE GOLDEN RATIO 74
5.6 FIBONACCI, RABBITS AND COMPUTERS 78
5.7 FIBONACCI AND DIVISIBILITY 81
5.8 GENERALIZED FIBONACCI AND LUCAS NUMBERS 81
5.9 EGYPTIAN FRACTIONS, INHERITANCE
AND SOME UNSOLVED PROBLEMS 85
5.10 FAREY FRACTIONS 86
5.10.1 FAREY TREES 88
5.10.2 LOCKED PALLAS 92
5.11 FIBONACCI AND TH
E PROBLEM OF BANK DEPOSITS 93
5.12 ERROR-FREE COMPUTING 94
PART III
. CONGRUENCE
S AN
D TH
E LIKE
6. LINEAR CONGRUENCE
S
99
6.1 RESIDUES 99
6.2 SOME SIMPLE FIELDS 102
6.3 POWERS AND CONGRUENCES 103
7. DIOPHANTIN
E EQUATION
S
106
7.1 RELATION WITH CONGRUENCES 106
7.2 A GAUSSIAN TRICK 107
7.3 NONLINEAR DIOPHANTINE EQUATIONS 109
CONTENTS XXI
7.4 TRIANGULAR NUMBERS 110
7.5 PYTHAGOREAN NUMBERS 112
7.6 EXPONENTIAL DIOPHANTINE EQUATIONS 113
7.7 FORMAT S LAST THEOREM 113
7.8 THE DEMISE OF A CONJECTURE BY EULER 115
7.9 A NONLINEAR DIOPHANTINE EQUATION IN PHYSICS
AND THE GEOMETRY OF NUMBERS 116
7.10 NORMAL-MODE DEGENERACY IN ROOM ACOUSTICS
(A NUMBER-THEORETIC APPLICATION) 120
7.11 WARING S PROBLEM 121
8. TH
E THEOREM
S OF FERMAT, WILSO
N AND EULER
122
8.1 FERMAT S THEOREM 122
8.2 WILSON S THEOREM 123
8.3 EULER S THEOREM 124
8.4 THE IMPOSSIBLE STAR OF DAVID 125
8.5 DIRICHLET AND LINEAR PROGRESSION 127
PART IV
. CRYPTOGRAPHY AND DIVISOR
S
9. EULER TRAP DOOR
S AND PUBLIC-KE
Y ENCRYPTIO
N
129
9.1 A NUMERICAL TRAP DOOR 131
9.2 DIGITAL ENCRYPTION 132
9.3 PUBLIC-KEY ENCRYPTION 133
9.4 A SIMPLE EXAMPLE 135
9.5 REPEATED ENCRYPTION 136
9.6 SUMMARY AND ENCRYPTION REQUIREMENTS 137
10. TH
E DIVISOR FUNCTIONS
139
10.1 THE NUMBER OF DIVISORS 139
10.2 THE AVERAGE OF THE DIVISOR FUNCTION 142
10.3 THE GEOMETRIC MEAN OF THE DIVISORS 142
10.4 THE SUMMATORY FUNCTION OF THE DIVISOR FUNCTION 143
10.5 THE GENERALIZED DIVISOR FUNCTIONS 143
10.6 THE AVERAGE VALUE OF EULER S FUNCTION 144
11
. TH
E PRIM
E DIVISOR FUNCTIONS
146
11.1 THE NUMBER OF DIFFERENT PRIME DIVISORS 146
11.2 THE DISTRIBUTION OF
U(N)
150
11.3 THE NUMBER OF PRIME DIVISORS 151
11.4 THE HARMONIC MEAN OF
Q(N)
154
11.5 MEDIANS AND PERCENTILES OF
Q(N)
156
11.6 IMPLICATIONS FOR PUBLIC-KEY ENCRYPTION 157
XXII CONTENTS
12. CERTIFIED SIGNATURE
S
158
12.1 A STORY OF CREATIVE FINANCING 158
12.2 CERTIFIED SIGNATURE FOR PUBLIC-KEY ENCRYPTION 158
13. PRIMITIV
E ROOT
S
160
13.1 ORDERS 160
13.2 PERIODS OF DECIMAL AND BINARY FRACTIONS 163
13.3 A PRIMITIVE PROOF OF WILSON S THEOREM 166
13.4 THE INDEX - A NUMBER-THEORETIC LOGARITHM 166
13.5 SOLUTION OF EXPONENTIAL CONGRUENCES 167
13.6 WHAT IS TH
E ORDER
T
M
OF AN INTEGER
M
MODULO A PRIME P? . . 169
13.7 INDEX ENCRYPTION 170
13.8 A FOURIER PROPERTY OF PRIMITIVE ROOTS
AND CONCERT HALL ACOUSTICS 170
13.9 MORE SPACIOUS-SOUNDING SOUND 172
13.10 GALOIS ARRAYS FOR X-RAY ASTRONOMY 174
13.11 A NEGATIVE PROPERTY OF TH
E FERMAT PRIMES 175
14. KNAPSACK ENCRYPTIO
N
177
14.1 AN EASY KNAPSACK 177
14.2 A HARD KNAPSACK 178
PAR
T V
. RESIDUE
S AND DIFFRACTION
15. QUADRATI
C RESIDUE
S
181
15.1 QUADRATIC CONGRUENCES 181
15.2 EULER S CRITERION 182
15.3 THE LEGENDRE SYMBOL 183
15.4 A FOURIER PROPERTY OF LEGENDRE SEQUENCES 185
15.5 GAUSS SUMS 185
15.6 PRETT
Y DIFFRACTION 187
15.7 QUADRATIC RECIPROCITY 187
15.8 A FOURIER PROPERTY OF QUADRATIC-RESIDUE SEQUENCES 188
15.9 SPREAD SPECTRUM COMMUNICATION 190
15.10 GENERALIZED LEGENDRE SEQUENCES OBTAINED
THROUGH COMPLEXIFICATION OF THE EULER CRITERION 191
PAR
T VI
. CHINES
E AND OTHE
R FAST ALGORITHM
S
16. TH
E CHINES
E REMAINDE
R THEORE
M
AND SIMULTANEOU
S CONGRUENCE
S
194
16.1 SIMULTANEOUS CONGRUENCES 194
16.2 THE SINO-REPRESENTATION: A CHINESE NUMBER SYSTEM 195
CONTENTS XXIII
16.3 APPLICATIONS OF THE SINO-REPRESENTATION 196
16.4 DISCRETE FOURIER TRANSFORMATION IN SINO 198
16.5 A SINO-OPTICAL FOURIER TRANSFORMER 199
16.6 GENERALIZED SINO-REPRESENTATION 200
16.7 FAST PRIME-LENGTH FOURIER TRANSFORM 201
17. FAST TRANSFORMATIO
N AN
D KRONECKE
R PRODUCT
S
203
17.1 A FAST HADAMARD TRANSFORM 203
17.2 THE BASIC PRINCIPLE OF THE FAST FOURIER TRANSFORMS 206
18. QUADRATI
C CONGRUENCE
S
207
18.1 APPLICATION OF THE CHINESE REMAINDER THEOREM (CRT) 207
PAR
T VII
. PSEUDOPRIMES
, MOBIU
S TRANSFORM, AND PARTITION
S
19. PSEUDOPRIMES
, POKE
R AND REMOT
E COIN TOSSIN
G
209
19.1 PULLING ROOTS TO FERRET OUT COMPOSITES 209
19.2 FACTORS FROM A SQUARE ROOT 210
19.3 COIN TOSSING BY TELEPHONE 212
19.4 ABSOLUTE AND STRONG PSEUDOPRIMES 214
19.5 FERMAT AND STRONG PSEUDOPRIMES 216
19.6 DETERMINISTIC PRIMALITY TESTING 216
19.7 A VERY SIMPLE FACTORING ALGORITHM 218
19.8 FACTORING WITH ELLIPTIC CURVES 218
19.9 QUANTUM FACTORING 219
20. TH
E MOBIU
S FUNCTIO
N AND TH
E MOBIU
S TRANSFORM
220
20.1 THE MOBIUS TRANSFORM AND ITS INVERSE 220
20.2 PROOF OF THE INVERSION FORMULA 222
20.3 SECOND INVERSION FORMULA 223
20.4 THIRD INVERSION FORMULA 223
20.5 FOURTH INVERSION FORMULA 224
20.6 RIEMANN S HYPOTHESIS
AND THE DISPROOF OF THE MERTENS CONJECTURE 224
20.7 DIRICHLET SERIES AND THE MOBIUS FUNCTION 225
21
. GENERATIN
G FUNCTION
S AND PARTITION
S
228
21.1 GENERATING FUNCTIONS 228
21.2 PARTITIONS OF INTEGERS 230
21.3 GENERATING FUNCTIONS OF PARTITIONS 231
21.4 RESTRICTED PARTITIONS 232
XXIV CONTENTS
PART VIII. CYCLOTOM
Y AND POLYNOMIAL
S
22. CYCLOTOMIC POLYNOMIAL
S
236
22.1 HOW TO DIVIDE A CIRCLE INTO EQUAL PARTS 236
22.2 GAUSS S GREAT INSIGHT 239
22.3 FACTORING IN DIFFERENT FIELDS 243
22.4 CYCLOTOMY IN THE COMPLEX PLANE 243
22.5 HOW TO DIVIDE A CIRCLE WITH COMPASS AND STRAIGHTEDGE 244
22.5.1 RATIONAL FACTORS OF
Z
N
-
1 246
22.6 AN ALTERNATIVE RATIONAL FACTORIZATION 247
22.7 RELATION BETWEEN RATIONAL FACTORS AND COMPLEX ROOTS 248
22.8 HOW TO CALCULATE WITH CYCLOTOMIC POLYNOMIALS 249
23. LINEAR SYSTEM
S AND POLYNOMIAL
S
251
23.1 IMPULSE RESPONSES 251
23.2 TIME-DISCRETE SYSTEMS AND THE Z TRANSFORM 252
23.3 DISCRETE CONVOLUTION 252
23.4 CYCLOTOMIC POLYNOMIALS AND Z TRANSFORM 253
24. POLYNOMIA
L THEOR
Y
254
24.1 SOME BASIC FACTS OF POLYNOMIAL LIFE 254
24.2 POLYNOMIAL RESIDUES 255
24.3 CHINESE REMAINDERS FOR POLYNOMIALS 256
24.4 EUCLID S ALGORITHM FOR POLYNOMIALS 257
PART IX. GALOIS FIELD
S AND MORE APPLICATION
S
25. GALOIS FIELD
S
260
25.1 PRIME ORDER 260
25.2 PRIME POWER ORDER 260
25.3 GENERATION OF GF(2
4
) 262
25.4 HOW MANY PRIMITIVE ELEMENTS? 264
25.5 RECURSIVE RELATIONS 264
25.6 HOW TO CALCULATE IN GF(P
M
) 266
25.7 ZECH LOGARITHM, DOPPLER RADAR
AND OPTIMUM AMBIGUITY FUNCTIONS 267
25.8 A UNIQUE PHASE-ARRAY BASED ON THE ZECH LOGARITHM 270
25.9 SPREAD-SPECTRUM COMMUNICATION AND ZECH LOGARITHMS 272
26. SPECTRAL PROPERTIE
S OF GALOIS SEQUENCE
S
273
26.1 CIRCULAR CORRELATION 273
26.2 APPLICATION TO ERROR-CORRECTING CODES
AND SPEECH RECOGNITION 275
CONTENTS XXV
26.3 APPLICATION TO PRECISION MEASUREMENTS 277
26.4 CONCERT HALL MEASUREMENTS 278
26.5 THE FOURTH EFFECT OF GENERAL RELATIVITY 279
26.6 TOWARD BETTOR CONCERT HALL ACOUSTICS 280
26.7 HIGHER-DIMENSIONAL DIFFUSORS 285
26.8 ACTIVE ARRAY APPLICATIONS 286
27. RANDO
M NUMBE
R GENERATOR
S
287
27.1 PSEUDORANDOM GALOIS SEQUENCES 288
27.2 RANDOMNESS FROM CONGRUENCES 289
27.3 CONTINUOUS DISTRIBUTIONS 290
27.4 FOUR WAYS TO GENERATE A GAUSSIAN VARIABLE 291
27.5 PSEUDORANDOM SEQUENCES IN CRYPTOGRAPHY 292
28. WAVEFORMS AND RADIATIO
N PATTERN
S
293
28.1 SPECIAL PHASES 294
28.2 THE RUDIN-SHAPIRO POLYNOMIALS 296
28.3 GAUSS SUMS AND PEAK FACTORS 297
28.4 GALOIS SEQUENCES AND THE SMALLEST PEAK FACTORS 299
28.5 MINIMUM REDUNDANCY ANTENNAS 301
28.6 GOLOMB RULERS 303
29. NUMBE
R THEORY, RANDOMNES
S AND ART
305
29.1 NUMBER THEORY AND GRAPHIC DESIGN 305
29.2 THE PRIMES OF GAUSS AND EISENSTEIN 307
29.3 GALOIS FIELDS AND IMPOSSIBLE NECKLACES 308
29.4 BAROQUE INTEGERS 312
PART X
. SELF-SIMILARITY, FRACTALS AND ART
30
. SELF-SIMILARITY, FRACTALS, DETERMINISTI
C CHAOS
AND A NE
W STAT
E OF MATTE
R
315
30.1 FIBONACCI, NOBLE NUMBERS AND A NEW STATE OF MATTER 318
30.2 CANTOR SETS, FRACTALS AND A MUSICAL PARADOX 324
30.3 THE TWIN DRAGON:
A FRACTAL FROM A COMPLEX NUMBER SYSTEM 329
30.4 STATISTICAL FRACTALS 331
30.5 SOME CRAZY MAPPINGS 333
30.6 THE LOGISTIC PARABOLA AND STRANGE ATTRACTORS 336
30.7 CONCLUSION 339
|
| adam_txt |
M. R. SCHROEDER
NUMBER THEORY
IN SCIENCE
AND COMMUNICATION
WITH APPLICATIONS IN CRYPTOGRAPHY,
PHYSICS, DIGITAL INFORMATION, COMPUTING,
AND SELF-SIMILARITY
FOURTH EDITION
WITH 99 FIGURES
4Y SPRINGER
CONTENTS
PAR
T I. A FEW FUNDAMENTAL
S
1. INTRODUCTIO
N
1
THE FAMILY OF NUMBERS 4
1.1 FIBONACCI, CONTINUED FRACTIONS AND THE GOLDEN RATIO 7
1.2 FERMAT, PRIMES AND CYCLOTOMY 9
1.3 EULER, TOTIENTS AND CRYPTOGRAPHY 11
1.4 GAUSS, CONGRUENCES AND DIFFRACTION 13
1.5 GALOIS, FIELDS AND CODES 14
2. TH
E NATURA
L NUMBER
S
19
2.1 THE FUNDAMENTAL THEOREM 19
2.2 TH
E LEAST COMMON MULTIPLE 20
2.3 PLANETARY "GEARS" 21
2.4 THE GREATEST COMMON DIVISOR 21
2.5 HUMAN PITCH PERCEPTION 23
2.6 OCTAVES, TEMPERAMENT, KILOS AND DECIBELS 24
2.7 COPRIMES 26
2.8 EUCLID'S ALGORITHM 26
2.9 THE DECIMAL SYSTEM DECIMATED 27
3
. PRIME
S
28
3.1 HOW MANY PRIMES ARE THERE? 28
3.2 THE SIEVE OF ERATOSTHENES 29
3.3 A CHINESE THEOREM IN ERROR 30
3.4 A FORMULA FOR PRIMES 31
3.5 MERSENNE PRIMES 32
3.6 REPUNITS 36
3.7 PERFECT NUMBERS 37
3.8 FERMAT PRIMES 38
3.9 GAUSS AND THE IMPOSSIBLE HEPTAGON 39
4. TH
E PRIM
E DISTRIBUTIO
N
41
4.1 A PROBABILISTIC ARGUMENT 41
4.2 THE PRIME-COUNTING FUNCTION
N(X)
43
XX CONTENTS
4.3 DAVID HILBERT AND LARGE NUCLEI 47
4.4 COPRIME PROBABILITIES 48
4.5 PRIMES IN PROGRESSIONS 51
4.6 PRIMCLESS EXPANSES 53
4.7 SQUAREFREE AND COPRIME INTEGERS 54
4.8 TWIN PRIMES 54
4.9 PRIME TRIPLETS 56
4.10 PRIME QUADRUPLETS AND QUINTUPLETS 57
4.11 PRIMES AT ANY DISTANCE 58
4.12 SPACING DISTRIBUTION BETWEEN ADJACENT PRIMES 61
4.13 GOLDBACH'S CONJECTURE 61
4.14 SUM OF THREE PRIMES 63
PART II. SOM
E SIMPL
E APPLICATION
S
5. FRACTIONS
: CONTINUED
, EGYPTIA
N AN
D FAREY
65
5.1 A NEGLECTED SUBJECT 65
5.2 RELATIONS WITH MEASURE THEORY 69
5.3 PERIODIC CONTINUED FRACTIONS 70
5.4 ELECTRICAL NETWORKS AND SQUARED SQUARES 73
5.5 FIBONACCI NUMBERS AND THE GOLDEN RATIO 74
5.6 FIBONACCI, RABBITS AND COMPUTERS 78
5.7 FIBONACCI AND DIVISIBILITY 81
5.8 GENERALIZED FIBONACCI AND LUCAS NUMBERS 81
5.9 EGYPTIAN FRACTIONS, INHERITANCE
AND SOME UNSOLVED PROBLEMS 85
5.10 FAREY FRACTIONS 86
5.10.1 FAREY TREES 88
5.10.2 LOCKED PALLAS 92
5.11 FIBONACCI AND TH
E PROBLEM OF BANK DEPOSITS 93
5.12 ERROR-FREE COMPUTING 94
PART III
. CONGRUENCE
S AN
D TH
E LIKE
6. LINEAR CONGRUENCE
S
99
6.1 RESIDUES 99
6.2 SOME SIMPLE FIELDS 102
6.3 POWERS AND CONGRUENCES 103
7. DIOPHANTIN
E EQUATION
S
106
7.1 RELATION WITH CONGRUENCES 106
7.2 A GAUSSIAN TRICK 107
7.3 NONLINEAR DIOPHANTINE EQUATIONS 109
CONTENTS XXI
7.4 TRIANGULAR NUMBERS 110
7.5 PYTHAGOREAN NUMBERS 112
7.6 EXPONENTIAL DIOPHANTINE EQUATIONS 113
7.7 FORMAT'S LAST "THEOREM" 113
7.8 THE DEMISE OF A CONJECTURE BY EULER 115
7.9 A NONLINEAR DIOPHANTINE EQUATION IN PHYSICS
AND THE GEOMETRY OF NUMBERS 116
7.10 NORMAL-MODE DEGENERACY IN ROOM ACOUSTICS
(A NUMBER-THEORETIC APPLICATION) 120
7.11 WARING'S PROBLEM 121
8. TH
E THEOREM
S OF FERMAT, WILSO
N AND EULER
122
8.1 FERMAT'S THEOREM 122
8.2 WILSON'S THEOREM 123
8.3 EULER'S THEOREM 124
8.4 THE IMPOSSIBLE STAR OF DAVID 125
8.5 DIRICHLET AND LINEAR PROGRESSION 127
PART IV
. CRYPTOGRAPHY AND DIVISOR
S
9. EULER TRAP DOOR
S AND PUBLIC-KE
Y ENCRYPTIO
N
129
9.1 A NUMERICAL TRAP DOOR 131
9.2 DIGITAL ENCRYPTION 132
9.3 PUBLIC-KEY ENCRYPTION 133
9.4 A SIMPLE EXAMPLE 135
9.5 REPEATED ENCRYPTION 136
9.6 SUMMARY AND ENCRYPTION REQUIREMENTS 137
10. TH
E DIVISOR FUNCTIONS
139
10.1 THE NUMBER OF DIVISORS 139
10.2 THE AVERAGE OF THE DIVISOR FUNCTION 142
10.3 THE GEOMETRIC MEAN OF THE DIVISORS 142
10.4 THE SUMMATORY FUNCTION OF THE DIVISOR FUNCTION 143
10.5 THE GENERALIZED DIVISOR FUNCTIONS 143
10.6 THE AVERAGE VALUE OF EULER'S FUNCTION 144
11
. TH
E PRIM
E DIVISOR FUNCTIONS
146
11.1 THE NUMBER OF DIFFERENT PRIME DIVISORS 146
11.2 THE DISTRIBUTION OF
U(N)
150
11.3 THE NUMBER OF PRIME DIVISORS 151
11.4 THE HARMONIC MEAN OF
Q(N)
154
11.5 MEDIANS AND PERCENTILES OF
Q(N)
156
11.6 IMPLICATIONS FOR PUBLIC-KEY ENCRYPTION 157
XXII CONTENTS
12. CERTIFIED SIGNATURE
S
158
12.1 A STORY OF CREATIVE FINANCING 158
12.2 CERTIFIED SIGNATURE FOR PUBLIC-KEY ENCRYPTION 158
13. PRIMITIV
E ROOT
S
160
13.1 ORDERS 160
13.2 PERIODS OF DECIMAL AND BINARY FRACTIONS 163
13.3 A PRIMITIVE PROOF OF WILSON'S THEOREM 166
13.4 THE INDEX - A NUMBER-THEORETIC LOGARITHM 166
13.5 SOLUTION OF EXPONENTIAL CONGRUENCES 167
13.6 WHAT IS TH
E ORDER
T
M
OF AN INTEGER
M
MODULO A PRIME P? . . 169
13.7 INDEX "ENCRYPTION" 170
13.8 A FOURIER PROPERTY OF PRIMITIVE ROOTS
AND CONCERT HALL ACOUSTICS 170
13.9 MORE SPACIOUS-SOUNDING SOUND 172
13.10 GALOIS ARRAYS FOR X-RAY ASTRONOMY 174
13.11 A NEGATIVE PROPERTY OF TH
E FERMAT PRIMES 175
14. KNAPSACK ENCRYPTIO
N
177
14.1 AN EASY KNAPSACK 177
14.2 A HARD KNAPSACK 178
PAR
T V
. RESIDUE
S AND DIFFRACTION
15. QUADRATI
C RESIDUE
S
181
15.1 QUADRATIC CONGRUENCES 181
15.2 EULER'S CRITERION 182
15.3 THE LEGENDRE SYMBOL 183
15.4 A FOURIER PROPERTY OF LEGENDRE SEQUENCES 185
15.5 GAUSS SUMS 185
15.6 PRETT
Y DIFFRACTION 187
15.7 QUADRATIC RECIPROCITY 187
15.8 A FOURIER PROPERTY OF QUADRATIC-RESIDUE SEQUENCES 188
15.9 SPREAD SPECTRUM COMMUNICATION 190
15.10 GENERALIZED LEGENDRE SEQUENCES OBTAINED
THROUGH COMPLEXIFICATION OF THE EULER CRITERION 191
PAR
T VI
. CHINES
E AND OTHE
R FAST ALGORITHM
S
16. TH
E CHINES
E REMAINDE
R THEORE
M
AND SIMULTANEOU
S CONGRUENCE
S
194
16.1 SIMULTANEOUS CONGRUENCES 194
16.2 THE SINO-REPRESENTATION: A CHINESE NUMBER SYSTEM 195
CONTENTS XXIII
16.3 APPLICATIONS OF THE SINO-REPRESENTATION 196
16.4 DISCRETE FOURIER TRANSFORMATION IN SINO 198
16.5 A SINO-OPTICAL FOURIER TRANSFORMER 199
16.6 GENERALIZED SINO-REPRESENTATION 200
16.7 FAST PRIME-LENGTH FOURIER TRANSFORM 201
17. FAST TRANSFORMATIO
N AN
D KRONECKE
R PRODUCT
S
203
17.1 A FAST HADAMARD TRANSFORM 203
17.2 THE BASIC PRINCIPLE OF THE FAST FOURIER TRANSFORMS 206
18. QUADRATI
C CONGRUENCE
S
207
18.1 APPLICATION OF THE CHINESE REMAINDER THEOREM (CRT) 207
PAR
T VII
. PSEUDOPRIMES
, MOBIU
S TRANSFORM, AND PARTITION
S
19. PSEUDOPRIMES
, POKE
R AND REMOT
E COIN TOSSIN
G
209
19.1 PULLING ROOTS TO FERRET OUT COMPOSITES 209
19.2 FACTORS FROM A SQUARE ROOT 210
19.3 COIN TOSSING BY TELEPHONE 212
19.4 ABSOLUTE AND STRONG PSEUDOPRIMES 214
19.5 FERMAT AND STRONG PSEUDOPRIMES 216
19.6 DETERMINISTIC PRIMALITY TESTING 216
19.7 A VERY SIMPLE FACTORING ALGORITHM 218
19.8 FACTORING WITH ELLIPTIC CURVES 218
19.9 QUANTUM FACTORING 219
20. TH
E MOBIU
S FUNCTIO
N AND TH
E MOBIU
S TRANSFORM
220
20.1 THE MOBIUS TRANSFORM AND ITS INVERSE 220
20.2 PROOF OF THE INVERSION FORMULA 222
20.3 SECOND INVERSION FORMULA 223
20.4 THIRD INVERSION FORMULA 223
20.5 FOURTH INVERSION FORMULA 224
20.6 RIEMANN'S HYPOTHESIS
AND THE DISPROOF OF THE MERTENS CONJECTURE 224
20.7 DIRICHLET SERIES AND THE MOBIUS FUNCTION 225
21
. GENERATIN
G FUNCTION
S AND PARTITION
S
228
21.1 GENERATING FUNCTIONS 228
21.2 PARTITIONS OF INTEGERS 230
21.3 GENERATING FUNCTIONS OF PARTITIONS 231
21.4 RESTRICTED PARTITIONS 232
XXIV CONTENTS
PART VIII. CYCLOTOM
Y AND POLYNOMIAL
S
22. CYCLOTOMIC POLYNOMIAL
S
236
22.1 HOW TO DIVIDE A CIRCLE INTO EQUAL PARTS 236
22.2 GAUSS'S GREAT INSIGHT 239
22.3 FACTORING IN DIFFERENT FIELDS 243
22.4 CYCLOTOMY IN THE COMPLEX PLANE 243
22.5 HOW TO DIVIDE A CIRCLE WITH COMPASS AND STRAIGHTEDGE 244
22.5.1 RATIONAL FACTORS OF
Z
N
-
1 246
22.6 AN ALTERNATIVE RATIONAL FACTORIZATION 247
22.7 RELATION BETWEEN RATIONAL FACTORS AND COMPLEX ROOTS 248
22.8 HOW TO CALCULATE WITH CYCLOTOMIC POLYNOMIALS 249
23. LINEAR SYSTEM
S AND POLYNOMIAL
S
251
23.1 IMPULSE RESPONSES 251
23.2 TIME-DISCRETE SYSTEMS AND THE Z TRANSFORM 252
23.3 DISCRETE CONVOLUTION 252
23.4 CYCLOTOMIC POLYNOMIALS AND Z TRANSFORM 253
24. POLYNOMIA
L THEOR
Y
254
24.1 SOME BASIC FACTS OF POLYNOMIAL LIFE 254
24.2 POLYNOMIAL RESIDUES 255
24.3 CHINESE REMAINDERS FOR POLYNOMIALS 256
24.4 EUCLID'S ALGORITHM FOR POLYNOMIALS 257
PART IX. GALOIS FIELD
S AND MORE APPLICATION
S
25. GALOIS FIELD
S
260
25.1 PRIME ORDER 260
25.2 PRIME POWER ORDER 260
25.3 GENERATION OF GF(2
4
) 262
25.4 HOW MANY PRIMITIVE ELEMENTS? 264
25.5 RECURSIVE RELATIONS 264
25.6 HOW TO CALCULATE IN GF(P
M
) 266
25.7 ZECH LOGARITHM, DOPPLER RADAR
AND OPTIMUM AMBIGUITY FUNCTIONS 267
25.8 A UNIQUE PHASE-ARRAY BASED ON THE ZECH LOGARITHM 270
25.9 SPREAD-SPECTRUM COMMUNICATION AND ZECH LOGARITHMS 272
26. SPECTRAL PROPERTIE
S OF GALOIS SEQUENCE
S
273
26.1 CIRCULAR CORRELATION 273
26.2 APPLICATION TO ERROR-CORRECTING CODES
AND SPEECH RECOGNITION 275
CONTENTS XXV
26.3 APPLICATION TO PRECISION MEASUREMENTS 277
26.4 CONCERT HALL MEASUREMENTS 278
26.5 THE FOURTH EFFECT OF GENERAL RELATIVITY 279
26.6 TOWARD BETTOR CONCERT HALL ACOUSTICS 280
26.7 HIGHER-DIMENSIONAL DIFFUSORS 285
26.8 ACTIVE ARRAY APPLICATIONS 286
27. RANDO
M NUMBE
R GENERATOR
S
287
27.1 PSEUDORANDOM GALOIS SEQUENCES 288
27.2 RANDOMNESS FROM CONGRUENCES 289
27.3 "CONTINUOUS" DISTRIBUTIONS 290
27.4 FOUR WAYS TO GENERATE A GAUSSIAN VARIABLE 291
27.5 PSEUDORANDOM SEQUENCES IN CRYPTOGRAPHY 292
28. WAVEFORMS AND RADIATIO
N PATTERN
S
293
28.1 SPECIAL PHASES 294
28.2 THE RUDIN-SHAPIRO POLYNOMIALS 296
28.3 GAUSS SUMS AND PEAK FACTORS 297
28.4 GALOIS SEQUENCES AND THE SMALLEST PEAK FACTORS 299
28.5 MINIMUM REDUNDANCY ANTENNAS 301
28.6 GOLOMB RULERS 303
29. NUMBE
R THEORY, RANDOMNES
S AND "ART"
305
29.1 NUMBER THEORY AND GRAPHIC DESIGN 305
29.2 THE PRIMES OF GAUSS AND EISENSTEIN 307
29.3 GALOIS FIELDS AND IMPOSSIBLE NECKLACES 308
29.4 "BAROQUE" INTEGERS 312
PART X
. SELF-SIMILARITY, FRACTALS AND ART
30
. SELF-SIMILARITY, FRACTALS, DETERMINISTI
C CHAOS
AND A NE
W STAT
E OF MATTE
R
315
30.1 FIBONACCI, NOBLE NUMBERS AND A NEW STATE OF MATTER 318
30.2 CANTOR SETS, FRACTALS AND A MUSICAL PARADOX 324
30.3 THE TWIN DRAGON:
A FRACTAL FROM A COMPLEX NUMBER SYSTEM 329
30.4 STATISTICAL FRACTALS 331
30.5 SOME CRAZY MAPPINGS 333
30.6 THE LOGISTIC PARABOLA AND STRANGE ATTRACTORS 336
30.7 CONCLUSION 339 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Schröder, Manfred |
| author_facet | Schröder, Manfred |
| author_role | aut |
| author_sort | Schröder, Manfred |
| author_variant | m s ms |
| building | Verbundindex |
| bvnumber | BV021247785 |
| classification_tum | MAT 100f |
| ctrlnum | (OCoLC)66425112 (DE-599)BVBBV021247785 |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| edition | 4. ed. |
| format | Book |
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| id | DE-604.BV021247785 |
| illustrated | Illustrated |
| index_date | 2024-07-02T13:38:36Z |
| indexdate | 2024-07-09T20:33:47Z |
| institution | BVB |
| isbn | 9783540265962 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014569242 |
| oclc_num | 66425112 |
| open_access_boolean | |
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| owner_facet | DE-29T DE-91G DE-BY-TUM DE-11 |
| physical | XXII, 363 S. Ill., graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Springer |
| record_format | marc |
| series | Springer series in information sciences |
| series2 | Springer series in information sciences |
| spelling | Schröder, Manfred Verfasser aut Number theory in science and communication with applications in cryptography, physics, digital information, computing, and self-similarity M. R. Schroeder 4. ed. Berlin [u.a.] Springer 2005 XXII, 363 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in information sciences 7 Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Anwendung (DE-588)4196864-5 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s Anwendung (DE-588)4196864-5 s 1\p DE-604 Springer series in information sciences 7 (DE-604)BV000008063 7 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2669071&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014569242&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 14 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Schröder, Manfred Number theory in science and communication with applications in cryptography, physics, digital information, computing, and self-similarity Springer series in information sciences Zahlentheorie (DE-588)4067277-3 gnd Anwendung (DE-588)4196864-5 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4196864-5 |
| 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_exact_search_txtP | 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 | Zahlentheorie (DE-588)4067277-3 gnd Anwendung (DE-588)4196864-5 gnd |
| topic_facet | Zahlentheorie Anwendung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2669071&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014569242&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000008063 |
| work_keys_str_mv | AT schrodermanfred numbertheoryinscienceandcommunicationwithapplicationsincryptographyphysicsdigitalinformationcomputingandselfsimilarity |