An introduction to number theory with cryptography:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton
CRC Press, Taylor & Francis Group
[2018]
|
| Edition: | Second edition |
| Series: | Textbooks in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | "A Chapman & Hall book.". - Includes index |
| Physical Description: | xxii, 578 pages Illustrationen 24 cm |
| ISBN: | 9781138063471 |
Staff View
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV044918877 | ||
| 003 | DE-604 | ||
| 005 | 20180524 | ||
| 007 | t | ||
| 008 | 180424s2018 xxua||| |||| 00||| eng d | ||
| 010 | |a 017044405 | ||
| 020 | |a 9781138063471 |9 978-1-138-06347-1 | ||
| 035 | |a (OCoLC)1030344174 | ||
| 035 | |a (DE-599)BVBBV044918877 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 044 | |a xxu |c US | ||
| 049 | |a DE-739 | ||
| 050 | 0 | |a QA241 | |
| 082 | 0 | |a 512.7 |2 23 | |
| 084 | |a SK 180 |0 (DE-625)143222: |2 rvk | ||
| 100 | 1 | |a Kraft, James S. |4 aut | |
| 245 | 1 | 0 | |a An introduction to number theory with cryptography |c James S. Kraft, Lawrence C. Washington |
| 250 | |a Second edition | ||
| 264 | 1 | |a Boca Raton |b CRC Press, Taylor & Francis Group |c [2018] | |
| 300 | |a xxii, 578 pages |b Illustrationen |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Textbooks in mathematics | |
| 500 | |a "A Chapman & Hall book.". - Includes index | ||
| 650 | 4 | |a Number theory | |
| 650 | 4 | |a Cryptography | |
| 650 | 0 | 7 | |a Zahlentheorie |0 (DE-588)4067277-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Kryptologie |0 (DE-588)4033329-2 |2 gnd |9 rswk-swf |
| 655 | 7 | |8 1\p |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
| 689 | 0 | 0 | |a Zahlentheorie |0 (DE-588)4067277-3 |D s |
| 689 | 0 | 1 | |a Kryptologie |0 (DE-588)4033329-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Washington, Lawrence C. |d 1951- |0 (DE-588)1033730076 |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030312215&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030312215&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-030312215 | ||
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
Record in the Search Index
| _version_ | 1804178486513893376 |
|---|---|
| adam_text | Contents
Preface xix
1 Introduction 1
1.1 Diophantine Equations .................................. 2
1.2 Modular Arithmetic...................................... 4
1.3 Primes and the Distribution of Primes................... 5
1.4 Cryptography ........................................... 7
2 Divisibility 9
2.1 Divisibility ........................................... 9
2.2 Euclid’s Theorem ...................................... 11
2.3 Euclid’s Original Proof ............................... 13
2.4 The Sieve of Eratosthenes.............................. 15
2.5 The Division Algorithm................................. 17
2.5.1 A Cryptographic Application..................... 19
2.6 The Greatest Common Divisor............................ 20
2.7 The Euclidean Algorithm................................ 23
2.7.1 The Extended Euclidean Algorithm ........... 25
2.8 Other Bases ........................................... 31
2.9 Fermat and Mersenne Numbers ................ 34
2.10 Chapter Highlights .................................... 38
2.11 Problems .............................................. 38
2.11.1 Exercises ...................................... 38
2.11.2 Projects........................................ 45
2.11.3 Computer Explorations........................... 47
2.11.4 Answers to “Check Your Understanding” .... 48
IX
X
Contents
3 Linear Diophantine Equations 51
3.1 ax + by = c............................................ 51
3.2 The Postage Stamp Problem.............................. 57
3.3 Chapter Highlights .................................... 60
3.4 Problems............................................... 60
3.4.1 Exercises....................................... 60
3.4.2 Answers to “Check Your Understanding” .... 62
4 Unique Factorization 63
4.1 The Starting Point..................................... 63
4.2 The Fundamental Theorem of Arithmetic.................. 64
4.3 Euclid and the Fundamental Theorem of Arithmetic . . 69
4.4 Chapter Highlights .................................... 70
4.5 Problems ............................................. 71
4.5.1 Exercises....................................... 71
4.5.2 Projects........................................ 73
4.5.3 Answers to “Check Your Understanding” .... 73
5 Applications of Unique Factorization 75
5.1 A Puzzle .............................................. 75
5.2 Irrationality Proofs .................................. 77
5.2.1 Four More Proofs That a/2 Is Irrational......... 79
5.3 The Rational Root Theorem ............................. 81
5.4 Pythagorean Triples ................................... 84
5.5 Differences of Squares................................. 90
5.6 Prime Factorization of Factorials...................... 92
5.7 The Riemann Zeta Function ............................. 94
5.7.1 T l/P Diverges..................................100
5.8 Chapter Highlights ....................................105
5.9 Problems ..............................................106
5.8.1 Exercises ......................................106
5.9.2 Projects........................................108
5.9.3 Computer Explorations...........................112
XI
5.9.4 Answers to “Check Your Understanding” .... 112
6 Congruences 113
6.1 Definitions and Examples..............................113
6.2 Modular Exponentiation ...............................122
6.3 Divisibility Tests....................................124
6.4 Linear Congruences ...................................129
6.5 The Chinese Remainder Theorem ........................136
6.6 Fractions mod m.......................................141
6.7 Queens on a Chessboard ...............................143
6.8 Chapter Highlights ...................................145
6.9 Problems ........................................... 145
6.9.1 Exercises .....................................145
6.9.2 Projects.......................................152
6.9.3 Computer Explorations..........................153
6.9.4 Answers to “Check Your Understanding” .... 154
7 Classical Cryptosystems 155
7.1 Introduction ...................................... . 155
7.2 Shift and Affine Ciphers..............................156
7.3 Vigenère Ciphers......................................161
7.4 Transposition Ciphers.................................167
7.5 Stream Ciphers .......................................170
7.5.1 One-Time Pad.................................. 171
7.5.2 Linear Feedback Shift Registers (LFSR).........172
7.6 Block Ciphers ........................................175
7.7 Secret Sharing .......................................179
7.8 Generating Random Numbers ............................181
7.9 Chapter Highlights ...................................183
7.10 Problems .............................................183
7.10.1 Exercises .....................................183
7.10.2 Answers to “Check Your Understanding” .... 186
xii Contents
8 Fermat, Euler, and Wilson 189
8.1 Fermat’s Theorem......................................189
8.2 Euler’s Theorem.......................................194
8.3 Wilson’s Theorem .....................................200
8.4 Chapter Highlights ...................................202
8.5 Problems .............................................203
8.5.1 Exercises.................................203
8.5.2 Projects..................................206
8.5.3 Computer Explorations.....................207
8.5.4 Answers to “Check Your Understanding” .... 207
9 RSA 209
9.1 RSA Encryption........................................210
9.2 Digital Signatures ...................................217
9.3 Chapter Highlights ...................................219
9.4 Problems..............................................219
9.3.1 Exercises.................................219
9.4.2 Projects..................................224
9.4.3 Computer Explorations.....................225
9.4.4 Answers to “Check Your Understanding” .... 226
10 Polynomial Congruences 227
10.1 Polynomials Mod Primes ...............................227
10.2 Solutions Modulo Prime Powers ........................230
10.3 Composite Moduli......................................234
10.4 Chapter Highlights ...................................235
10.5 Problems..............................................235
10.4.1 Exercises .................................... 235
10.5.2 Projects..................................236
10.5.3 Computer Explorations.....................237
10.5.4 Answers to “Check Your Understanding” .... 238
Xlll
11 Order and Primitive Roots 239
11.1 Orders of Elements ..................................239
11.1.1 Fermat Numbers................................241
11.1.2 Mersenne Numbers..............................243
11.2 Primitive Roots .....................................244
11.3 Decimals ............................................250
11.3.1 Midy’s Theorem................................253
11.4 Card Shuffling ......................................255
11.5 The Discrete Log Problem ............................257
11.5.1 Baby Step-Giant Step Method...................258
11.5.2 The Index Calculus ...........................260
11.6 Existence of Primitive Roots ........................263
11.7 Chapter Highlights ..................................266
11.8 Problems.............................................266
11.7.1 Exercises ....................................266
11.8.2 Projects.................................... 269
11.8.3 Computer Explorations.........................271
11.8.4 Answers to “Check Your Understanding” .... 271
12 More Cryptographic Applications 273
12.1 Diffie-Hellman Key Exchange .........................273
12.2 Coin Flipping over the Telephone ....................275
12.3 Mental Poker.........................................277
12.4 The ElGamal Public Key Cryptosystem..................282
12.5 Chapter Highlights ..................................285
12.6 Problems ............................................285
12.6.1 Exercises ....................................285
12.6.2 Projects......................................287
12.6.3 Computer Explorations.........................287
12.6.4 Answers to “Check Your Understanding” .... 288
XIV
Contents
13 Quadratic Reciprocity 289
13.1 Squares and Square Roots Mod Primes ..................289
13.2 Computing Square Roots Mod p..........................296
13.3 Quadratic Equations ..................................298
13.4 The Jacobi Symbol ....................................300
13.5 Proof of Quadratic Reciprocity........................305
13.6 Chapter Highlights ...................................312
13.7 Problems..............................................312
13.7.1 Exercises .....................................312
13.7.2 Projects.......................................316
13.7.3 Answers to “Check Your Understanding” .... 318
14 Primality and Factorization 319
14.1 Trial Division and Fermat Factorization...............319
14.2 Primality Testing ....................................323
14.2.1 Pseudoprimes...................................323
14.2.2 The Pocklington—Lehmer Primality Test.........328
14.2.3 The AKS Primality Test.........................331
14.2.4 Fermat Numbers.................................333
14.2.5 Mersenne Numbers...............................335
14.3 Factorization.........................................335
14.3.1 x2 = y2........................336
14.3.2 Factoring Pseudoprimes and Factoring Using
RSA Exponents................................339
14.3.3 Pollard’s p — 1 Method.......................340
14.3.4 The Quadratic Sieve............................342
14.4 Coin Flipping over the Telephone .....................350
14.5 Chapter Highlights ...................................352
14.6 Problems .............................................352
14.6.1 Exercises .....................................352
14.6.2 Projects.......................................355
14.6.3 Computer Explorations..........................355
14.6.4 Answers to “Check Your Understanding” .... 356
XV
15 Geometry of Numbers 357
15.1 Volumes and Minkowski’s Theorem ....................357
15.2 Sums of Two Squares ................................362
15.2.1 Algorithm for Writing = 1 (mod 4) as a Sum
of Two Squares................................366
15.3 Sums of Four Squares.................................368
15.4 Pell’s Equation......................................370
15.4.1 Bhaskara’s Chakravala Method..................373
15.5 Chapter Highlights ..............................376
15.6 Problems.............................................376
15.6.1 Exercises ....................................376
15.6.2 Projects......................................380
15.6.3 Answers to “Check Your Understanding” .... 384
16 Arithmetic Functions 385
16.1 Perfect Numbers..................................... 385
16.2 Multiplicative Functions .......................... 389
16.3 Chapter Highlights ..............................395
16.4 Problems.............................................395
16.3.1 Exercises.....................................395
16.4.2 Projects......................................397
16.4.3 Computer Explorations.........................398
16.4.4 Answers to “Check Your Understanding” .... 399
17 Continued Fractions 401
17.1 Rational Approximations; Pell’s Equation............402
17.1.1 Evaluating Continued Fractions................405
17.1.2 Pell’s Equation...............................407
17.2 Basic Theory.........................................410
17.3 Rational Numbers.....................................418
17.4 Periodic Continued Fractions.........................420
17.4.1 Purely Periodic Continued Fractions...........422
17.4.2 Eventually Periodic Continued Fractions.......427
xvi Contents
17.5 Square Roots of Integers ............................ 429
17.6 Some Irrational Numbers............................... 432
17.7 Chapter Highlights ....................................438
17.8 Problems ..............................................438
17.8.1 Exercises..................................438
17.8.2 Projects...................................439
17.8.3 Computer Explorations......................441
17.8.4 Answers to “Check Your Understanding” .... 441
18 Gaussian Integers 443
18.1 Complex Arithmetic.....................................443
18.2 Gaussian Irreducibles .................................445
18.3 The Division Algorithm.................................449
18.4 Unique Factorization ..................................452
18.5 Applications ..........................................458
18.5.1 Sums of Two Squares .......................... 458
18.5.2 Pythagorean Triples...................... 461
18.5.3 y2 = x3- 1.....................462
18.6 Chapter Highlights .................................. 464
18.7 Problems............................................. 464
18.7.1 Exercises ......................................464
18.7.2 Projects........................................465
18.7.3 Computer Explorations......................465
18.7.4 Answers to “Check Your Understanding” .... 465
19 Algebraic Integers 467
19.1 Quadratic Fields and Algebraic Integers................467
19.2 Units ............................................... 472
19.3 Z[y/=2]................................................476
19.4 Z[VS].......................................479
19.4.1 The Lucas-Lehmer Test......................482
19.5 Non-Unique Factorization...............................486
19.6 Chapter Highlights ....................................488
XVII
19.7 Problems ............................................488
19.7.1 Exercises ....................................488
19.7.2 Projects.................................489
19.7.3 Answers to “Check Your Understanding” .... 491
20 The Distribution of Primes 493
20.1 Bertrand’s Postulate.................................493
20.2 Chebyshev’s Approximate Prime Number Theorem . . 502
20.3 Chapter Highlights ..................................507
20.4 Problems ............................................508
20.4.1 Exercises ....................................508
20.4.2 Projects.................................509
20.4.3 Computer Explorations....................510
21 Epilogue: Fermat’s Last Theorem 511
21.1 Introduction ........................................511
21.2 Elliptic Curves......................................514
21.3 Modularity...........................................517
A Supplementary Topics 521
A.l What Is a Proof? .....................................521
A.1.1 Proof by Contradiction .............. 527
A.2 Geometric Series......................................530
A.3 Mathematical Induction.............................. 531
A.4 Pascal’s Triangle and the Binomial Theorem ...........537
A.5 Fibonacci Numbers ....................................543
A.6 Matrices .............................................546
A.7 Problems .............................................550
A. 7.1 Exercises ....................................550
A.7.2 Answers to “Check Your Understanding” .... 552
B Answers and Hints for Odd-Numbered Exercises 555
Index
573
Contents
Preface xix
1 Introduction 1
1.1 Diophantine Equations .................................. 2
1.2 Modular Arithmetic...................................... 4
1.3 Primes and the Distribution of Primes................... 5
1.4 Cryptography ........................................... 7
2 Divisibility 9
2.1 Divisibility ........................................... 9
2.2 Euclid’s Theorem ...................................... 11
2.3 Euclid’s Original Proof ............................... 13
2.4 The Sieve of Eratosthenes.............................. 15
2.5 The Division Algorithm................................. 17
2.5.1 A Cryptographic Application..................... 19
2.6 The Greatest Common Divisor............................ 20
2.7 The Euclidean Algorithm................................ 23
2.7.1 The Extended Euclidean Algorithm ........... 25
2.8 Other Bases ........................................... 31
2.9 Fermat and Mersenne Numbers ................ 34
2.10 Chapter Highlights .................................... 38
2.11 Problems .............................................. 38
2.11.1 Exercises ...................................... 38
2.11.2 Projects........................................ 45
2.11.3 Computer Explorations........................... 47
2.11.4 Answers to “Check Your Understanding” .... 48
IX
X
Contents
3 Linear Diophantine Equations 51
3.1 ax + by = c............................................ 51
3.2 The Postage Stamp Problem.............................. 57
3.3 Chapter Highlights .................................... 60
3.4 Problems............................................... 60
3.4.1 Exercises....................................... 60
3.4.2 Answers to “Check Your Understanding” .... 62
4 Unique Factorization 63
4.1 The Starting Point..................................... 63
4.2 The Fundamental Theorem of Arithmetic.................. 64
4.3 Euclid and the Fundamental Theorem of Arithmetic . . 69
4.4 Chapter Highlights .................................... 70
4.5 Problems ............................................. 71
4.5.1 Exercises....................................... 71
4.5.2 Projects........................................ 73
4.5.3 Answers to “Check Your Understanding” .... 73
5 Applications of Unique Factorization 75
5.1 A Puzzle .............................................. 75
5.2 Irrationality Proofs .................................. 77
5.2.1 Four More Proofs That a/2 Is Irrational......... 79
5.3 The Rational Root Theorem ............................. 81
5.4 Pythagorean Triples ................................... 84
5.5 Differences of Squares................................. 90
5.6 Prime Factorization of Factorials...................... 92
5.7 The Riemann Zeta Function ............................. 94
5.7.1 T l/P Diverges..................................100
5.8 Chapter Highlights ....................................105
5.9 Problems ..............................................106
5.8.1 Exercises ......................................106
5.9.2 Projects........................................108
5.9.3 Computer Explorations...........................112
XI
5.9.4 Answers to “Check Your Understanding” .... 112
6 Congruences 113
6.1 Definitions and Examples..............................113
6.2 Modular Exponentiation ...............................122
6.3 Divisibility Tests....................................124
6.4 Linear Congruences ...................................129
6.5 The Chinese Remainder Theorem ........................136
6.6 Fractions mod m.......................................141
6.7 Queens on a Chessboard ...............................143
6.8 Chapter Highlights ...................................145
6.9 Problems ........................................... 145
6.9.1 Exercises .....................................145
6.9.2 Projects.......................................152
6.9.3 Computer Explorations..........................153
6.9.4 Answers to “Check Your Understanding” .... 154
7 Classical Cryptosystems 155
7.1 Introduction ...................................... . 155
7.2 Shift and Affine Ciphers..............................156
7.3 Vigenère Ciphers......................................161
7.4 Transposition Ciphers.................................167
7.5 Stream Ciphers .......................................170
7.5.1 One-Time Pad.................................. 171
7.5.2 Linear Feedback Shift Registers (LFSR).........172
7.6 Block Ciphers ........................................175
7.7 Secret Sharing .......................................179
7.8 Generating Random Numbers ............................181
7.9 Chapter Highlights ...................................183
7.10 Problems .............................................183
7.10.1 Exercises .....................................183
7.10.2 Answers to “Check Your Understanding” .... 186
xii Contents
8 Fermat, Euler, and Wilson 189
8.1 Fermat’s Theorem......................................189
8.2 Euler’s Theorem.......................................194
8.3 Wilson’s Theorem .....................................200
8.4 Chapter Highlights ...................................202
8.5 Problems .............................................203
8.5.1 Exercises.................................203
8.5.2 Projects..................................206
8.5.3 Computer Explorations.....................207
8.5.4 Answers to “Check Your Understanding” .... 207
9 RSA 209
9.1 RSA Encryption........................................210
9.2 Digital Signatures ...................................217
9.3 Chapter Highlights ...................................219
9.4 Problems..............................................219
9.3.1 Exercises.................................219
9.4.2 Projects..................................224
9.4.3 Computer Explorations.....................225
9.4.4 Answers to “Check Your Understanding” .... 226
10 Polynomial Congruences 227
10.1 Polynomials Mod Primes ...............................227
10.2 Solutions Modulo Prime Powers ........................230
10.3 Composite Moduli......................................234
10.4 Chapter Highlights ...................................235
10.5 Problems..............................................235
10.4.1 Exercises .................................... 235
10.5.2 Projects..................................236
10.5.3 Computer Explorations.....................237
10.5.4 Answers to “Check Your Understanding” .... 238
Xlll
11 Order and Primitive Roots 239
11.1 Orders of Elements ..................................239
11.1.1 Fermat Numbers................................241
11.1.2 Mersenne Numbers..............................243
11.2 Primitive Roots .....................................244
11.3 Decimals ............................................250
11.3.1 Midy’s Theorem................................253
11.4 Card Shuffling ......................................255
11.5 The Discrete Log Problem ............................257
11.5.1 Baby Step-Giant Step Method...................258
11.5.2 The Index Calculus ...........................260
11.6 Existence of Primitive Roots ........................263
11.7 Chapter Highlights ..................................266
11.8 Problems.............................................266
11.7.1 Exercises ....................................266
11.8.2 Projects.................................... 269
11.8.3 Computer Explorations.........................271
11.8.4 Answers to “Check Your Understanding” .... 271
12 More Cryptographic Applications 273
12.1 Diffie-Hellman Key Exchange .........................273
12.2 Coin Flipping over the Telephone ....................275
12.3 Mental Poker.........................................277
12.4 The ElGamal Public Key Cryptosystem..................282
12.5 Chapter Highlights ..................................285
12.6 Problems ............................................285
12.6.1 Exercises ....................................285
12.6.2 Projects......................................287
12.6.3 Computer Explorations.........................287
12.6.4 Answers to “Check Your Understanding” .... 288
XIV
Contents
13 Quadratic Reciprocity 289
13.1 Squares and Square Roots Mod Primes ..................289
13.2 Computing Square Roots Mod p..........................296
13.3 Quadratic Equations ..................................298
13.4 The Jacobi Symbol ....................................300
13.5 Proof of Quadratic Reciprocity........................305
13.6 Chapter Highlights ...................................312
13.7 Problems..............................................312
13.7.1 Exercises .....................................312
13.7.2 Projects.......................................316
13.7.3 Answers to “Check Your Understanding” .... 318
14 Primality and Factorization 319
14.1 Trial Division and Fermat Factorization...............319
14.2 Primality Testing ....................................323
14.2.1 Pseudoprimes...................................323
14.2.2 The Pocklington—Lehmer Primality Test.........328
14.2.3 The AKS Primality Test.........................331
14.2.4 Fermat Numbers.................................333
14.2.5 Mersenne Numbers...............................335
14.3 Factorization.........................................335
14.3.1 x2 = y2........................336
14.3.2 Factoring Pseudoprimes and Factoring Using
RSA Exponents................................339
14.3.3 Pollard’s p — 1 Method.......................340
14.3.4 The Quadratic Sieve............................342
14.4 Coin Flipping over the Telephone .....................350
14.5 Chapter Highlights ...................................352
14.6 Problems .............................................352
14.6.1 Exercises .....................................352
14.6.2 Projects.......................................355
14.6.3 Computer Explorations..........................355
14.6.4 Answers to “Check Your Understanding” .... 356
XV
15 Geometry of Numbers 357
15.1 Volumes and Minkowski’s Theorem ....................357
15.2 Sums of Two Squares ................................362
15.2.1 Algorithm for Writing = 1 (mod 4) as a Sum
of Two Squares................................366
15.3 Sums of Four Squares.................................368
15.4 Pell’s Equation......................................370
15.4.1 Bhaskara’s Chakravala Method..................373
15.5 Chapter Highlights ..............................376
15.6 Problems.............................................376
15.6.1 Exercises ....................................376
15.6.2 Projects......................................380
15.6.3 Answers to “Check Your Understanding” .... 384
16 Arithmetic Functions 385
16.1 Perfect Numbers..................................... 385
16.2 Multiplicative Functions .......................... 389
16.3 Chapter Highlights ..............................395
16.4 Problems.............................................395
16.3.1 Exercises.....................................395
16.4.2 Projects......................................397
16.4.3 Computer Explorations.........................398
16.4.4 Answers to “Check Your Understanding” .... 399
17 Continued Fractions 401
17.1 Rational Approximations; Pell’s Equation............402
17.1.1 Evaluating Continued Fractions................405
17.1.2 Pell’s Equation...............................407
17.2 Basic Theory.........................................410
17.3 Rational Numbers.....................................418
17.4 Periodic Continued Fractions.........................420
17.4.1 Purely Periodic Continued Fractions...........422
17.4.2 Eventually Periodic Continued Fractions.......427
xvi Contents
17.5 Square Roots of Integers ............................ 429
17.6 Some Irrational Numbers............................... 432
17.7 Chapter Highlights ....................................438
17.8 Problems ..............................................438
17.8.1 Exercises..................................438
17.8.2 Projects...................................439
17.8.3 Computer Explorations......................441
17.8.4 Answers to “Check Your Understanding” .... 441
18 Gaussian Integers 443
18.1 Complex Arithmetic.....................................443
18.2 Gaussian Irreducibles .................................445
18.3 The Division Algorithm.................................449
18.4 Unique Factorization ..................................452
18.5 Applications ..........................................458
18.5.1 Sums of Two Squares .......................... 458
18.5.2 Pythagorean Triples...................... 461
18.5.3 y2 = x3- 1.....................462
18.6 Chapter Highlights .................................. 464
18.7 Problems............................................. 464
18.7.1 Exercises ......................................464
18.7.2 Projects........................................465
18.7.3 Computer Explorations......................465
18.7.4 Answers to “Check Your Understanding” .... 465
19 Algebraic Integers 467
19.1 Quadratic Fields and Algebraic Integers................467
19.2 Units ............................................... 472
19.3 Z[y/=2]................................................476
19.4 Z[VS].......................................479
19.4.1 The Lucas-Lehmer Test......................482
19.5 Non-Unique Factorization...............................486
19.6 Chapter Highlights ....................................488
XVII
19.7 Problems ............................................488
19.7.1 Exercises ....................................488
19.7.2 Projects.................................489
19.7.3 Answers to “Check Your Understanding” .... 491
20 The Distribution of Primes 493
20.1 Bertrand’s Postulate.................................493
20.2 Chebyshev’s Approximate Prime Number Theorem . . 502
20.3 Chapter Highlights ..................................507
20.4 Problems ............................................508
20.4.1 Exercises ....................................508
20.4.2 Projects.................................509
20.4.3 Computer Explorations....................510
21 Epilogue: Fermat’s Last Theorem 511
21.1 Introduction ........................................511
21.2 Elliptic Curves......................................514
21.3 Modularity...........................................517
A Supplementary Topics 521
A.l What Is a Proof? .....................................521
A.1.1 Proof by Contradiction .............. 527
A.2 Geometric Series......................................530
A.3 Mathematical Induction.............................. 531
A.4 Pascal’s Triangle and the Binomial Theorem ...........537
A.5 Fibonacci Numbers ....................................543
A.6 Matrices .............................................546
A.7 Problems .............................................550
A. 7.1 Exercises ....................................550
A.7.2 Answers to “Check Your Understanding” .... 552
B Answers and Hints for Odd-Numbered Exercises 555
Index
573
|
| any_adam_object | 1 |
| author | Kraft, James S. Washington, Lawrence C. 1951- |
| author_GND | (DE-588)1033730076 |
| author_facet | Kraft, James S. Washington, Lawrence C. 1951- |
| author_role | aut aut |
| author_sort | Kraft, James S. |
| author_variant | j s k js jsk l c w lc lcw |
| building | Verbundindex |
| bvnumber | BV044918877 |
| callnumber-first | Q - Science |
| callnumber-label | QA241 |
| callnumber-raw | QA241 |
| callnumber-search | QA241 |
| callnumber-sort | QA 3241 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)1030344174 (DE-599)BVBBV044918877 |
| dewey-full | 512.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7 |
| dewey-search | 512.7 |
| dewey-sort | 3512.7 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02144nam a2200493 c 4500</leader><controlfield tag="001">BV044918877</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20180524 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">180424s2018 xxua||| |||| 00||| eng d</controlfield><datafield tag="010" ind1=" " ind2=" "><subfield code="a">017044405</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9781138063471</subfield><subfield code="9">978-1-138-06347-1</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)1030344174</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV044918877</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">xxu</subfield><subfield code="c">US</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-739</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA241</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">512.7</subfield><subfield code="2">23</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 180</subfield><subfield code="0">(DE-625)143222:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Kraft, James S.</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">An introduction to number theory with cryptography</subfield><subfield code="c">James S. Kraft, Lawrence C. Washington</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">Second edition</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Boca Raton</subfield><subfield code="b">CRC Press, Taylor & Francis Group</subfield><subfield code="c">[2018]</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">xxii, 578 pages</subfield><subfield code="b">Illustrationen</subfield><subfield code="c">24 cm</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">Textbooks in mathematics</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">"A Chapman & Hall book.". - Includes index</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Number theory</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Cryptography</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Zahlentheorie</subfield><subfield code="0">(DE-588)4067277-3</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Kryptologie</subfield><subfield code="0">(DE-588)4033329-2</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="655" ind1=" " ind2="7"><subfield code="8">1\p</subfield><subfield code="0">(DE-588)4123623-3</subfield><subfield code="a">Lehrbuch</subfield><subfield code="2">gnd-content</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Zahlentheorie</subfield><subfield code="0">(DE-588)4067277-3</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Kryptologie</subfield><subfield code="0">(DE-588)4033329-2</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Washington, Lawrence C.</subfield><subfield code="d">1951-</subfield><subfield code="0">(DE-588)1033730076</subfield><subfield code="4">aut</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Passau - ADAM Catalogue Enrichment</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030312215&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Passau - ADAM Catalogue Enrichment</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030312215&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-030312215</subfield></datafield><datafield tag="883" ind1="1" ind2=" "><subfield code="8">1\p</subfield><subfield code="a">cgwrk</subfield><subfield code="d">20201028</subfield><subfield code="q">DE-101</subfield><subfield code="u">https://d-nb.info/provenance/plan#cgwrk</subfield></datafield></record></collection> |
| genre | 1\p (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV044918877 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T08:04:44Z |
| institution | BVB |
| isbn | 9781138063471 |
| language | English |
| lccn | 017044405 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030312215 |
| oclc_num | 1030344174 |
| open_access_boolean | |
| owner | DE-739 |
| owner_facet | DE-739 |
| physical | xxii, 578 pages Illustrationen 24 cm |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | CRC Press, Taylor & Francis Group |
| record_format | marc |
| series2 | Textbooks in mathematics |
| spelling | Kraft, James S. aut An introduction to number theory with cryptography James S. Kraft, Lawrence C. Washington Second edition Boca Raton CRC Press, Taylor & Francis Group [2018] xxii, 578 pages Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier Textbooks in mathematics "A Chapman & Hall book.". - Includes index Number theory Cryptography Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Kryptologie (DE-588)4033329-2 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Zahlentheorie (DE-588)4067277-3 s Kryptologie (DE-588)4033329-2 s DE-604 Washington, Lawrence C. 1951- (DE-588)1033730076 aut 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=030312215&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 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=030312215&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Kraft, James S. Washington, Lawrence C. 1951- An introduction to number theory with cryptography Number theory Cryptography Zahlentheorie (DE-588)4067277-3 gnd Kryptologie (DE-588)4033329-2 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4033329-2 (DE-588)4123623-3 |
| title | An introduction to number theory with cryptography |
| title_auth | An introduction to number theory with cryptography |
| title_exact_search | An introduction to number theory with cryptography |
| title_full | An introduction to number theory with cryptography James S. Kraft, Lawrence C. Washington |
| title_fullStr | An introduction to number theory with cryptography James S. Kraft, Lawrence C. Washington |
| title_full_unstemmed | An introduction to number theory with cryptography James S. Kraft, Lawrence C. Washington |
| title_short | An introduction to number theory with cryptography |
| title_sort | an introduction to number theory with cryptography |
| topic | Number theory Cryptography Zahlentheorie (DE-588)4067277-3 gnd Kryptologie (DE-588)4033329-2 gnd |
| topic_facet | Number theory Cryptography Zahlentheorie Kryptologie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030312215&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030312215&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kraftjamess anintroductiontonumbertheorywithcryptography AT washingtonlawrencec anintroductiontonumbertheorywithcryptography |
There is no print copy available.
Indexes