An introduction to number theory with cryptography:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, FL
CRC Press
2014
|
| Series: | A Chapman & Hall book
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 554 S. graph. Darst. |
| ISBN: | 9781482214413 1482214415 |
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Record in the Search Index
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| adam_text | Titel: An introduction to number theory with cryptography
Autor: Kraft, James S
Jahr: 2014
Contents
Preface xv
0 Introduction 1
0.1 Diophantine Equations ..............................2
0.2 Modular Arithmetic ..................................4
0.3 Primes and the Distribution of Primes..............5
0.4 Cryptography..........................................7
1 Divisibility 9
1.1 Divisibility ............................................9
1.2 Euclid s Theorem......................................11
1.3 Euclid s Original Proof ..............................13
1.4 The Sieve of Eratosthenes............................15
1.5 The Division Algorithm ..............................17
1.5.1 A Cryptographic Application ................19
1.6 The Greatest Common Divisor ......................20
1.7 The Euclidean Algorithm ............................22
1.7.1 The Extended Euclidean Algorithm..........25
1.8 Other Bases ..........................................30
1.9 Linear Diophantine Equations........................32
1.10 The Postage Stamp Problem ........................38
1.11 Fermât and Mersenne Numbers......................41
1.12 Chapter Highlights....................................46
1.13 Problems ..............................................46
1.13.1 Exercises........................................46
1.13.2 Projects..............................53
1.13.3 Computer Explorations........................55
vii
viii Contents
1.13.4 Answers to Check Your Understanding . . 57
2 Unique Factorization 59
2.1 Preliminary Results ................. 59
2.2 The Fundamental Theorem of Arithmetic ..... 61
2.3 Euclid and the Fundamental Theorem of Arithmetic 66
2.4 Chapter Highlights.................. 67
2.5 Problems ....................... 67
2.5.1 Exercises........................................67
2.5.2 Projects........................................68
2.5.3 Answers to Check Your Understanding . . 70
3 Applications of Unique Factorization 71
3.1 A Puzzle ..............................................71
3.2 Irrationality Proofs....................................73
3.2.1 Four More Proofs That y/2 Is Irrational . . 75
3.3 The Rational Root Theorem ........................77
3.4 Pythagorean Triples ..................................80
3.5 Differences of Squares ................................86
3.6 Prime Factorization of Factorials ....................88
3.7 The Riemann Zeta Function..........................90
3.8 Chapter Highlights....................................96
3.9 Problems ..............................................96
3.9.1 Exercises........................................96
3.9.2 Projects....................100
3.9.3 Computer Explorations............104
3.9.4 Answers to Check Your Understanding . . 105
4 Congruences 107
4.1 Definitions and Examples ..............107
4.2 Modular Exponentiation...............115
4.3 Divisibility Tests...................116
4.4 Linear Congruences .................120
4.5 The Chinese Remainder Theorem..........127
ix
4.6 Fractions mod m...................132
4.7 Fermat s Theorem ..................134
4.8 Euler s Theorem ...................139
4.9 Wilson s Theorem ..................147
4.10 Queens on a Chessboard...............149
4.11 Chapter Highlights..................151
4.12 Problems .......................151
4.12.1 Exercises....................151
4.12.2 Projects....................159
4.12.3 Computer Explorations............163
4.12.4 Answers to Check Your Understanding . . 164
5 Cryptographic Applications 167
5.1 Introduction .....................167
5.2 Shift and Affine Ciphers...............170
5.3 Secret Sharing ....................175
5.4 RSA..........................177
5.5 Chapter Highlights..................184
5.6 Problems .......................184
5.6.1 Exercises....................184
5.6.2 Projects....................188
5.6.3 Computer Explorations............191
5.6.4 Answers to Check Your Understanding . . 192
6 Polynomial Congruences 193
6.1 Polynomials Mod Primes ..............193
6.2 Solutions Modulo Prime Powers...........196
6.3 Composite Moduli ..................202
6.4 Chapter Highlights..................203
6.5 Problems .......................203
6.5.1 Exercises....................203
6.5.2 Projects....................204
6.5.3 Computer Explorations............205
X Contents
6.5.4 Answers to Check Your Understanding . . 206
7 Order and Primitive Roots 207
7.1 Orders of Elements..................207
7.1.1 Fermât Numbers...............209
7.1.2 Mersenne Numbers..............211
7.2 Primitive Roots ...................211
7.3 Decimals .......................217
7.3.1 Midy s Theorem................220
7.4 Card Shuffling ....................222
7.5 The Discrete Log Problem..............224
7.5.1 Baby Step-Giant Step Method........226
7.5.2 The Index Calculus..............228
7.6 Existence of Primitive Roots ............231
7.7 Chapter Highlights..................233
7.8 Problems .......................234
7.8.1 Exercises....................234
7.8.2 Projects....................238
7.8.3 Computer Explorations............239
7.8.4 Answers to Check Your Understanding . . 240
8 More Cryptographic Applications 241
8.1 Diffie-Hellman Key Exchange ............241
8.2 Coin Flipping over the Telephone..........243
8.3 Mental Poker.....................246
8.4 The ElGamal Public Key Cryptosystem ......250
8.5 Digital Signatures ..................253
8.6 Chapter Highlights..................255
8.7 Problems .......................255
8.7.1 Exercises....................255
8.7.2 Projects....................259
8.7.3 Computer Explorations............260
8.7.4 Answers to Check Your Understanding . . 260
xi
9 Quadratic Reciprocity 263
9.1 Squares and Square Roots Mod Primes ......263
9.2 Computing Square Roots Mod p ..........270
9.3 Quadratic Equations.................272
9.4 The Jacobi Symbol..................274
9.5 Proof of Quadratic Reciprocity ...........278
9.6 Chapter Highlights..................285
9.7 Problems .......................286
9.7.1 Exercises....................286
9.7.2 Projects....................291
9.7.3 Answers to Check Your Understanding . . 293
10 Primality and Factorization 295
10.1 Trial Division and Fermât Factorization ......295
10.2 Primality Testing...................299
10.2.1 Pseudoprimes.................299
10.2.2 The Pocklington-Lehmer Primality Test . . 304
10.2.3 The AKS Primality Test...........307
10.2.4 Fermât Numbers...............309
10.2.5 Mersenne Numbers..............311
10.3 Factorization .....................312
10.3.1 x2 = y2 ....................312
10.3.2 Factoring Pseudoprimes and Factoring Us-
ing RSA Exponents..............315
10.3.3 Pollard s p — 1 Method............316
10.3.4 The Quadratic Sieve.............318
10.4 Coin Flipping over the Telephone..........326
10.5 Chapter Highlights..................328
10.6 Problems .......................329
10.6.1 Exercises....................329
10.6.2 Projects....................332
10.6.3 Computer Explorations............333
10.6.4 Answers to Check Your Understanding . . 334
xii Contents
11 Geometry of Numbers 337
11.1 Volumes and Minkowski s Theorem.........337
11.2 Sums of Two Squares ................342
11.2.1 Algorithm for Writing p = 1 (mod 4) as a
Sum of Two Squares.............345
11.3 Sums of Four Squares ................347
11.4 Pell s Equation....................349
11.4.1 Bhâskara s Chakravala Method.......353
11.5 Chapter Highlights..................355
11.6 Problems .......................356
11.6.1 Exercises....................356
11.6.2 Projects....................359
11.6.3 Answers to Check Your Understanding . . 365
12 Arithmetic Functions 367
12.1 Perfect Numbers ...................367
12.2 Multiplicative Functions...............371
12.3 Chapter Highlights..................378
12.4 Problems .......................378
12.4.1 Exercises....................378
12.4.2 Projects....................381
12.4.3 Computer Explorations............381
12.4.4 Answers to Check Your Understanding . . 382
13 Continued Fractions 383
13.1 Rational Approximations; Pell s Equation .....384
13.1.1 Evaluating Continued Fractions.......387
13.1.2 Pell s Equation................389
13.2 Basic Theory .....................392
13.3 Rational Numbers ..................400
13.4 Periodic Continued Fractions ............402
13.4.1 Purely Periodic Continued Fractions .... 404
13.4.2 Eventually Periodic Continued Fractions . . 409
xiii
13.5 Square Roots of Integers...............411
13.6 Some Irrational Numbers ..............414
13.7 Chapter Highlights..................420
13.8 Problems .......................421
13.8.1 Exercises....................421
13.8.2 Projects....................422
13.8.3 Computer Explorations............425
13.8.4 Answers to Check Your Understanding . . 425
14 Gaussian Integers 427
14.1 Complex Arithmetic.................427
14.2 Gaussian Irreducibles ................429
14.3 The Division Algorithm ...............433
14.4 Unique Factorization.................436
14.5 Applications .....................442
14.5.1 Sums of Two Squares.............442
14.5.2 Pythagorean Triples .............445
14.5.3 y2 = Xs — 1 ..................447
14.6 Chapter Highlights..................448
14.7 Problems .......................449
14.7.1 Exercises....................449
14.7.2 Projects....................450
14.7.3 Computer Explorations............450
14.7.4 Answers to Check Your Understanding . . 450
15 Algebraic Integers 453
15.1 Quadratic Fields and Algebraic Integers ......453
15.2 Units .........................458
15.3 Z[^/^2]........................462
15.4 Z[ /3].........................466
15.4.1 The Lucas-Lehmer Test ...........469
15.5 Non-unique Factorization ..............472
15.6 Chapter Highlights..................474
xiv Contents
15.7 Problems .......................475
15.7.1 Exercises....................475
15.7.2 Projects....................476
15.7.3 Answers to Check Your Understanding . . 478
16 Analytic Methods 479
16.1 VP Diverges....................479
16.2 Bertrand s Postulate.................485
16.3 Chebyshev s Approximate Prime Number Theorem 493
16.4 Chapter Highlights..................499
16.5 Problems .......................499
16.5.1 Exercises....................499
16.5.2 Projects....................500
16.5.3 Computer Explorations............501
17 Epilogue: Fermat s Last Theorem 503
17.1 Introduction .....................503
17.2 Elliptic Curves ....................506
17.3 Modularity ......................510
A Supplementary Topics 513
A.l Geometric Series ...................513
A.2 Mathematical Induction...............515
A.3 Pascal s Triangle and the Binomial Theorem .... 521
A.4 Fibonacci Numbers .................526
A.5 Problems .......................530
A.5.1 Exercises....................530
A.5.2 Answers to Check Your Understanding . . 532
B Answers and Hints for Odd-Numbered Exercises 535
Index
549
|
| any_adam_object | 1 |
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| spelling | Kraft, James S. Verfasser aut An introduction to number theory with cryptography James S. Kraft ; Lawrence C. Washington Boca Raton, FL CRC Press 2014 XVIII, 554 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier A Chapman & Hall book Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Kryptologie (DE-588)4033329-2 gnd rswk-swf (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- Verfasser (DE-588)1033730076 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026190400&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| 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 |
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