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Introduction to Number Theory with Cryptography 2nd edition [Kõva köide]

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, (Gilman School, Baltimore, Maryland, USA)
  • Formaat: Hardback, 578 pages, kõrgus x laius: 234x156 mm, kaal: 1000 g, 18 Tables, black and white; 28 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 31-Jan-2018
  • Kirjastus: CRC Press
  • ISBN-10: 1138063479
  • ISBN-13: 9781138063471
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Introduction to Number Theory with Cryptography 2nd editionSuurem pilt
  • Formaat: Hardback, 578 pages, kõrgus x laius: 234x156 mm, kaal: 1000 g, 18 Tables, black and white; 28 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 31-Jan-2018
  • Kirjastus: CRC Press
  • ISBN-10: 1138063479
  • ISBN-13: 9781138063471
Teised raamatud teemal:
Teised raamatud sellest sooduspakkumisest: Taylor & Francis teadusraamatud International Student Editions hindadega
Püsilink: https://www.kriso.ee/db/9781138063471.html
Märksõnad:
Building on the success of the first edition, An Introduction to Number Theory with Cryptography, Second Edition, increases coverage of the popular and important topic of cryptography, integrating it with traditional topics in number theory.

The authors have written the text in an engaging style to reflect number theory's increasing popularity. The book is designed to be used by sophomore, junior, and senior undergraduates, but it is also accessible to advanced high school students and is appropriate for independent study. It includes a few more advanced topics for students who wish to explore beyond the traditional curriculum.

Features of the second edition include





Over 800 exercises, projects, and computer explorations





Increased coverage of cryptography, including Vigenere, Stream, Transposition,and Block ciphers, along with RSA and discrete log-based systems





"Check Your Understanding" questions for instant feedback to students





New Appendices on "What is a proof?" and on Matrices





Select basic (pre-RSA) cryptography now placed in an earlier chapter so that the topic can be covered right after the basic material on congruences





Answers and hints for odd-numbered problems

About the Authors:

Jim Kraft received his Ph.D. from the University of Maryland in 1987 and has published several research papers in algebraic number theory. His previous teaching positions include the University of Rochester, St. Mary's College of California, and Ithaca College, and he has also worked in communications security. Dr. Kraft currently teaches mathematics at the Gilman School.

Larry Washington received his Ph.D. from Princeton University in 1974 and has published extensively in number theory, including books on cryptography (with Wade Trappe), cyclotomic fields, and elliptic curves. Dr. Washington is currently Professor of Mathematics and Distinguished Scholar-Teacher at the University of Maryland.

Arvustused

" provides a fine history of number theory and surveys its applications. College-level undergrads will appreciate the number theory topics, arranged in a format suitable for any standard course in the topic, and will also appreciate the inclusion of many exercises and projects to support all the theory provided. In providing a foundation text with step-by-step analysis, examples, and exercises, this is a top teaching tool recommended for any cryptography student or instructor." California Bookwatch

Preface xix
1 Introduction
1(8)
1.1 Diophantine Equations
2(2)
1.2 Modular Arithmetic
4(1)
1.3 Primes and the Distribution of Primes
5(2)
1.4 Cryptography
7(2)
2 Divisibility
9(42)
2.1 Divisibility
9(2)
2.2 Euclid's Theorem
11(2)
2.3 Euclid's Original Proof
13(2)
2.4 The Sieve of Eratosthenes
15(2)
2.5 The Division Algorithm
17(3)
2.5.1 A Cryptographic Application
19(1)
2.6 The Greatest Common Divisor
20(3)
2.7 The Euclidean Algorithm
23(8)
2.7.1 The Extended Euclidean Algorithm
25(6)
2.8 Other Bases
31(3)
2.9 Fermat and Mersenne Numbers
34(4)
2.10
Chapter Highlights
38(1)
2.11 Problems
38(13)
2.11.1 Exercises
38(7)
2.11.2 Projects
45(2)
2.11.3 Computer Explorations
47(1)
2.11.4 Answers to "Check Your Understanding"
48(3)
3 Linear Diophantine Equations
51(12)
3.1 ax + by = c
51(6)
3.2 The Postage Stamp Problem
57(3)
3.3
Chapter Highlights
60(1)
3.4 Problems
60(3)
3.4.1 Exercises
60(2)
3.4.2 Answers to "Check Your Understanding"
62(1)
4 Unique Factorization
63(12)
4.1 The Starting Point
63(1)
4.2 The Fundamental Theorem of Arithmetic
64(5)
4.3 Euclid and the Fundamental Theorem of Arithmetic
69(1)
4.4
Chapter Highlights
70(1)
4.5 Problems
71(4)
4.5.1 Exercises
71(2)
4.5.2 Projects
73(1)
4.5.3 Answers to "Check Your Understanding"
73(2)
5 Applications of Unique Factorization
75(38)
5.1 A Puzzle
75(2)
5.2 Irrationality Proofs
77(4)
5.2.1 Four More Proofs That √2 Is Irrational
79(2)
5.3 The Rational Root Theorem
81(3)
5.4 Pythagorean Triples
84(6)
5.5 Differences of Squares
90(2)
5.6 Prime Factorization of Factorials
92(2)
5.7 The Riemann Zeta Function
94(11)
5.7.1 Σ1/p Diverges
100(5)
5.8
Chapter Highlights
105(1)
5.9 Problems
106(7)
5.8.1 Exercises
106(2)
5.9.2 Projects
108(4)
5.9.3 Computer Explorations
112(1)
5.9.4 Answers to "Check Your Understanding"
112(1)
6 Congruences
113(42)
6.1 Definitions and Examples
113(9)
6.2 Modular Exponentiation
122(2)
6.3 Divisibility Tests
124(5)
6.4 Linear Congruences
129(7)
6.5 The Chinese Remainder Theorem
136(5)
6.6 Fractions mod m
141(2)
6.7 Queens on a Chessboard
143(2)
6.8
Chapter Highlights
145(1)
6.9 Problems
145(10)
6.9.1 Exercises
145(7)
6.9.2 Projects
152(1)
6.9.3 Computer Explorations
153(1)
6.9.4 Answers to "Check Your Understanding"
154(1)
7 Classical Cryptosystems
155(34)
7.1 Introduction
155(1)
7.2 Shift and Affine Ciphers
156(5)
7.3 Vigenere Ciphers
161(6)
7.4 Transposition Ciphers
167(3)
7.5 Stream Ciphers
170(5)
7.5.1 One-Time Pad
171(1)
7.5.2 Linear Feedback Shift Registers (LFSR)
172(3)
7.6 Block Ciphers
175(4)
7.7 Secret Sharing
179(2)
7.8 Generating Random Numbers
181(2)
7.9
Chapter Highlights
183(1)
7.10 Problems
183(6)
7.10.1 Exercises
183(3)
7.10.2 Answers to "Check Your Understanding"
186(3)
8 Fermat, Euler, and Wilson
189(20)
8.1 Fermat's Theorem
189(5)
8.2 Euler's Theorem
194(6)
8.3 Wilson's Theorem
200(2)
8.4
Chapter Highlights
202(1)
8.5 Problems
203(6)
8.5.1 Exercises
203(3)
8.5.2 Projects
206(1)
8.5.3 Computer Explorations
207(1)
8.5.4 Answers to "Check Your Understanding"
207(2)
9 RSA
209(18)
9.1 RSA Encryption
210(7)
9.2 Digital Signatures
217(2)
9.3
Chapter Highlights
219(1)
9.4 Problems
219(8)
9.3.1 Exercises
219(5)
9.4.2 Projects
224(1)
9.4.3 Computer Explorations
225(1)
9.4.4 Answers to "Check Your Understanding"
226(1)
10 Polynomial Congruences
227(12)
10.1 Polynomials Mod Primes
227(3)
10.2 Solutions Modulo Prime Powers
230(4)
10.3 Composite Moduli
234(1)
10.4
Chapter Highlights
235(1)
10.5 Problems
235(4)
10.4.1 Exercises
235(1)
10.5.2 Projects
236(1)
10.5.3 Computer Explorations
237(1)
10.5.4 Answers to "Check Your Understanding"
238(1)
11 Order and Primitive Roots
239(34)
11.1 Orders of Elements
239(5)
11.1.1 Fermat Numbers
241(2)
11.1.2 Mersenne Numbers
243(1)
11.2 Primitive Roots
244(6)
11.3 Decimals
250(5)
11.3.1 Midy's Theorem
253(2)
11.4 Card Shuffling
255(2)
11.5 The Discrete Log Problem
257(6)
11.5.1 Baby Step--Giant Step Method
258(2)
11.5.2 The Index Calculus
260(3)
11.6 Existence of Primitive Roots
263(3)
11.7
Chapter Highlights
266(1)
11.8 Problems
266(7)
11.7.1 Exercises
266(3)
11.8.2 Projects
269(2)
11.8.3 Computer Explorations
271(1)
11.8.4 Answers to "Check Your Understanding"
271(2)
12 More Cryptographic Applications
273(16)
12.1 Diffie--Hellman Key Exchange
273(2)
12.2 Coin Flipping over the Telephone
275(2)
12.3 Mental Poker
277(5)
12.4 The ElGamal Public Key Cryptosystem
282(3)
12.5
Chapter Highlights
285(1)
12.6 Problems
285(4)
12.6.1 Exercises
285(2)
12.6.2 Projects
287(1)
12.6.3 Computer Explorations
287(1)
12.6.4 Answers to "Check Your Understanding"
288(1)
13 Quadratic Reciprocity
289(30)
13.1 Squares and Square Roots Mod Primes
289(7)
13.2 Computing Square Roots Mod p
296(2)
13.3 Quadratic Equations
298(2)
13.4 The Jacobi Symbol
300(5)
13.5 Proof of Quadratic Reciprocity
305(7)
13.6
Chapter Highlights
312(1)
13.7 Problems
312(7)
13.7.1 Exercises
312(4)
13.7.2 Projects
316(2)
13.7.3 Answers to "Check Your Understanding"
318(1)
14 Primality and Factorization
319(38)
14.1 Trial Division and Fermat Factorization
319(4)
14.2 Primality Testing
323(12)
14.2.1 Pseudoprimes
323(5)
14.2.2 The Pocklington--Lehmer Primality Test
328(3)
14.2.3 The AKS Primality Test
331(2)
14.2.4 Fermat Numbers
333(2)
14.2.5 Mersenne Numbers
335(1)
14.3 Factorization
335(15)
14.3.1 x2 ≡ y2
336(3)
14.3.2 Factoring Pseudoprimes and Factoring Using RSA Exponents
339(1)
14.3.3 Pollard's p -- 1 Method
340(2)
14.3.4 The Quadratic Sieve
342(8)
14.4 Coin Flipping over the Telephone
350(2)
14.5
Chapter Highlights
352(1)
14.6 Problems
352(5)
14.6.1 Exercises
352(3)
14.6.2 Projects
355(1)
14.6.3 Computer Explorations
355(1)
14.6.4 Answers to "Check Your Understanding"
356(1)
15 Geometry of Numbers
357(28)
15.1 Volumes and Minkowski's Theorem
357(5)
15.2 Sums of Two Squares
362(6)
15.2.1 Algorithm for Writing p ≡ 1 (mod 4) as a Sum of Two Squares
366(2)
15.3 Sums of Four Squares
368(2)
15.4 Pell's Equation
370(6)
15.4.1 Bhaskara's Chakravala Method
373(3)
15.5
Chapter Highlights
376(1)
15.6 Problems
376(9)
15.6.1 Exercises
376(4)
15.6.2 Projects
380(4)
15.6.3 Answers to "Check Your Understanding"
384(1)
16 Arithmetic Functions
385(16)
16.1 Perfect Numbers
385(4)
16.2 Multiplicative Functions
389(6)
16.3
Chapter Highlights
395(1)
16.4 Problems
395(6)
16.3.1 Exercises
395(2)
16.4.2 Projects
397(1)
16.4.3 Computer Explorations
398(1)
16.4.4 Answers to "Check Your Understanding"
399(2)
17 Continued Fractions
401(42)
17.1 Rational Approximations; Pell's Equation
402(8)
17.1.1 Evaluating Continued Fractions
405(2)
17.1.2 Pell's Equation
407(3)
17.2 Basic Theory
410(8)
17.3 Rational Numbers
418(2)
17.4 Periodic Continued Fractions
420(9)
17.4.1 Purely Periodic Continued Fractions
422(5)
17.4.2 Eventually Periodic Continued Fractions
427(2)
17.5 Square Roots of Integers
429(3)
17.6 Some Irrational Numbers
432(6)
17.7
Chapter Highlights
438(1)
17.8 Problems
438(5)
17.8.1 Exercises
438(1)
17.8.2 Projects
439(2)
17.8.3 Computer Explorations
441(1)
17.8.4 Answers to "Check Your Understanding"
441(2)
18 Gaussian Integers
443(24)
18.1 Complex Arithmetic
443(2)
18.2 Gaussian Irreducibles
445(4)
18.3 The Division Algorithm
449(3)
18.4 Unique Factorization
452(6)
18.5 Applications
458(6)
18.5.1 Sums of Two Squares
458(3)
18.5.2 Pythagorean Triples
461(1)
18.5.3 y2 = x3 - 1
462(2)
18.6
Chapter Highlights
464(1)
18.7 Problems
464(3)
18.7.1 Exercises
464(1)
18.7.2 Projects
465(1)
18.7.3 Computer Explorations
465(1)
18.7.4 Answers to "Check Your Understanding"
465(2)
19 Algebraic Integers
467(26)
19.1 Quadratic Fields and Algebraic Integers
467(5)
19.2 Units
472(4)
19.3 Z[ √-2]
476(3)
19.4 Z[ √3]
479(7)
19.4.1 The Lucas--Lehmer Test
482(4)
19.5 Non-Unique Factorization
486(2)
19.6
Chapter Highlights
488(1)
19.7 Problems
488(5)
19.7.1 Exercises
488(1)
19.7.2 Projects
489(2)
19.7.3 Answers to "Check Your Understanding"
491(2)
20 The Distribution of Primes
493(18)
20.1 Bertrand's Postulate
493(9)
20.2 Chebyshev's Approximate Prime Number Theorem
502(5)
20.3
Chapter Highlights
507(1)
20.4 Problems
508(3)
20.4.1 Exercises
508(1)
20.4.2 Projects
509(1)
20.4.3 Computer Explorations
510(1)
21 Epilogue: Fermat's Last Theorem
511(10)
21.1 Introduction
511(3)
21.2 Elliptic Curves
514(3)
21.3 Modularity
517(4)
A Supplementary Topics
521(34)
A.1 What Is a Proof?
521(9)
A.1.1 Proof by Contradiction
527(3)
A.2 Geometric Series
530(1)
A.3 Mathematical Induction
531(6)
A.4 Pascal's Triangle and the Binomial Theorem
537(6)
A.5 Fibonacci Numbers
543(3)
A.6 Matrices
546(4)
A.7 Problems
550(5)
A.7.1 Exercises
550(2)
A.7.2 Answers to "Check Your Understanding"
552(3)
B Answers and Hints for Odd-Numbered Exercises
555(18)
Index 573
Jim Kraft received his Ph.D. from the University of Maryland in 1987 and has published several research papers in algebraic number theory. His previous teaching positions include the University of Rochester, St. Mary's College of California, and Ithaca College, and he has also worked in communications security. Dr. Kraft currently teaches mathematics at the Gilman School.









Larry Washington received his Ph.D. from Princeton University in 1974 and has published extensively in number theory, including books on cryptography (with Wade Trappe), cyclotomic fields, and elliptic curves. Dr. Washington is currently Professor of Mathematics and Distinguished Scholar-Teacher at the University of Maryland.