A computational introduction to number theory and algebra:
"Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasizes algorithms and applications, such as cryptography and...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2009
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Volltext Inhaltsverzeichnis |
| Zusammenfassung: | "Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory." -- Publisher's description. |
| Beschreibung: | Literaturverz. S. 566 - 571 |
| Beschreibung: | XVII, 580 S. Diagramme. - graph. Darst. |
| ISBN: | 9780521516440 |
Internformat
MARC
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| 100 | 1 | |a Shoup, Victor |d ca. 20./21. Jh. |e Verfasser |0 (DE-588)125580517X |4 aut | |
| 245 | 1 | 0 | |a A computational introduction to number theory and algebra |c Victor Shoup |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2009 | |
| 300 | |a XVII, 580 S. |b Diagramme. - graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 566 - 571 | ||
| 520 | 3 | |a "Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory." -- Publisher's description. | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Number theory |x Data processing | |
| 650 | 0 | 7 | |a Algebraische Zahlentheorie |0 (DE-588)4001170-7 |2 gnd |9 rswk-swf |
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| 856 | 4 | 1 | |u http://www.shoup.net/ntb/ |x Verlag |z kostenfrei |3 Volltext |
| 856 | 4 | 2 | |m Digitalisierung UB Bamberg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016683585&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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| 883 | 1 | |8 2\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 912 | |a ebook | ||
Datensatz im Suchindex
| _version_ | 1805076175704293376 |
|---|---|
| adam_text |
Contents
Preface
page
x
Preliminaries
xiv
1
Basic properties of the integers
1
1.1
Divisibility and primality
1
1.2
Ideals and greatest common divisors
5
1.3
Some consequences of unique factorization
10
2
Congruences
15
2.1
Equivalence relations
15
2.2
Definitions and basic properties of congruences
16
2.3
Solving linear congruences
19
2.4
The Chinese remainder theorem
22
2.5
Residue classes
25
2.6
Euler's phi function
31
2.7
Euler's theorem and Fermat's little theorem
32
2.8
Quadratic residues
35
2.9
Summations over divisors
45
3
Computing with large integers
50
3.1
Asymptotic notation
50
3.2
Machine models and complexity theory
53
3.3
Basic integer arithmetic
55
3.4
Computing in
Z„
64
3.5
Faster integer arithmetic
(*) 69
3.6
Notes
71
4
Euclid's algorithm
74
4.1
The basic Euclidean algorithm
74
4.2
The extended Euclidean algorithm
77
4.3
Computing modular inverses and Chinese remaindering
82
vi
Contents
4.4
Speeding up algorithms via modular computation
84
4.5
An effective version of Fermat's two squares theorem
86
4.6
Rational reconstruction and applications
89
4.7
The RSA cryptosystem
99
4.8
Notes
102
The distribution of primes
104
5.1
Chebyshev's theorem on the density of primes
104
5.2
Bertrand's postulate
108
5.3
Mertens' theorem
110
5.4
The sieve of Eratosthenes
115
5.5
The prime number theorem
.
and beyond
116
5.6
Notes
124
Abelian groups
126
6.1
Definitions, basic properties, and examples
126
6.2
Subgroups
132
6.3
Cosets and quotient groups
137
6.4
Group homomorphisms and isomorphisms
142
6.5
Cyclic groups
153
6.6
The structure of finite abelian groups
(*)
163
Rings
166
7.1
Definitions, basic properties, and examples
166
7.2
Polynomial rings
176
7.3
Ideals and quotient rings
185
7.4
Ring homomorphisms and isomorphisms
192
7.5
The structure of Z*
203
Finite
and discrete probability distributions
207
8.1
Basic definitions
207
8.2
Conditional probability and independence
213
8.3
Random variables
221
8.4
Expectation and variance
233
8.5
Some useful bounds
241
8.6
Balls and bins
245
8.7
Hash functions
252
8.8
Statistical distance
260
8.9
Measures of randomness and the leftover hash lemma
(*)
266
8.10
Discrete probability distributions
270
8.11
Notes
275
Contents
vii
9
Probabilistic
algorithms
277
9.1
Basic
definitions
278
9.2
Generating a random number from a given interval
285
9.3
The generate and test paradigm
287
9.4
Generating a random prime
292
9.5
Generating a random non-increasing sequence
295
9.6
Generating a random factored number
298
9.7
Some complexity theory
302
9.8
Notes
304
10
Probabilistic primality testing
306
10.1
Trial division
306
10.2
The Miller-Rabin test
307
10.3
Generating random primes using the Miller-Rabin test
311
10.4
Factoring and computing
Euler'
s
phi function
320
10.5
Notes
324
11
Finding generators and discrete logarithms in Z*
327
11.1
Finding a generator for Z*
327
11.2
Computing discrete logarithms in Z*
329
11.3
The Diffie-Hellman key establishment protocol
334
11.4
Notes
340
12
Quadratic reciprocity and computing modular square roots
342
12.1
The Legendre symbol
342
12.2
The Jacobi symbol
346
12.3
Computing the Jacobi symbol
348
12.4
Testing quadratic residuosity
349
12.5
Computing modular square roots
350
12.6
The quadratic residuosity assumption
355
12.7
Notes
357
13
Modules and vector spaces
358
13.1
Definitions, basic properties, and examples
358
13.2
Submodules
and quotient modules
360
13.3
Module homomorphisms and isomorphisms
363
13.4
Linear independence and bases
367
13.5
Vector spaces and dimension
370
14
Matrices
377
14.1
Basic definitions and properties
377
14.2
Matrices and linear maps
381
14.3
The inverse of a matrix
386
viii Contents
14.4
Gaussian elimination
14.5
Applications of Gaussian elimination
14.6
Notes
15
Subexponential-time discrete logarithms and factoring
15.1
S mooth
numbers
15.2
An algorithm for discrete logarithms
15.3
An algorithm for factoring integers
15.4
Practical improvements
15.5
Notes
16
More rings
16.1
Algebras
16.2
The field of fractions of an integral domain
16.3
Unique factorization of polynomials
16.4
Polynomial congruences
16.5
Minimal polynomial
s
16.6
General properties of extension fields
16.7
Formal derivatives
16.8
Formal power series and Laurent series
16.9
Unique factorization domains
(*)
16.10
Notes
17
Polynomial arithmetic and applications
17.1
В
asie
arithmetic
17.2
Computing minimal polynomials in F[X]/(/)(I)
17.3
Euclid's algorithm
17.4
Computing modular inverses and Chinese remaindering
17.5
Rational function reconstruction and applications
17.6
Faster polynomial arithmetic
(*)
17.7
Notes
18
Linearly generated sequences and applications
18.1
Basic definitions and properties
18.2
Computing minimal polynomials: a special case
18.3
Computing minimal polynomials: a more general case
18.4
Solving sparse linear systems
18.5
Computing minimal polynomials in F[X]/(f)(U)
18.6
The algebra of linear transformations
(*)
18.7
Notes
19
Finite fields
19.1
Preliminaries
Contents ix
19.2
The existence of finite fields
511
19.3
The subfield structure and uniqueness of finite fields
515
19.4
Conjugates, norms and traces
516
20
Algorithms for finite fields
522
20.1
Tests for and constructing irreducible polynomials
522
20.2
Computing minimal polynomials in F[X]/(f)(lll)
525
20.3
Factoring polynomials: square-free decomposition
526
20.4
Factoring polynomials: the Cantor-Zassenhaus algorithm
530
20.5
Factoring polynomials: Berlekamp's algorithm
538
20.6
Deterministic factorization algorithms
(*) 544
20.7
Notes
546
21
Deterministic primality testing
548
21.1
The basic idea
548
21.2
The algorithm and its analysis
549
21.3
Notes
558
Appendix: Some useful facts
561
Bibliography
566
Index of notation
572
Index
574 |
| adam_txt |
Contents
Preface
page
x
Preliminaries
xiv
1
Basic properties of the integers
1
1.1
Divisibility and primality
1
1.2
Ideals and greatest common divisors
5
1.3
Some consequences of unique factorization
10
2
Congruences
15
2.1
Equivalence relations
15
2.2
Definitions and basic properties of congruences
16
2.3
Solving linear congruences
19
2.4
The Chinese remainder theorem
22
2.5
Residue classes
25
2.6
Euler's phi function
31
2.7
Euler's theorem and Fermat's little theorem
32
2.8
Quadratic residues
35
2.9
Summations over divisors
45
3
Computing with large integers
50
3.1
Asymptotic notation
50
3.2
Machine models and complexity theory
53
3.3
Basic integer arithmetic
55
3.4
Computing in
Z„
64
3.5
Faster integer arithmetic
(*) 69
3.6
Notes
71
4
Euclid's algorithm
74
4.1
The basic Euclidean algorithm
74
4.2
The extended Euclidean algorithm
77
4.3
Computing modular inverses and Chinese remaindering
82
vi
Contents
4.4
Speeding up algorithms via modular computation
84
4.5
An effective version of Fermat's two squares theorem
86
4.6
Rational reconstruction and applications
89
4.7
The RSA cryptosystem
99
4.8
Notes
102
The distribution of primes
104
5.1
Chebyshev's theorem on the density of primes
104
5.2
Bertrand's postulate
108
5.3
Mertens' theorem
110
5.4
The sieve of Eratosthenes
115
5.5
The prime number theorem
.
and beyond
116
5.6
Notes
124
Abelian groups
126
6.1
Definitions, basic properties, and examples
126
6.2
Subgroups
132
6.3
Cosets and quotient groups
137
6.4
Group homomorphisms and isomorphisms
142
6.5
Cyclic groups
153
6.6
The structure of finite abelian groups
(*)
163
Rings
166
7.1
Definitions, basic properties, and examples
166
7.2
Polynomial rings
176
7.3
Ideals and quotient rings
185
7.4
Ring homomorphisms and isomorphisms
192
7.5
The structure of Z*
203
Finite
and discrete probability distributions
207
8.1
Basic definitions
207
8.2
Conditional probability and independence
213
8.3
Random variables
221
8.4
Expectation and variance
233
8.5
Some useful bounds
241
8.6
Balls and bins
245
8.7
Hash functions
252
8.8
Statistical distance
260
8.9
Measures of randomness and the leftover hash lemma
(*)
266
8.10
Discrete probability distributions
270
8.11
Notes
275
Contents
vii
9
Probabilistic
algorithms
277
9.1
Basic
definitions
278
9.2
Generating a random number from a given interval
285
9.3
The generate and test paradigm
287
9.4
Generating a random prime
292
9.5
Generating a random non-increasing sequence
295
9.6
Generating a random factored number
298
9.7
Some complexity theory
302
9.8
Notes
304
10
Probabilistic primality testing
306
10.1
Trial division
306
10.2
The Miller-Rabin test
307
10.3
Generating random primes using the Miller-Rabin test
311
10.4
Factoring and computing
Euler'
s
phi function
320
10.5
Notes
324
11
Finding generators and discrete logarithms in Z*
327
11.1
Finding a generator for Z*
327
11.2
Computing discrete logarithms in Z*
329
11.3
The Diffie-Hellman key establishment protocol
334
11.4
Notes
340
12
Quadratic reciprocity and computing modular square roots
342
12.1
The Legendre symbol
342
12.2
The Jacobi symbol
346
12.3
Computing the Jacobi symbol
348
12.4
Testing quadratic residuosity
349
12.5
Computing modular square roots
350
12.6
The quadratic residuosity assumption
355
12.7
Notes
357
13
Modules and vector spaces
358
13.1
Definitions, basic properties, and examples
358
13.2
Submodules
and quotient modules
360
13.3
Module homomorphisms and isomorphisms
363
13.4
Linear independence and bases
367
13.5
Vector spaces and dimension
370
14
Matrices
377
14.1
Basic definitions and properties
377
14.2
Matrices and linear maps
381
14.3
The inverse of a matrix
386
viii Contents
14.4
Gaussian elimination
14.5
Applications of Gaussian elimination
14.6
Notes
15
Subexponential-time discrete logarithms and factoring
15.1
S mooth
numbers
15.2
An algorithm for discrete logarithms
15.3
An algorithm for factoring integers
15.4
Practical improvements
15.5
Notes
16
More rings
16.1
Algebras
16.2
The field of fractions of an integral domain
16.3
Unique factorization of polynomials
16.4
Polynomial congruences
16.5
Minimal polynomial
s
16.6
General properties of extension fields
16.7
Formal derivatives
16.8
Formal power series and Laurent series
16.9
Unique factorization domains
(*)
16.10
Notes
17
Polynomial arithmetic and applications
17.1
В
asie
arithmetic
17.2
Computing minimal polynomials in F[X]/(/)(I)
17.3
Euclid's algorithm
17.4
Computing modular inverses and Chinese remaindering
17.5
Rational function reconstruction and applications
17.6
Faster polynomial arithmetic
(*)
17.7
Notes
18
Linearly generated sequences and applications
18.1
Basic definitions and properties
18.2
Computing minimal polynomials: a special case
18.3
Computing minimal polynomials: a more general case
18.4
Solving sparse linear systems
18.5
Computing minimal polynomials in F[X]/(f)(U)
18.6
The algebra of linear transformations
(*)
18.7
Notes
19
Finite fields
19.1
Preliminaries
Contents ix
19.2
The existence of finite fields
511
19.3
The subfield structure and uniqueness of finite fields
515
19.4
Conjugates, norms and traces
516
20
Algorithms for finite fields
522
20.1
Tests for and constructing irreducible polynomials
522
20.2
Computing minimal polynomials in F[X]/(f)(lll)
525
20.3
Factoring polynomials: square-free decomposition
526
20.4
Factoring polynomials: the Cantor-Zassenhaus algorithm
530
20.5
Factoring polynomials: Berlekamp's algorithm
538
20.6
Deterministic factorization algorithms
(*) 544
20.7
Notes
546
21
Deterministic primality testing
548
21.1
The basic idea
548
21.2
The algorithm and its analysis
549
21.3
Notes
558
Appendix: Some useful facts
561
Bibliography
566
Index of notation
572
Index
574 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Shoup, Victor ca. 20./21. Jh |
| author_GND | (DE-588)125580517X |
| author_facet | Shoup, Victor ca. 20./21. Jh |
| author_role | aut |
| author_sort | Shoup, Victor ca. 20./21. Jh |
| author_variant | v s vs |
| building | Verbundindex |
| bvnumber | BV035014392 |
| 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 ST 600 |
| classification_tum | MAT 110f |
| collection | ebook |
| ctrlnum | (OCoLC)277069279 (DE-599)BVBBV035014392 |
| dewey-full | 004.0151 |
| dewey-hundreds | 000 - Computer science, information, general works |
| dewey-ones | 004 - Computer science |
| dewey-raw | 004.0151 |
| dewey-search | 004.0151 |
| dewey-sort | 14.0151 |
| dewey-tens | 000 - Computer science, information, general works |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV035014392 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:44:55Z |
| indexdate | 2024-07-20T05:53:07Z |
| institution | BVB |
| isbn | 9780521516440 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016683585 |
| oclc_num | 277069279 |
| open_access_boolean | 1 |
| owner | DE-473 DE-BY-UBG DE-29T DE-20 DE-91G DE-BY-TUM DE-83 DE-739 DE-706 |
| owner_facet | DE-473 DE-BY-UBG DE-29T DE-20 DE-91G DE-BY-TUM DE-83 DE-739 DE-706 |
| physical | XVII, 580 S. Diagramme. - graph. Darst. |
| psigel | ebook |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Shoup, Victor ca. 20./21. Jh. Verfasser (DE-588)125580517X aut A computational introduction to number theory and algebra Victor Shoup 2. ed. Cambridge [u.a.] Cambridge Univ. Press 2009 XVII, 580 S. Diagramme. - graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 566 - 571 "Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory." -- Publisher's description. Datenverarbeitung Number theory Data processing Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Computeralgebra (DE-588)4010449-7 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 s DE-604 Computeralgebra (DE-588)4010449-7 s 1\p DE-604 Zahlentheorie (DE-588)4067277-3 s 2\p DE-604 Erscheint auch als Online-Ausgabe, kostenpflichtig 978-0-511-81454-9 http://www.shoup.net/ntb/ Verlag kostenfrei Volltext Digitalisierung UB Bamberg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016683585&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Shoup, Victor ca. 20./21. Jh A computational introduction to number theory and algebra Datenverarbeitung Number theory Data processing Algebraische Zahlentheorie (DE-588)4001170-7 gnd Zahlentheorie (DE-588)4067277-3 gnd Computeralgebra (DE-588)4010449-7 gnd |
| subject_GND | (DE-588)4001170-7 (DE-588)4067277-3 (DE-588)4010449-7 |
| title | A computational introduction to number theory and algebra |
| title_auth | A computational introduction to number theory and algebra |
| title_exact_search | A computational introduction to number theory and algebra |
| title_exact_search_txtP | A computational introduction to number theory and algebra |
| title_full | A computational introduction to number theory and algebra Victor Shoup |
| title_fullStr | A computational introduction to number theory and algebra Victor Shoup |
| title_full_unstemmed | A computational introduction to number theory and algebra Victor Shoup |
| title_short | A computational introduction to number theory and algebra |
| title_sort | a computational introduction to number theory and algebra |
| topic | Datenverarbeitung Number theory Data processing Algebraische Zahlentheorie (DE-588)4001170-7 gnd Zahlentheorie (DE-588)4067277-3 gnd Computeralgebra (DE-588)4010449-7 gnd |
| topic_facet | Datenverarbeitung Number theory Data processing Algebraische Zahlentheorie Zahlentheorie Computeralgebra |
| url | http://www.shoup.net/ntb/ http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016683585&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT shoupvictor acomputationalintroductiontonumbertheoryandalgebra |