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计算代数数论研究生课程指南
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更新于2024-07-17
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《计算代数数论课程》是Henri Cohen编著的一本专著,它源于作者在波尔多第一大学为研究生们开设的计算数论课程的教学笔记。这本书的独特之处在于,它不仅包含了详尽的算法细节,而且涵盖了比常规课程更为深入的主题,旨在满足两个不同层次的数学家需求:那些可能只需要特定算法详细步骤的数学家,以及希望在算法数论领域进一步探索的数学家。 该书作为 Graduate Texts in Mathematics 系列的第138卷,与一系列其他经典数学教材并列,如J.H.Ewing、F.W.Gehring和P.R.Halmos等编辑的系列作品,涵盖了从基本的函数理论到高级分析、模块论、稳定映射理论、功能分析、代数结构、逻辑理论、线性代数、几何泛函分析、实数和抽象分析、抽象代数、一般拓扑以及 Commutative Algebra 的基础和进阶内容。 Cohen通过精心设计,确保了《计算代数数论课程》既能作为一本实用工具书,提供实用的算法指导,又能作为一门学科的入门指南,激发对算法数论更深层次的理解。书中可能包含诸如素数测试、扩展欧几里得算法、费马小定理的高效实现、椭圆曲线加密算法的基础理论等内容,这些都是现代密码学和信息安全中的核心组成部分。 对于想要系统学习计算数论的研究生或研究者来说,这是一本不可或缺的参考书籍,它不仅提供了扎实的理论基础,还提供了实际操作中的关键技巧和策略。同时,它也展示了数学与计算机科学的紧密融合,是那些寻求将理论转化为实践的数学家的理想读物。 《计算代数数论课程》是一本结合教学与研究价值的深度著作,适合不同层次的数学专业人士,无论是在学术界还是工业界,都能从中收获丰富的知识和技能。
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Contents
XVII
3.5.5. Discussion . . . . . . . . . . . . . . . . .
140
3.6. Additional Polynomial Algorithms. . . . . . . .
141
3.6.1. Modular Methods for Computing GCD's in
Z[X)
141
3.6.2. Factorization of Polynomials over a Number Field
142
3.6.3. A Root Finding Algorithm over C
145
3.7. Exercises for Chapter 3
.....
147
Chapter
4.
Algorithms
for
Algebraic
Number
Theory
I
151
4.1. Algebraic Numbers and Number Fields
.......
151
4.1.1. Basic Definitions and Properties of Algebraic Numbers
151
4.1.2. Number Fields
.................
152
4.2. Representation and Operations on Algebraic Numbers .
156
4.2.1. Algebraic Numbers as Roots of their Minimal Polynomial.
156
4.2.2. The Standard Representation of an Algebraic Number . .
157
4.2.3. The Matrix (or Regular) Representation of an Algebraic Number
158
4.2.4. The Conjugate Vector Representation of an Algebraic Number
159
4.3. Trace, Norm and Characteristic Polynomial
.....
160
4.4.
Discriminants, Integral Bases and Polynomial Reduction
163
4.4.1. Discriminants and Integral Bases
163
4.4.2. The Polynomial Reduction Algorithm
166
4.5. The Subfield Problem and Applications
172
4.5.1. The Subfield Problem Using the
LLL
Algorithm
172
4.5.2. The Subfield Problem Using Linear Algebra over C
173
4.5.3. The Subfield Problem Using Algebraic Algorithms .
175
4.5.4. Applications of
the
Solutions to the Subfield Problem
177
4.6. Orders and Ideals .
179
4.6.1. Basic Definitions
179
4.6.2. Ideals of
ZK
184
4.7. Representation of Modules and Ideals
186
4.7.1. Modules and the Hermite Normal Form
186
4.7.2. Representation of Ideals . . . .
188
4.8. Decomposition of Prime Numbers I . . .
193
4.8.1. Definitions and Main Results . . . . .
194
4.8.2. A Simple Algorithm
for
the Decomposition of Primes
196
4.8.3. Computing Valuations . . . . .
198
4.8.4. Ideal Inversion and
the
Different .
202
4.9. Units and Ideal Classes
205
4.9.1. The Class Group
205
4.9.2. Units and the Regulator
206
4.9.3. Conclusion: the Main Computational Tasks of Algebraic
Number Theory . .
214
4.10. Exercises
for
Chapter 4 . . . . . . . . . . . . . . .
215
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XVIII
Chapter
5.
AlgorithIlls
for
Quadratic
Fields
.
5.1. Discriminant, Integral Basis and Decomposition of Primes
5.2. Ideals and Quadratic Forms
.........
.
5.3. Class Numbers of Imaginary Quadratic Fields
5.3.1. Computing Class Numbers Using Reduced Forms
5.3.2. Computing Class Numbers Using Modular Forms
5.3.3. Computing Class Numbers Using Analytic Formulas
5.4. Class Groups of Imaginary Quadratic Fields
5.4.1. Shanks's Baby Step Giant Step Method
5.4.2. Reduction and Composition of Quadratic Forms
5.4.3. Class Groups Using Shanks's Method
5.5. McCurley's Sub-exponential Algorithm
5.5.1. Outline of
the
Algorithm . . . . . .
5.5.2. Detailed Description of
the
Algorithm
5.5.3. Atkin's Variant . . . . . . . . . .
5.6. Class Groups of Real Quadratic Fields .
5.6.1. Computing Class Numbers Using Reduced Forms
5.6.2. Computing Class Numbers Using Analytic Formulas
5.6.3. A Heuristic Method of
Shanks.
. . . . . . . . .
Contents
218
218
220
226
226
229
232
235
235
238
245
247
247
250
255
257
257
261
263
5.7. Computation of
the
Fundamental Unit and of the Regulator 264
5.7.1. Description of
the
Algorithms . . . . . . . 264
5.7.2. Analysis of the Continued Fraction Algorithm
266
5.7.3. Computation of
the
Regulator.
. .
273
5.8.
The
Infrastructure Method of Shanks
274
5.8.1.
The
Distance Function . . . . . . 274
5.8.2. Description of
the
Algorithm . . .
278
5.8.3. Compact Representation of the Fundamental Unit 280
5.8.4. Other Application and Generalization of the Distance Function
282
5.9. Buchmann's Sub-exponential Algorithm . . . . . . . . .
..
283
5.9.1. Outline of
the
Algorithm . . . . . . . . . . . . . . .
..
284
5.9.2. Detailed Description of Buchmann's Sub-exponential Algorithm
286
5.10.
The
Cohen-Lenstra Heuristics . . . . . . . . . . . . 289
5.10.1. Results and Heuristics for Imaginary Quadratic
Fields.
290
5.10.2. Results and Heuristics for Real Quadratic Fields
292
5.11. Exercises for
Chapter
5 . . . . . . . . . . . . . . .
293
Chapter
6. AlgorithIllS for
Algebraic
NUIllber
Theory
II
297
6.1. Computing
the
Maximal Order . 297
6.1.1.
The
Pohst-Zassenhaus Theorem . 297
6.1.2.
The
Dedekind Criterion
..
. .
299
6.1.3. Outline of
the
Round 2 Algorithm 302
6.1.4. Detailed Description of the Round 2 Algorithm 305
6.2. Decomposition of Prime Numbers II . . . . . . 306
6.2.1. Newton Polygons . . . . . . . . . . . . . 307
6.2.2. Theoretical Description of the Buchmann-Lenstra Method 309
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Contents
XIX
6.2.3. Multiplying and Dividing Ideals Modulo p . . . . . . .
..
311
6.2.4. Splitting of Separable Algebras over F p . • • . . . . . .
.•
312
6.2.5. Detailed Description of the Algorithm
for
Prime Decomposition 314
6.3. Computing Galois Groups
316
6.3.1. The Resolvent Method
316
6.3.2. Degree 3
319
6.3.3. Degree 4 319
6.3.4. Degree 5
322
6.3.5. Degree 6 323
6.3.6. Degree 7
325
6.3.7. A List of Test Polynomials 327
6.4. Examples of Families of Number Fields
328
6.4.1. Making Tables of Number Fields 328
6.4.2. Cyclic Cubic Fields
..
. . . . . . 330
6.4.3. Pure Cubic Fields . . . . . . . . . 337
6.4.4. Decomposition of Primes in Pure Cubic Fields
341
6.4.5. General Cubic
Fields.
. . . . . . . . . . .
345
6.5. Computing the Class Group, Regulator and Fundamental Units 346
6.5.1. Ideal Reduction . . . . . . . . . . . . . . . . . .
..
346
6.5.2. Computing the Relation Matrix . . . . . . . . . . .
..
348
6.5.3. Computing the Regulator and a System of Fundamental Units
351
6.5.4. The General Class Group and Unit
Algorithm.
352
6.5.5. The Principal Ideal Problem 354
6.6. Exercises for Chapter 6 . . . . . . . . . 356
Chapter
7.
Introduction
to
Elliptic
Curves
360
7.1. Basic Definitions
7.1.1. Introduction
..........
.
7.1.2. Elliptic Integrals and Elliptic Functions
7.1.3. Elliptic Curves over a Field . . . . . .
7.1.4. Points on Elliptic Curves . . . . . . .
7.2. Complex Multiplication and Class Numbers
7.2.1. Maps Between Complex Elliptic Curves
7.2.2. Isogenies . . . . . . . . . . . . . . .
7.2.3. Complex Multiplication
.......
.
7.2.4. Complex Multiplication and Hilbert Class Fields
7.2.5. Modular Equations
.....
7.3. Rank and L-functions . . . . .
7.3.1. The Zeta Function of a Variety
7.3.2. L-functions of Elliptic Curves .
7.3.3. The Taniyama-Weil Conjecture
7.3.4. The Birch and Swinnerton-Dyer Conjecture
7.4. Algorithms for Elliptic Curves
.....
7.4.1. Algorithms for Elliptic Curves over C
7.4.2. Algorithm
for
Reducing a General Cubic
360
360
360
362
365
369
370
372
374
377
378
379
380
381
383
385
387
387
392
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xx
7.4.3. Algorithms for Elliptic Curves over
IFp
7.5. Algorithms for Elliptic Curves over Q
7.5.1. Tate's algorithm . . . . . . . . . .
7.5.2. Computing rational points . . . . .
7.5.3. Algorithms for computing
the
L-function
7.6. Algorithms for Elliptic Curves with Complex Multiplication
7.6.1. Computing the Complex Values of
j(r)
..
7.6.2. Computing the Hilbert Class Polynomials .
7.6.3. Computing Weber Class Polynomials.
7.7. Exercises
for
Chapter 7
......
.
Chapter
8.
Factoring
in
the
Dark
Ages
8.1. Factoring and Primality
Testing.
8.2. Compositeness
Tests.
. . . . . . .
8.3. Primality Tests . . . . . . . . . .
8.3.1. The Pocklington-Lehmer
N - 1 Test
8.3.2. Briefly, Other Tests
8.4. Lehmann's Method . .
8.5. Pollard's
p Method . .
8.5.1. Outline of the Method
8.5.2. Methods for Detecting Periodicity
8.5.3. Brent's Modified Algorithm .
8.5.4. Analysis of the Algorithm
8.6. Shanks's Class Group Method
8.7. Shanks's SQUFOF
8.8. The
p - I-method .
8.8.1. The First Stage .
8.8.2. The Second Stage
8.8.3. Other Algorithms of
the
Same Type
8.9. Exercises for Chapter 8
.....
Chapter
9.
Modern
Primality
Tests
9.1. The Jacobi Sum Test
9.1.1. Group Rings of Cyclotomic Extensions
9.1.2. Characters, Gauss Sums and Jacobi Sums
9.1.3. The Basic Test . . . .
9.1.4. Checking Condition
C
p
•••••••
9.1.5. The Use of Jacobi Sums . . . . . .
9.1.6. Detailed Description of
the
Algorithm
9.1.7. Discussion . . .
....
9.2. The Elliptic Curve Test
9.2.1. The Goldwasser-Kilian Test
9.2.2. Atkin's Test
9.3. Exercises for Chapter 9
Contents
396
399
399
402
405
407
407
408
408
409
412
412
414
416
416
417
418
419
419
420
422
423
426
427
431
432
433
434
435
437
438
438
440
442
447
449
455
457
459
459
463
467
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Contents
XXI
Chapter
10.
Modern
Factoring
Methods
469
10.1. The Continued Fraction Method
469
10.2. The Class Group Method . . . . . . .
473
10.2.1. Sketch of the Method . . . . . . . .
473
10.2.2. The Schnorr-Lenstra Factoring Method 474
10.3. The Elliptic Curve Method
476
10.3.1. Sketch of the Method . . . . . . . .
476
10.3.2. Elliptic Curves Modulo N . . . . . . 477
10.3.3. The ECM Factoring Method of Lenstra
479
10.3.4. Practical Considerations . . . . . . .
481
10.4. The Multiple Polynomial Quadratic Sieve
482
10.4.1. The Basic Quadratic Sieve Algorithm .
483
10.4.2. The Multiple Polynomial Quadratic Sieve
484
10.4.3. Improvements to the MPQS Algorithm
486
10.5. The Number Field Sieve
........
487
10.5.1. Introduction . . . . . . . . . . . . .
487
10.5.2. Description of the Special NFS when
h(K)
= 1
488
10.5.3. Description of the Special NFS when
h(K)
> 1
492
10.5.4. Description of the General NFS
......
493
10.5.5. Miscellaneous Improvements
to
the Number Field Sieve
495
10.6. Exercises
for
Chapter
10
...
. . . . .
496
Appendix
A.
Packages for
Number
Theory
498
Appendix
B.
Some
Useful
Tables . . . . .
503
B.1. Table of Class Numbers of Complex Quadratic Fields
503
B.2. Table of Class Numbers and Units of Real Quadratic Fields
505
B.3. Table of Class Numbers and Units of Complex Cubic Fields 509
B.4. Table of Class Numbers and Units of Totally Real Cubic
Fields.
511
B.5. Table of Elliptic Curves
514
Bibliography.
517
Index.
. . . . 529
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