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Prime numbers: a computational perspective

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Bibliographic Details
Main Authors:	Crandall, Richard E. 1947-2012 (Author), Pomerance, Carl 1944- (Author)
Format:	Book
Language:	English
Published:	New York [u.a.] Springer 2001
Subjects:	Numeri primi Primzahl Numbers, Prime Numerisches Verfahren Informatik Lehrbuch
Online Access:	Inhaltsverzeichnis
Physical Description:	XV, 545 S.
ISBN:	0387947779

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MARC


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adam_text	RICHARD CRANDALL CARL POMERANCE PRIME NUMBERS A COMPUTATIONAL PERSPECTIVE SPRINGER CONTENTS PREFACE VII 1 PRIMES! 1 1.1 PROBLEMS AND PROGRESS 1 1.1.1 FUNDAMENTAL THEOREM AND FUNDAMENTAL PROBLEM .... 1 1.1.2 TECHNOLOGICAL AND ALGORITHMIC PROGRESS 2 1.1.3 THE INFINITUDE OF PRIMES 5 1.1.4 ASYMPTOTIC RELATIONS AND ORDER NOMENCLATURE 7 1.1.5 HOW PRIMES ARE DISTRIBUTED 9 1.2 CELEBRATED CONJECTURES AND CURIOSITIES 12 1.2.1 TWIN PRIMES 12 1.2.2 PRIME FC-TUPLES AND HYPOTHESIS H 15 1.2.3 THE GOLDBACH CONJECTURE 16 1.2.4 THE CONVEXITY QUESTION 18 1.2.5 PRIME-PRODUCING FORMULAE 18 1.3 PRIMES OF SPECIAL FORM 19 1.3.1 MERSENNE PRIMES 20 1.3.2 FERMAT NUMBERS 24 1.3.3 CERTAIN PRESUMABLY RARE PRIMES 28 1.4 ANALYTIC NUMBER THEORY 30 1.4.1 THE RIEMANN ZETA FUNCTION . . . 30 1.4.2 COMPUTATIONAL SUCCESSES 35 1.4.3 DIRICHLET L-FUNCTIONS 36 1.4.4 EXPONENTIAL SUMS 40 1.4.5 SMOOTH NUMBERS 44 1.5 EXERCISES 45 1.6 RESEARCH PROBLEMS . . 69 2 NUMBER-THEORETICAL TOOLS 77 2.1 MODULAR ARITHMETIC 77 2.1.1 GREATEST COMMON DIVISOR AND INVERSE 77 2.1.2 POWERS 79 2.1.3 CHINESE REMAINDER THEOREM 81 2.2 POLYNOMIAL ARITHMETIC 83 2.2.1 GREATEST COMMON DIVISOR FOR POLYNOMIALS 83 2.2.2 FINITE FIELDS 85 CONTENTS 2.3 SQUARES AND ROOTS 89 2.3.1 QUADRATIC RESIDUES 89 2.3.2 SQUARE ROOTS 93 2.3.3 FINDING POLYNOMIAL ROOTS ,96 2.3.4 REPRESENTATION BY QUADRATIC FORMS 99 2.4 EXERCISES 101 2.5 RESEARCH PROBLEMS 106 RECOGNIZING PRIMES AND COMPOSITES 109 3.1 TRIAL DIVISION 109 3.1.1 DIVISIBILITY TESTS 109 3.1.2 TRIAL DIVISION 110 3.1.3 PRACTICAL CONSIDERATIONS ILL 3.1.4 THEORETICAL CONSIDERATIONS 112 3.2 SIEVING 113 3.2.1 SIEVING TO RECOGNIZE PRIMES 113 3.2.2 ERATOSTHENES PSEUDOCODE 114 3.2.3 SIEVING TO CONSTRUCT A FACTOR TABLE 114 3.2.4 SIEVING TO CONSTRUCT COMPLETE FACTORIZATIONS 115 3.2.5 SIEVING TO RECOGNIZE SMOOTH NUMBERS 115 3.2.6 SIEVING A POLYNOMIAL 116 3.2.7 THEORETICAL CONSIDERATIONS 118 3.3 PSEUDOPRIMES 119 3.3.1 FERMAT PSEUDOPRIMES 120 3.3.2 CARMICHAEL NUMBERS 121 3.4 PROBABLE PRIMES AND WITNESSES 123 3.4.1 THE LEAST WITNESS FOR N 128 3.5 LUCAS PSEUDOPRIMES 130 3.5.1 FIBONACCI AND LUCAS PSEUDOPRIMES 131 3.5.2 GRANTHAM S FROBENIUS TEST . . . 133 3.5.3 IMPLEMENTING THE LUCAS AND QUADRATIC FROBENIUS TESTS 134 3.5.4 THEORETICAL CONSIDERATIONS AND STRONGER TESTS 137 3.5.5 THE GENERAL FROBENIUS TEST 139 3.6 COUNTING PRIMES 140 3.6.1 COMBINATORIAL METHOD 141 3.6.2 ANALYTIC METHOD . . 146 3.7 EXERCISES 150 3.8 RESEARCH PROBLEMS 156 PRIMALITY PROVING 159 4.1 THE N - 1 TEST 159 4.1.1 THE LUCAS THEOREM AND PEPIN TEST 159 4.1.2 PARTIAL FACTORIZATION 160 4.1.3 SUCCINCT CERTIFICATES 164 4.2 THE N + 1 TEST 167 4.2.1 THE LUCAS-LEHMER TEST 167 CONTENTS XIII 4.2.2 AN IMPROVED N + 1 TEST, AND A COMBINED N 2 * 1 TEST . . 170 4.2.3 DIVISORS IN RESIDUE CLASSES 172 4.3 THE FINITE FIELD PRIMALITY TEST 175 4.4 GAUSS AND JACOBI SUMS 180 4.4.1 GAUSS SUMS TEST 180 4.4.2 JACOBI SUMS TEST 185 4.5 EXERCISES 185 4.6 RESEARCH PROBLEMS 189 5 EXPONENTIAL FACTORING ALGORITHMS 191 5.1 SQUARES 191 5.1.1 FERMAT METHOD 191 5.1.2 LEHMAN METHOD 193 5.1.3 FACTOR SIEVES 194 5.2 MONTE CARLO METHODS 195 5.2.1 POLLARD RHO METHOD FOR FACTORING 195 5.2.2 POLLARD RHO METHOD FOR DISCRETE LOGARITHMS 197 5.2.3 POLLARD LAMBDA METHOD FOR DISCRETE LOGARITHMS 199 5.3 BABY-STEPS, GIANT-STEPS 200 5.4 POLLARD P - 1 METHOD 202 5.5 POLYNOMIAL EVALUATION METHOD 204 5.6 BINARY QUADRATIC FORMS 204 5.6.1 QUADRATIC FORM FUNDAMENTALS 204 5.6.2 FACTORING WITH QUADRATIC FORM REPRESENTATIONS 207 5.6.3 COMPOSITION AND THE CLASS GROUP 210 5.6.4 AMBIGUOUS FORMS AND FACTORIZATION 213 5.7 EXERCISES 216 5.8 RESEARCH PROBLEMS 220 6 SUBEXPONENTIAL FACTORING ALGORITHMS 225 6.1 THE QUADRATIC SIEVE FACTORIZATION METHOD 225 6.1.1 BASIC QS 225 6.1.2 BASIC QS: A SUMMARY 230 6.1.3 FAST MATRIX METHODS 232 6.1.4 LARGE PRIME VARIATIONS 234 6.1.5 MULTIPLE POLYNOMIALS 237 6.1.6 SELF INITIALIZATION .238 6.1.7 ZHANG S SPECIAL QUADRATIC SIEVE 240 6.2 NUMBER FIELD SIEVE 242 6.2.1 BASIC NFS: STRATEGY 243 6.2.2 BASIC NFS: EXPONENT VECTORS 244 6.2.3 BASIC NFS: COMPLEXITY 249 6.2.4 BASIC NFS: OBSTRUCTIONS 252 6.2.5 BASIC NFS: SQUARE ROOTS 255 6.2.6 BASIC NFS: SUMMARY ALGORITHM 256 6.2.7 NFS: FURTHER CONSIDERATIONS 258 R CONTENTS 6.3 RIGOROUS FACTORING 265 6.4 INDEX-CALCULUS METHOD FOR DISCRETE LOGARITHMS 266 6.4.1 DISCRETE LOGARITHMS IN PRIME FINITE FIELDS 267 6.4.2 DISCRETE LOGARITHMS VIA SMOOTH POLYNOMIALS AND SMOOTH ALGEBRAIC INTEGERS 269 6.5 EXERCISES 270 6.6 RESEARCH PROBLEMS 279 ELLIPTIC CURVE ARITHMETIC 283 7.1 ELLIPTIC CURVE FUNDAMENTALS 283 7.2 ELLIPTIC ARITHMETIC 287 7.3 THE THEOREMS OF HASSE, DEURING, AND LENSTRA 297 7.4 ELLIPTIC CURVE METHOD 299 7.4.1 BASIC ECM ALGORITHM 300 7.4.2 OPTIMIZATION OF ECM 303 7.5 COUNTING POINTS ON ELLIPTIC CURVES 311 7.5.1 SHANKS-MESTRE METHOD 311 7.5.2 SCHOOF METHOD 315 7.5.3 ATKIN-MORAIN METHOD 321 7.6 ELLIPTIC CURVE PRIMALITY PROVING (ECPP) 332 7.6.1 GOLDWASSER-KILIAN PRIMALITY TEST 332 7.6.2 ATKIN-MORAIN PRIMALITY TEST 336 7.7 EXERCISES 338 7.8 RESEARCH PROBLEMS 344 THE UBIQUITY OF PRIME NUMBERS 351 8.1 CRYPTOGRAPHY 351 8.1.1 DIFFIE-HELLMAN KEY EXCHANGE 351 8.1.2 RSA CRYPTOSYSTEM 353 8.1.3 ELLIPTIC CURVE CRYPTOSYSTEMS (ECCS) 355 8.1.4 COIN-FLIP PROTOCOL 360 8.2 RANDOM-NUMBER GENERATION 361 8.2.1 MODULAR METHODS 361 8.3 QUASI-MONTE CARLO (QMC) METHODS 368 8.3.1 DISCREPANCY THEORY 368 8.3.2 SPECIFIC QMC SEQUENCES 371 8.3.3 PRIMES ON WALL STREET? 373 8.4 DIOPHANTINE ANALYSIS 379 8.5 QUANTUM COMPUTATION 382 8.5.1 INTUITION ON QUANTUM TURING MACHINES (QTMS) 383 8.5.2 THE SHOR QUANTUM ALGORITHM FOR FACTORING 386 8.6 CURIOUS, ANECDOTAL, AND INTERDISCIPLINARY REFERENCES TO PRIMES 388 8.7 EXERCISES 394 8.8 RESEARCH PROBLEMS 398 CONTENTS XV 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC 405 9.1 TOUR OF GRAMMAR-SCHOOL METHODS 405 9.1.1 MULTIPLICATION 405 9.1.2 SQUARING 406 9.1.3 DIV AND MOD 407 9.2 ENHANCEMENTS TO MODULAR ARITHMETIC 409 9.2.1 MONTGOMERY METHOD 409 9.2.2 NEWTON METHODS 412 9.2.3 MODULI OF SPECIAL FORM 416 9.3 EXPONENTIATION 419 9.3.1 BASIC BINARY LADDERS 420 9.3.2 ENHANCEMENTS TO LADDERS 422 9.4 ENHANCEMENTS FOR GCD AND INVERSE 425 9.4.1 BINARY GCD ALGORITHMS 425 9.4.2 SPECIAL INVERSION ALGORITHMS 427 9.4.3 RECURSIVE GCD FOR VERY LARGE OPERANDS 428 9.5 LARGE-INTEGER MULTIPLICATION 431 9.5.1 KARATSUBA AND TOOM-COOK METHODS 431 9.5.2 FOURIER TRANSFORM ALGORITHMS 434 9.5.3 CONVOLUTION THEORY 443 9.5.4 DISCRETE WEIGHTED TRANSFORM (DWT) METHODS 448 9.5.5 NUMBER-THEORETICAL TRANSFORM METHODS 454 9.5.6 SCHONHAGE METHOD 457 9.5.7 NUSSBAUMER METHOD 459 9.5.8 COMPLEXITY OF MULTIPLICATION ALGORITHMS 462 9.5.9 APPLICATION TO THE CHINESE REMAINDER THEOREM 463 9.6 POLYNOMIAL ARITHMETIC 465 9.6.1 POLYNOMIAL MULTIPLICATION 465 9.6.2 FAST POLYNOMIAL INVERSION AND REMAINDERING 466 9.6.3 POLYNOMIAL EVALUATION * . . . 469 9.7 EXERCISES 473 9.8 RESEARCH PROBLEMS 489 BOOK PSEUDOCODE 495 REFERENCES 501 INDEX 527
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spellingShingle	Crandall, Richard E. 1947-2012 Pomerance, Carl 1944- Prime numbers a computational perspective Numeri primi sbt Primzahl Numbers, Prime Numerisches Verfahren (DE-588)4128130-5 gnd Informatik (DE-588)4026894-9 gnd Primzahl (DE-588)4047263-2 gnd
subject_GND	(DE-588)4128130-5 (DE-588)4026894-9 (DE-588)4047263-2 (DE-588)4123623-3
title	Prime numbers a computational perspective
title_auth	Prime numbers a computational perspective
title_exact_search	Prime numbers a computational perspective
title_full	Prime numbers a computational perspective Richard Crandall ; Carl Pomerance
title_fullStr	Prime numbers a computational perspective Richard Crandall ; Carl Pomerance
title_full_unstemmed	Prime numbers a computational perspective Richard Crandall ; Carl Pomerance
title_short	Prime numbers
title_sort	prime numbers a computational perspective
title_sub	a computational perspective
topic	Numeri primi sbt Primzahl Numbers, Prime Numerisches Verfahren (DE-588)4128130-5 gnd Informatik (DE-588)4026894-9 gnd Primzahl (DE-588)4047263-2 gnd
topic_facet	Numeri primi Primzahl Numbers, Prime Numerisches Verfahren Informatik Lehrbuch
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