Verfügbarkeit: Prime numbers

Prime numbers: a computational perspective

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Bibliographische Detailangaben
Hauptverfasser:	Crandall, Richard E. 1947-2012 (VerfasserIn), Pomerance, Carl 1944- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	New York [u.a.] Springer 2005
Ausgabe:	2. ed.
Schlagworte:	Getallen Números primos Teoria dos números Numbers, Prime Informatik Numerisches Verfahren Primzahl Lehrbuch
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	Auch als Internetausgabe
Beschreibung:	XV, 597 S.
ISBN:	9780387252827

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Datensatz im Suchindex

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adam_text	Contents Preface vu 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 .................. 6 1.1.4 Asymptotic relations and order nomenclature ...... 8 1.1.5 How primes are distributed ................ 10 1.2 Celebrated conjectures and curiosities .............. 14 1.2.1 Twin primes ........................ 14 1.2.2 Prime fc-tuples and hypothesis H ............. 17 1.2.3 The Goldbach conjecture ................. 18 1.2.4 The convexity question .................. 20 1.2.5 Prime-producing formulae ................. 21 1.3 Primes of special form ....................... 22 1.3.1 Mersenne primes ...................... 22 1.3.2 Fermat numbers ...................... 27 1.3.3 Certain presumably rare primes .............. 31 1.4 Analytic number theory ...................... 33 1.4.1 The Riemann zeta function ................ 33 1.4.2 Computational successes .................. 38 1.4.3 Dirichlet ¿-functions .................... 39 1.4.4 Exponential sums ...................... 43 1.4.5 Smooth numbers ...................... 48 1.5 Exercises .............................. 49 1.6 Research problems ......................... 75 2 NUMBER-THEORETICAL TOOLS 83 2.1 Modular arithmetic ......................... 83 2.1.1 Greatest common divisor and inverse ........... 83 2.1.2 Powers ............................ 85 2.1.3 Chinese remainder theorem ................ 87 2.2 Polynomial arithmetic ....................... 89 2.2.1 Greatest common divisor for polynomials ........ 89 2.2.2 Finite fields ......................... 91 2.3 Squares and roots .......................... 96 2.3.1 Quadratic residues ..................... 96 2.3.2 Square roots ........................ 99 2.3.3 Finding polynomial roots ................. 103 2.3.4 Representation by quadratic forms ............ 106 2.4 Exercises .............................. 108 2.5 Research problems ......................... 113 RECOGNIZING PRIMES AND COMPOSITES 117 3.1 Trial division ............................117 3.1.1 Divisibility tests ......................117 3.1.2 Trial division ........................118 3.1.3 Practical considerations ..................119 3.1.4 Theoretical considerations .................120 3.2 Sieving ................................121 3.2.1 Sieving to recognize primes ................121 3.2.2 Eratosthenes pseudocode .................122 3.2.3 Sieving to construct a factor table ............122 3.2.4 Sieving to construct complete factorizations .......123 3.2.5 Sieving to recognize smooth numbers ...........123 3.2.6 Sieving a polynomial ....................124 3.2.7 Theoretical considerations .................126 3.3 Recognizing smooth numbers ...................128 3.4 Pseudoprimes ............................131 3.4.1 Fermat pseudoprimes ...................131 3.4.2 Carmichael numbers ....................133 3.5 Probable primes and witnesses ..................135 3.5.1 The least witness for η ...................140 3.6 Lucas pseudoprimes ........................142 3.6.1 Fibonacci and Lucas pseudoprimes ............142 3.6.2 Grantham s Frobenius test ................145 3.6.3 Implementing the Lucas and quadratic Frobenius tests . 146 3.6.4 Theoretical considerations and stronger tests ......149 3.6.5 The general Frobenius test ................151 3.7 Counting primes ..........................152 3.7.1 Combinatorial method ...................152 3.7.2 Analytic method ......................158 3.8 Exercises ..............................162 3.9 Research problems .........................168 PRIM ALIT Y PROVING 173 4.1 The η - 1 test ...........................173 4.1.1 The Lucas theorem and Pepin test ............173 4.1.2 Partial factorization ....................174 4.1.3 Succinct certificates ....................179 4.2 The », + 1 test ...........................181 4.2.1 The Lucas-Lehmer test ..................181 4.2.2 An improved η + 1 test, and a combined n2 - 1 test . . 184 4.2.3 Divisors in residue classes ................. 186 4.3 The finite field primality test ................... 190 4.4 Gauss and Jacobi sums ...................... 194 4.4.1 Gauss sums test ...................... 194 4.4.2 Jacobi sums test ...................... 199 4.5 The primality test of Agrawal, Kayal, and Saxena (AKS test) . 200 4.5.1 Primality testing with roots of unity ........... 201 4.5.2 The complexity of Algorithm 4.5.1............ 205 4.5.3 Primality testing with Gaussian periods ......... 207 4.5.4 A quartic time primality test ............... 213 4.6 Exercises .............................. 217 4.7 Research problems ......................... 222 EXPONENTIAL FACTORING ALGORITHMS 225 5.1 Squares ...............................225 5.1.1 Fermat method .......................225 5.1.2 Lehman method ......................227 5.1.3 Factor sieves ........................228 5.2 Monte Carlo methods .......................229 5.2.1 Pollard rho method for factoring .............229 5.2.2 Pollard rho method for discrete logarithms .......232 5.2.3 Pollard lambda method for discrete logarithms .....233 5.3 Baby-steps, giant-steps .......................235 5.4 Pollard ρ - 1 method ........................236 5.5 Polynomial evaluation method ..................238 5.6 Binary quadratic forms .......................239 5.6.1 Quadratic form fundamentals ...............239 5.6.2 Factoring with quadratic form representations ......242 5.6.3 Composition and the class group .............245 5.6.4 Ambiguous forms and factorization ............248 5.7 Exercises ..............................251 5.8 Research problems .........................255 SUBEXPONENTIAL FACTORING ALGORITHMS 261 6.1 The quadratic sieve factorization method ............261 6.1.1 Basic QS ..........................261 6.1.2 Basic QS: A summary ...................266 6.1.3 Fast matrix methods ....................268 6.1.4 Large prime variations ...................270 6.1.5 Multiple polynomials ....................273 6.1.6 Self initialization ......................274 6.1.7 Zhang s special quadratic sieve ..............276 6.2 Number field sieve .........................278 6.2.1 Basic NFS: Strategy ....................279 6.2.2 Basic NFS: Exponent vectors ...............280 6.2.3 Basic NFS: Complexity .................. 285 6.2.4 Basic NFS: Obstructions .................. 288 6.2.5 Basic NFS: Square roots .................. 291 6.2.6 Basic NFS: Summary algorithm .............. 292 6.2.7 NFS: Further considerations ................ 294 6.3 Rigorous factoring ......................... 301 6.4 Index-calculus method for discrete logarithms .......... 302 6.4.1 Discrete logarithms in prime finite fields ......... 303 6.4.2 Discrete logarithms via smooth polynomials and smooth algebraic integers ...................... 305 6.5 Exercises .............................. 306 6.6 Research problems ......................... 315 ELLIPTIC CURVE ARITHMETIC 319 7.1 Elliptic curve fundamentals ....................319 7.2 Elliptic arithmetic .........................323 7.3 The theorems of Hasse, Deuring, and Lenstra ..........333 7.4 Elliptic curve method .......................335 7.4.1 Basic ECM algorithm ...................336 7.4.2 Optimization of ECM ...................339 7.5 Counting points on elliptic curves .................347 7.5.1 Shanks-Mestre method ..................347 7.5.2 Schoof method .......................351 7.5.3 Atkin-Morain method ...................358 7.6 Elliptic curve primality proving (ECPP) .............368 7.6.1 Goldwasser-Kilian primality test .............368 7.6.2 Atkin-Morain primality test ................371 7.6.3 Fast primality-proving via ellpitic curves (fastECPP) . 373 7.7 Exercises ..............................374 7.8 Research problems .........................380 THE UBIQUITY OF PRIME NUMBERS 387 8.1 Cryptography ............................387 8.1.1 Diffie-Hellman key exchange ...............387 8.1.2 RSA cryptosystem .....................389 8.1.3 Elliptic curve cryptosystems (ECCs) ...........391 8.1.4 Coin-flip protocol ......................396 8.2 Random-number generation ....................397 8.2.1 Modular methods ......................398 8.3 Quasi-Monte Carlo (qMC) methods ...............404 8.3.1 Discrepancy theory .....................404 8.3.2 Specific qMC sequences ..................407 8.3.3 Primes on Wall Street? ..................409 8.4 Diophantine analysis ........................415 8.5 Quantum computation .......................418 8.5.1 Intuition on quantum Turing machines (QTMs) .....419 8.5.2 The Shoř quantum algorithm for factoring ........422 8.6 Curious, anecdotal, and interdisciplinary references to primes . 424 8.7 Exercises ..............................431 8.8 Research problems .........................436 9 PAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC 443 9.1 Tour of grammar-school methods ................443 9.1.1 Multiplication ........................443 9.1.2 Squaring ...........................444 9.1.3 Div and mod ........................445 9.2 Enhancements to modular arithmetic ...............447 9.2.1 Montgomery method ....................447 9.2.2 Newton methods ......................450 9.2.3 Moduli of special form ...................454 9.3 Exponentiation ...........................457 9.3.1 Basic binary ladders ....................458 9.3.2 Enhancements to ladders .................460 9.4 Enhancements for gcd and inverse ................463 9.4.1 Binary gcd algorithms ...................463 9.4.2 Special inversion algorithms ................465 9.4.3 Recursive-gcd schemes for very large operands .....466 9.5 Large-integer multiplication ....................473 9.5.1 Karatsuba and Toom-Cook methods ...........473 9.5.2 Fourier transform algorithms ...............476 9.5.3 Convolution theory .....................488 9.5.4 Discrete weighted transform (DWT) methods ......493 9.5.5 Number-theoretical transform methods ..........498 9.5.6 Schönhage method .....................502 9.5.7 Nussbaumer method ....................503 9.5.8 Complexity of multiplication algorithms .........506 9.5.9 Application to the Chinese remainder theorem .....508 9.6 Polynomial arithmetic .......................509 9.6.1 Polynomial multiplication .................510 9.6.2 Fast polynomial inversion and remaindering .......511 9.6.3 Polynomial evaluation ...................514 9.7 Exercises ..............................518 9.8 Research problems .........................535 Appendix: BOOK PSEUDOCODE 541 References 547 Index 577
adam_txt	Contents Preface vu 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 . 6 1.1.4 Asymptotic relations and order nomenclature . 8 1.1.5 How primes are distributed . 10 1.2 Celebrated conjectures and curiosities . 14 1.2.1 Twin primes . 14 1.2.2 Prime fc-tuples and hypothesis H . 17 1.2.3 The Goldbach conjecture . 18 1.2.4 The convexity question . 20 1.2.5 Prime-producing formulae . 21 1.3 Primes of special form . 22 1.3.1 Mersenne primes . 22 1.3.2 Fermat numbers . 27 1.3.3 Certain presumably rare primes . 31 1.4 Analytic number theory . 33 1.4.1 The Riemann zeta function . 33 1.4.2 Computational successes . 38 1.4.3 Dirichlet ¿-functions . 39 1.4.4 Exponential sums . 43 1.4.5 Smooth numbers . 48 1.5 Exercises . 49 1.6 Research problems . 75 2 NUMBER-THEORETICAL TOOLS 83 2.1 Modular arithmetic . 83 2.1.1 Greatest common divisor and inverse . 83 2.1.2 Powers . 85 2.1.3 Chinese remainder theorem . 87 2.2 Polynomial arithmetic . 89 2.2.1 Greatest common divisor for polynomials . 89 2.2.2 Finite fields . 91 2.3 Squares and roots . 96 2.3.1 Quadratic residues . 96 2.3.2 Square roots . 99 2.3.3 Finding polynomial roots . 103 2.3.4 Representation by quadratic forms . 106 2.4 Exercises . 108 2.5 Research problems . 113 RECOGNIZING PRIMES AND COMPOSITES 117 3.1 Trial division .117 3.1.1 Divisibility tests .117 3.1.2 Trial division .118 3.1.3 Practical considerations .119 3.1.4 Theoretical considerations .120 3.2 Sieving .121 3.2.1 Sieving to recognize primes .121 3.2.2 Eratosthenes pseudocode .122 3.2.3 Sieving to construct a factor table .122 3.2.4 Sieving to construct complete factorizations .123 3.2.5 Sieving to recognize smooth numbers .123 3.2.6 Sieving a polynomial .124 3.2.7 Theoretical considerations .126 3.3 Recognizing smooth numbers .128 3.4 Pseudoprimes .131 3.4.1 Fermat pseudoprimes .131 3.4.2 Carmichael numbers .133 3.5 Probable primes and witnesses .135 3.5.1 The least witness for η .140 3.6 Lucas pseudoprimes .142 3.6.1 Fibonacci and Lucas pseudoprimes .142 3.6.2 Grantham's Frobenius test .145 3.6.3 Implementing the Lucas and quadratic Frobenius tests . 146 3.6.4 Theoretical considerations and stronger tests .149 3.6.5 The general Frobenius test .151 3.7 Counting primes .152 3.7.1 Combinatorial method .152 3.7.2 Analytic method .158 3.8 Exercises .162 3.9 Research problems .168 PRIM ALIT Y PROVING 173 4.1 The η - 1 test .173 4.1.1 The Lucas theorem and Pepin test .173 4.1.2 Partial factorization .174 4.1.3 Succinct certificates .179 4.2 The », + 1 test .181 4.2.1 The Lucas-Lehmer test .181 4.2.2 An improved η + 1 test, and a combined n2 - 1 test . . 184 4.2.3 Divisors in residue classes . 186 4.3 The finite field primality test . 190 4.4 Gauss and Jacobi sums . 194 4.4.1 Gauss sums test . 194 4.4.2 Jacobi sums test . 199 4.5 The primality test of Agrawal, Kayal, and Saxena (AKS test) . 200 4.5.1 Primality testing with roots of unity . 201 4.5.2 The complexity of Algorithm 4.5.1. 205 4.5.3 Primality testing with Gaussian periods . 207 4.5.4 A quartic time primality test . 213 4.6 Exercises . 217 4.7 Research problems . 222 EXPONENTIAL FACTORING ALGORITHMS 225 5.1 Squares .225 5.1.1 Fermat method .225 5.1.2 Lehman method .227 5.1.3 Factor sieves .228 5.2 Monte Carlo methods .229 5.2.1 Pollard rho method for factoring .229 5.2.2 Pollard rho method for discrete logarithms .232 5.2.3 Pollard lambda method for discrete logarithms .233 5.3 Baby-steps, giant-steps .235 5.4 Pollard ρ - 1 method .236 5.5 Polynomial evaluation method .238 5.6 Binary quadratic forms .239 5.6.1 Quadratic form fundamentals .239 5.6.2 Factoring with quadratic form representations .242 5.6.3 Composition and the class group .245 5.6.4 Ambiguous forms and factorization .248 5.7 Exercises .251 5.8 Research problems .255 SUBEXPONENTIAL FACTORING ALGORITHMS 261 6.1 The quadratic sieve factorization method .261 6.1.1 Basic QS .261 6.1.2 Basic QS: A summary .266 6.1.3 Fast matrix methods .268 6.1.4 Large prime variations .270 6.1.5 Multiple polynomials .273 6.1.6 Self initialization .274 6.1.7 Zhang's special quadratic sieve .276 6.2 Number field sieve .278 6.2.1 Basic NFS: Strategy .279 6.2.2 Basic NFS: Exponent vectors .280 6.2.3 Basic NFS: Complexity . 285 6.2.4 Basic NFS: Obstructions . 288 6.2.5 Basic NFS: Square roots . 291 6.2.6 Basic NFS: Summary algorithm . 292 6.2.7 NFS: Further considerations . 294 6.3 Rigorous factoring . 301 6.4 Index-calculus method for discrete logarithms . 302 6.4.1 Discrete logarithms in prime finite fields . 303 6.4.2 Discrete logarithms via smooth polynomials and smooth algebraic integers . 305 6.5 Exercises . 306 6.6 Research problems . 315 ELLIPTIC CURVE ARITHMETIC 319 7.1 Elliptic curve fundamentals .319 7.2 Elliptic arithmetic .323 7.3 The theorems of Hasse, Deuring, and Lenstra .333 7.4 Elliptic curve method .335 7.4.1 Basic ECM algorithm .336 7.4.2 Optimization of ECM .339 7.5 Counting points on elliptic curves .347 7.5.1 Shanks-Mestre method .347 7.5.2 Schoof method .351 7.5.3 Atkin-Morain method .358 7.6 Elliptic curve primality proving (ECPP) .368 7.6.1 Goldwasser-Kilian primality test .368 7.6.2 Atkin-Morain primality test .371 7.6.3 Fast primality-proving via ellpitic curves (fastECPP) . 373 7.7 Exercises .374 7.8 Research problems .380 THE UBIQUITY OF PRIME NUMBERS 387 8.1 Cryptography .387 8.1.1 Diffie-Hellman key exchange .387 8.1.2 RSA cryptosystem .389 8.1.3 Elliptic curve cryptosystems (ECCs) .391 8.1.4 Coin-flip protocol .396 8.2 Random-number generation .397 8.2.1 Modular methods .398 8.3 Quasi-Monte Carlo (qMC) methods .404 8.3.1 Discrepancy theory .404 8.3.2 Specific qMC sequences .407 8.3.3 Primes on Wall Street? .409 8.4 Diophantine analysis .415 8.5 Quantum computation .418 8.5.1 Intuition on quantum Turing machines (QTMs) .419 8.5.2 The Shoř quantum algorithm for factoring .422 8.6 Curious, anecdotal, and interdisciplinary references to primes . 424 8.7 Exercises .431 8.8 Research problems .436 9 PAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC 443 9.1 Tour of "grammar-school" methods .443 9.1.1 Multiplication .443 9.1.2 Squaring .444 9.1.3 Div and mod .445 9.2 Enhancements to modular arithmetic .447 9.2.1 Montgomery method .447 9.2.2 Newton methods .450 9.2.3 Moduli of special form .454 9.3 Exponentiation .457 9.3.1 Basic binary ladders .458 9.3.2 Enhancements to ladders .460 9.4 Enhancements for gcd and inverse .463 9.4.1 Binary gcd algorithms .463 9.4.2 Special inversion algorithms .465 9.4.3 Recursive-gcd schemes for very large operands .466 9.5 Large-integer multiplication .473 9.5.1 Karatsuba and Toom-Cook methods .473 9.5.2 Fourier transform algorithms .476 9.5.3 Convolution theory .488 9.5.4 Discrete weighted transform (DWT) methods .493 9.5.5 Number-theoretical transform methods .498 9.5.6 Schönhage method .502 9.5.7 Nussbaumer method .503 9.5.8 Complexity of multiplication algorithms .506 9.5.9 Application to the Chinese remainder theorem .508 9.6 Polynomial arithmetic .509 9.6.1 Polynomial multiplication .510 9.6.2 Fast polynomial inversion and remaindering .511 9.6.3 Polynomial evaluation .514 9.7 Exercises .518 9.8 Research problems .535 Appendix: BOOK PSEUDOCODE 541 References 547 Index 577
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genre	(DE-588)4123623-3 Lehrbuch gnd-content
genre_facet	Lehrbuch
id	DE-604.BV023380567
illustrated	Not Illustrated
index_date	2024-07-02T21:16:14Z
indexdate	2024-07-09T21:17:18Z
institution	BVB
isbn	9780387252827
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016563679
oclc_num	61318559
open_access_boolean
owner	DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-634 DE-20 DE-83 DE-29T DE-11 DE-188 DE-898 DE-BY-UBR
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physical	XV, 597 S.
publishDate	2005
publishDateSearch	2005
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publisher	Springer
record_format	marc
spelling	Crandall, Richard E. 1947-2012 Verfasser (DE-588)11390732X aut Prime numbers a computational perspective Richard Crandall ; Carl Pomerance 2. ed. New York [u.a.] Springer 2005 XV, 597 S. txt rdacontent n rdamedia nc rdacarrier Auch als Internetausgabe Getallen gtt Números primos larpcal Teoria dos números larpcal Numbers, Prime Informatik (DE-588)4026894-9 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Primzahl (DE-588)4047263-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Primzahl (DE-588)4047263-2 s Numerisches Verfahren (DE-588)4128130-5 s DE-188 Informatik (DE-588)4026894-9 s 1\p DE-604 Pomerance, Carl 1944- Verfasser (DE-588)122920066 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016563679&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
spellingShingle	Crandall, Richard E. 1947-2012 Pomerance, Carl 1944- Prime numbers a computational perspective Getallen gtt Números primos larpcal Teoria dos números larpcal Numbers, Prime Informatik (DE-588)4026894-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Primzahl (DE-588)4047263-2 gnd
subject_GND	(DE-588)4026894-9 (DE-588)4128130-5 (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_exact_search_txtP	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	Getallen gtt Números primos larpcal Teoria dos números larpcal Numbers, Prime Informatik (DE-588)4026894-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Primzahl (DE-588)4047263-2 gnd
topic_facet	Getallen Números primos Teoria dos números Numbers, Prime Informatik Numerisches Verfahren Primzahl Lehrbuch
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016563679&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT crandallricharde primenumbersacomputationalperspective AT pomerancecarl primenumbersacomputationalperspective

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