Prime Numbers and Computer Methods for Factorization

480 Pages·1994·15.227 MB·English

by  Hans Riesel (auth.)

Progress in Mathematics
Volume 126
Series Editors
J. Oesterle
A. Weinstein
Hans Riesel
Prime Numbers
and Computer Methods
for Factorization
Second Edition
Springer Science+ Business Media, LLC
Hans Riesel
Department of Mathematics
The Royal Institute of Technology
S-I00 44 Stockholm
Sweden
Library of Congress Cataloging-in-Publication Data
Riesel, Hans, 1929-
Prime numbers and computer methods for factorization I Hans
Riesel. --2nd ed.
p. cm. --(Progress in mathematics ; v. 126)
Includes bibliographical references and index.
Contents: v. l. Fundamental algorithms
ISBN 978-1-4612-6681-5 ISBN 978-1-4612-0251-6 (eBook)
DOI 10.1007/978-1-4612-0251-6
l. Numbers, Prime--Data processing. 2. Factorization
(Mathematics)--Data processing. I. Title. II. Series : Progress in
mathematics (Boston, Mass.) ; vol. 126.
II. Title. III. Series.
QA246.R54 1994 94-27688
512'.72--dc20 CIP
Printed on acid-free paper
© Springer Science+Business Media New York 1994
Originally published by BirkMuser Boston in 1994
Softcover reprint oft he hardcover 2nd edition 1994
Copyright is not claimed for works of U.S. Government employees.
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ISBN 978-1-4612-6681-5
Typeset by the author.
9 8 7 654 3 2 1
PREFACE
In this book the author treats four fundamental and apparently simple prob
lems. They are: the number of primes below a given limit, the approximate
number of primes, the recognition of prime numbers and the factorization of large
numbers. A chapter on the details of the distribution of the primes is included as
well as a short description of a recent application of prime numbers, the so-called
RSA public-key cryptosystem. The author is also giving explicit algorithms and
computer programs. Whilst not claiming completeness, the author has tried to give
all important results known, including the latest discoveries. The use of comput
ers has in this area promoted a development which has enormously enlarged the
wealth of results known and that has made many older works and tables obsolete.
As is often the case in number theory, the problems posed are easy to under
stand but the solutions are theoretically advanced. Since this text is aimed at the
mathematicaIly inclined layman, as weIl as at the more advanced student, not all of
the proofs of the results given in this book are shown. Bibliographical references
in these cases serve those readers who wish to probe deeper. References to recent
original works are also given for those who wish to pursue some topic further.
Since number theory is seldom taught in basic mathematics courses, the author
has appended six sections containing all the algebra and number theory required
for the main body of the book. There are also two sections on multiple precision
computations and, finaIly, one section on Stieltjes integrals. This organization
of the subject-matter has been chosen in order not to disrupt the reader's line of
thought by long digressions into other areas. It is also the author's hope that the
text, organized in this way, becomes more readable for specialists in the field. Any
reader who gets stuck in the main text should first consult the appropriate appendix,
and then try to continue.
The six chapters of the main text are quite independent of each other, and
need not be read in order.
For those readers who have a computer (or even a programmable calculator)
available, computer programs have been provided for many of the methods de
scribed. In order to achieve a wide understanding, these programs are written in
the high-level programming language PASCAL. With this choice the author hopes
that most readers will be able to translate the programs into the language of the
computer they use with reasonable effort.
v
PREFACE
At the end of the book a large amount of results are collected in the form
of tables, originating partly from the author's own work in this field. All tables
have been verified and up-dated as far as possible. Also in connection with the
tables, bibliographical references are given to recent or to more extensive work in
the corresponding area.
The text is an up-dated version of an earlier book by the same author: "En
bok om primtal," existing in Swedish only.
The author is very much indebted to Richard Brent, Arne Fransen, Gunnar
Hellstrom, Hans Karlgren, D. H. Lehmer, Thorkil Naur, Andrzej Schinzel, Bob
Vaughan and many others for their reading of the manuscript and for suggest
ing many improvements. Thanks are also due to Beatrice Frock for revising the
English, and to the late Ken Clements for reading and correcting one of the last
versions of the manuscript.
The author wishes you a pleasant reading!
Stockholm, February 1985
PREFACE TO THE SECOND EDITION
During the last ten years the science of computational number theory has seen
great advances. Several important new methods have been introduced to recognize
prime numbers and to factor composite integers. These advances have stimulated
the author to add subsections on the applications of the elliptic curve method and on
the number field sieve to the main text as well as two new appendices covering the
basics of these new subjects.-Also, very many minor updatings and corrections
of the text have been made.
The author wishes to express his thanks to the many readers who have written
during the years and proposed improvements of the text. Special thanks are going
to Richard Brent, Fran~ois Morain, Peter Montgomery and in particular to Harvey
Dubner and Wilfrid Keller, who all helped during various stages of the preparation
of the new manuscript.
Stockholm, June 1994
vi
CONTENTS
Chapter 1. The Number of Primes Below a Given Limit
What Is a Prime Number? . . . . . . . . . . . 1
The Fundamental Theorem of Arithmetic ..... 2
Which Numbers Are Primes? The Sieve of Eratosthenes 2
General Remarks Concerning Computer Programs 4
A Sieve Program . . . . . . . 5
Compact Prime Tables . . . . . 7
Hexadecimal Compact Prime Tables 9
Difference Between Consecutive Primes 9
The Number of Primes Below x 10
Meissel's Formula 12
Evaluation of Pk (x, a) 12
Lehmer's Formula 13
Computations 14
A Computation Using Meissel's Formula 18
A Computation Using Lehmer's Formula 20
A Computer Program Using Lehmer's Formula 22
Mapes'Method 23
Deduction of Formulas 24
A Worked Example 26
Mapes' Algorithm 30
Programming Mapes' Algorithm 32
Recent Developments 33
Results ..... 34
Computational Complexity 35
Comparison Between the Methods Discussed 35
Bibliography ........ . 36
Chapter 2. The Primes Viewed at Large
Introduction . . . . . . . . . . 37
No Polynomial Can Produce Only Primes 37
Formulas Yielding All Primes .... 39
The Distribution of Primes Viewed at Large. Euclid's Theorem 40
The Formulas of Gauss and Legendre for JT(x). The Prime Number
Theorem ...... . 41
The Chebyshev Function O(x) 44
The Riemann Zeta-function 44
vii
CONTENTS
The Zeros of the Zeta-function 47
Conversion From f(x) Back to JT(x) 49
The Riemann Prime Number Formula 50
The Sign oflix - JT(x) • . . . . 52
The InHuence of the Complex Zeros of I;(s) on JT(x) 53
The Remainder Term in the Prime Number Theorem 56
Effective Inequalities for JT (x), P., and () (x) 56
The Number of Primes in Arithmetic Progressions 57
Bibliography ............ . 58
Chapter 3. Subtleties in the Distribution of Primes
The Distribution of Primes in Short Intervals 60
Twins and Some Other Constellations of Primes 60
Admissible Constellations of Primes 62
The Hardy-Littlewood Constants 64
The Prime k-Thples Conjecture 66
Theoretical Evidence in Favour of the Prime k-Thples Conjecture 67
Numerical Evidence in Favour of the Prime k-Thples Conjecture 68
The Second Hardy-Littlewood Conjecture 68
The Midpoint Sieve . . . . . . . . 70
Modification of the Midpoint Sieve 70
Construction of Superdense Admissible Constellations 71
Some Dense Clusters of Primes . . . . . . . . 73
The Distribution of Primes Between the 1\\'0 Series 4n + I and 4n + 3 73
Graph of the Function JT4.3(X) - JT4.I(X) 74
The Negative Regions ..... . 74
The Negative Blocks . . . . . . . 77
Large Gaps Between Consecutive Primes 78
The Cramer Conjecture 79
Bibliography 82
Chapter 4. The Recognition of Primes
Introduction . . . . . . . . . . . . . 84
Tests of Primality and of Compositeness . . . 84
Factorization Methods as Tests of Compositeness 85
Fermat's Theorem as Compositeness Test 85
Fermat's Theorem as Primality Test 85
Pseudoprimes and Probable Primes 86
A Computer Program for Fermat's Test 87
The Labor Involved in a Fermat Test 88
Carmichael Numbers 89
Euler Pseudo primes ..... 90
Strong Pseudoprimes and a Primality Test 91
A Computer Program for Strong Pseudoprime Tests 93
Counts of Pseudoprimes and Carmichael Numbers 94
viii
CONTENTS
Rigorous Primality Proofs . . . . . . 95
Lehmer's Converse of Fermat's Theorem 96
Formal Proof of Theorem 4.3 . . . 97
Ad Hoc Search for a Primitive Root 98
The Use of Several Bases 99
Fermat Numbers and Pepin's Theorem 100
Cofactors of Fermat Numbers 102
Generalized Fermat Numbers 102
A Relaxed Converse of Fermat's Theorem 103
Proth's Theorem 104
Tests of Compositeness for Numbers of the form N = h . 2" ± k 105
An Alternative Approach 105
Certificates of Primality 106
Primality Tests of Lucasian Type 107
Lucas Sequences 107
The Fibonacci Numbers 108
Large Subscripts 108
An Alternative Deduction 111
Divisibility Properties of the Numbers U" 112
Primality Proofs by Aid of Lucas Sequences 115
Lucas Tests for Mersenne Numbers 117
A Relaxation of Theorem 4.8 . 120
Pocklington's Theorem . . . . 121
Lehmer-Pocklington's Theorem 122
Pocklington-Type Theorems for Lucas Sequences 123
Primality Tests for Integers of the form N = h ·2" - 1, when 3th 124
Primality Tests for N = h . 2" - 1, when 31h 125
The Combined N - I and N + 1 Test 129
Lucas Pseudoprimes . . . . 130
Modem Primality Proofs 130
The Jacobi Sum Primality Test 131
Three Lemmas . 132
Lenstra's Theorem ... . 134
The Sets P and Q ... . 135
Running Time for the APRCL Test 136
Elliptic Curve Primality Proving, ECPP 136
The Goldwasser-Kilian Test 137
Atkin's Test 138
Bibliography 139
Chapter 5. Classical Methods of Factorization
Introduction . . . . . . . . . 141
When Do We Attempt Factorization? . . . 141
Trial Division . . . . . . . . . . . 141
A Computer Implementation of Trial Division 143
Euclid's Algorithm as an Aid to Factorization 145
ix
CONTENTS
Fermat's Factoring Method 147
Legendre's Congruence . 149
Euler's Factoring Method 151
Gauss' Factoring Method 152
Legendre's Factoring Method 155
The Number of Prime Factors of Large Numbers 156
How Does a Typical Factorization Look? 157
The Erdos-Kac Theorem ....... . 158
The Distribution of Prime Factors of Various Sizes 159
Dickman's Version of Theorem 5.4 161
A More Detailed Theory ...... . 161
The Size of the kth Largest Prime Factor of N 162
Smooth Integers ......... . 164
Searching for Factors of Certain Forms 165
Legendre's Theorem for the Factors of N = an ± bn 165
Adaptation to Search for Factors of the Form p = 2kn + I 169
Adaptation of Trial Division . . . . . . . . . . . 169
Adaptation of Fermat's Factoring Method . . . . . . 170
Adaptation of Euclid's Algorithm as an Aid to Factorization 171
Bibliography ............... . 171
Chapter 6. Modem Factorization Methods
Introduction . . . . . 173
Choice of Method . . . . . 173
Pollard's (p - I)-Method . . 174
Phase 2 of the (p - I)-Method 176
The (p + I)-Method . . . . 177
Pollard's rho Method 177
A Computer Program for Pollard's rho Method 180
An Algebraic Description of Pollard's rho Method 182
Brent's Modification of Pollard's rho Method 183
The Pollard-Brent Method for p = 2kn + 1 185
Shanks' Factoring Method SQUFOF 186
A Computer Program for SQUFOF 190
Comparison Between Pollard's rho Method and SQUFOF 193
Morrison and Brillhart's Continued Fraction Method CFRAC 193
The Factor Base ......... . 194
An Example of a Factorization with CFRAC 196
Further Details of CFRAC . . 200
The Early Abort Strategy 202
Results Achieved with CFRAC 203
Running Time Analysis of CFRAC 204
The Quadratic Sieve, QS 204
Smallest Solutions to Q(x) == 0 mod p 205
Special Factors . . . . 206
Results Achieved with QS . . . . 206
x