Computational Number Theory: Proceedings of the Colloquium on Computational Number Theory held at Kossuth Lajos University, Debrecen (Hungary), September 4-9, 1989 [Reprint 2011 ed.] 9783110865950, 9783110123944

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Computational Number Theory

Computational Number Theory Proceedings of the Colloquium on Computational Number Theory held at Kossuth Lajos University, Debrecen (Hungary), September 4-9,1989

Editors Attila Pethö Michael Ε. Pohst Hugh C.Williams Horst Günter Zimmer

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G Walter de Gruyter · Berlin · New York 1991

Editors Attila Pethö, Mathematical Institute, Kossuth Lajos University,Η-4010 Debrecen, Hungary Michael Ε. Pohst, Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstr. 1, D-4000 Düsseldorf 1, FRG Hugh C. Williams, Department of Computer Science, The University of Manitoba, Winnipeg, Mannitoba, Canada R3T 2N2 Horst Günter Zimmer, Fachbereich 9 - Mathematik, Universität des Saarlandes, D-6600 Saarbrücken, FRG 1991 Mathematics Subject Classification: 11-06,11-04; 12-06,12-04; 14-06,14-04. θ Printed on acid free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Colloquium on Computational Number Theory (1989 : Kossuth Lajos University) Computational number theory : proceedings of the Colloquium on Computational Number Theory held at Kossuth Lajos University, Debrecen (Hungary), September 4-9,1989 / editors, Attila Pethö... [et al.]. p. cm. Includes bibliographical references. ISBN 3-11-012394-0 (cloth : acid-free). ~ ISBN 0-89925-674-0 (cloth : acid-free) 1. Number theory—Data processing—Congresses. I. Pethö, Attila, 1950- . II. Title. QA241.C6874 1989 91-15924 512'.7~dc20 CIP

Die Deutsche Bibliothek - Cataloging-in-Publication Data Computational number theory : proceedings of the Colloquium on Computational Number Theory held at Kossuth Lajos University, Debrecen (Hungary), September 4-9,1989 / ed. Attila Pethö... - Berlin ; New York : de Gruyter, 1991 ISBN 3-11-012394-0 NE: Pethö, Attila [Hrsg.]; Colloquium on Computational Number Theory ; Kossuth Lajos Tudomänyegyetem

© Copyright 1991 by Walter de Gruyter & Co., D-1000 Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Printing: Gerike GmbH, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

Preface This volume contains original research and survey papers presented at the Colloquium on Computational Number Theory held in the facilities of the Regional Committee of the Hungarian Academy of Sciences at Debrecen, Hungary, from September 4 to September 9, 1989. There were 65 registered participants from 15 countries. Most of the 35 lectures which were delivered are included in these proceedings, and so are three additional papers by conference participants. The contents of these papers fall well within the areas covered by the colloquium. T h e contributions involve several aspects of computational number theory ranging from effective finiteness results to efficient algorithms in elementary, analytic and algebraic number theory. The topics treated, in particular, relate to finite fields, radix representations, quadratic forms, algebraic number fields, modular forms, elliptic curves and diophantine equations. In addition, two number theoretical software packages were demonstrated on various PCs. T h e wide range of themes of the colloquium was in complete agreement with the intentions of the organizers. The city of Debrecen, located at the eastern edge of Hungary, provided an auspicious environment for a meeting of number theorists coming from the eastern and the western world. Indeed, it was exactly at the time of this conference that Hungary punched the first holes into the iron curtain. This feature distinguished the colloquium from preceding conferences on computational number theory of which proceedings have been published, such as the ones held in 1969 at Oxford and in 1980 at Amsterdam. Another novelty was the above-mentioned presentation of number theoretical software packages, which allowed participants to solve some computational problems on the spot. Many people helped in preparing the conference or in editing the proceedings. We are especially indebted to K. Györy for his various activities as a member of the organizing committee. I. Gaal was a most reliable secretary who, together with I. Nemes, took care of the scientific as well as the social program. M. Pfeifer and J. Schmitt were of great assistance in setting up the various T^X versions of these proceedings. Finally, every paper had one or two referees. W e wish to thank them all. We are grateful also to the staff of Walter de Gruyter & Co. for the cooperative manner in which the publication of this volume was managed.

Saarbrücken. February 1991

Attila Pethö Michael Pohst Hugh C. Williams Horst G. Zimmer

List of contributors Bettina Arenz, Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, D-4000 Düsseldorf 1, Germany Bryan J. Birch, Mathematical Institute, University of Oxford, 24-29 St. Giles, Oxford, OX13LB, England Johannes Buchmann, Fachbereich 14 Informatik, Universität des Saarlandes, D-6600 Saarbrücken, Germany Harvey Cohn, Department of Mathematics, The City College of New York, Convent Ave. at 138th St., New York, N.Y, 10031, USA Reiner Creutzburg, Universität Karlsruhe, Institut für Algorithmen und Kognitive Systeme, P F 6980, D-7500 Karlsruhe, Germany Stephan Düllmann, Fachbereich 14 Informatik, Universität des Saarlandes, D-6600 Saarbrücken, Germany Veikko Ennola, Department of Mathematics, University of Turku, SF-20500 Turku 50, Finland Jan-Hendrik Evertse, Filips van Bourgondiestraat 41 A, NL-3117 SC Schiedam, The Netherlands David Ford, Department of Computer Science, Concordia University, Montreal, Quebec, Canada, H3G 1M8 Istvdn Gaal, Kossuth Lajos University, Mathematical Institute, H-4010 Debrecen Pf.12., Hungary Aleksander Grytczuk, Department of Mathematics, Pedagogical University, Zielona Gora, 65-562, ul.Sucharskiego 18.m.l4., Poland ΚάΙτηάη Györy, Kossuth Lajos University, Mathematical Institute, H-4010 Debrecen Pf.12., Hungary Franz Halter-Koch, Institut für Mathematik, Karl-Franzens Universität Graz, Halbärthgasse 1/1, A-8010 Graz, Österreich Christine Hollinger, Fachbereich 9 Mathematik, Universität des Saarlandes, D-6600 Saarbrücken, Germany

viii

ΒέΙα Kovdcs, Kossuth Lajos University, Mathematical Institute, H-4010 Debrecen Pf.12., Hungary Yuri V. Melnichuk, Lvovszkij Polityehnyicseszkij Insztitut, Kafedra Vicsiszlityelnoj Mat. i progrmmirovanyija, 290000 Lvov, ul. Mira 12, USSR Richard A. Mollin, Department of Mathematics and Statistics, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4 Ken Nakamula, Department of Mathematics, Tokyo Metropolitan University, 2-1-1 Fukazawa, Setagaya-ku, Tokyo, 158 Japan Istvdn Nemes, Kossuth Lajos University, Mathematical Institute, H-4010 Debrecen Pf.12., Hungary Attila Pethö, Kossuth Lajos University, Mathematical Institute, H-4010 Debrecen Pf.12., Hungary Michael Pohst, Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, D-4000 Düsseldorf 1, Germany Johannes Graf von Schmettow, Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, D-4000 Düsseldorf 1, Germany Ursula Schneiders, Fachbereich 9 Mathematik, Universität des Saarlandes, D-6600 Saarbrücken, Germany Ulrich Schröter, Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, D-4000 Düs-seldorf 1, Germany Nicole Schulte, Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, D-4000 Düsseldorf 1, Germany Pascale Serf., Fachbereich 9 Mathematik, Universität des Saarlandes, D-6600 Saarbrücken, Germany Igor E. Shparlinskiy, Institute of Radioengineering Electronics, Academy of Sciences of the USSR, K.Marx av.18, Moscow, GSP-3, 103907 USSR Gabriele Steidl, Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, 0-2500 Rostock, Germany S.A. Stepanov, Steklov Institute of Academy Nauk USSR , ul.Vavilova 42., Moscow GSP-1, 117966 USSR

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R.J. Stroeker, Econometric Institute, Erasmus University, P.O.Box 1738, 3000 DR Rotterdam, The Netherlands Manfred Tasche, Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, 0-2500 Rostock, Germany Bogdan Tropak, Department of Mathematics, Pedagogical University, Zielona Gora, 65-562, ul.Sucharskiego 18.m.l4., Poland Nikos Tzanakis, Department of Mathematics, University of Crete, P.O.Box 1470, Iraklion, Crete, Greece Benne M.M. de Weger, Faculty of Applied Mathematics, University of Twente, P.O.Box 217, 7500 AE Enschede, The Netherlands Hugh C. Williams, Department of Computer Science, The University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 Horst G. Zimmer, Fachbereich 9 Mathematik, Universität des Saarlandes, D-6600 Saarbrücken, Germany

Table of contents Preface List of contributors Table of contents On the construction of primitive elements and primitive normal bases in a finite field. S.A. Stepanov and I.E. Shparlinskiy

v

V11

X1

1

A numerical method for the determination of the cyclotomic polynomial coefficients. A. Grytczuk and B. Tropak Number systems. B. Kovdcs

21

Fast converging series representations of real numbers and their implementations in digital processing. Yu.V. Melnichuk

27

On a polynomial transformation and its application to the construction of a public key cryptosystem. A. Pethö

31

Number-theoretic transforms and a theorem of Sylvester Kronecker Zsigmondy. M. Tasche, G. Steidl and R. Creutzburg

45

A probabilistic class group and regulator algorithm and its implementation. J. Buchmann and S. Düllmann

53

Prime-producing quadratic polynomials and class numbers of quadratic orders. F. Halter-Koch

73

Applications of a new class number two criterion for real quadratic fields. R.A. Mollin

§3

On a solution of a class number two problem for a family of real quadratic fields. R.A. Mollin and H.C. Williams

95

xii

Cubic number fields with exceptional units. V. Ennola

103

Enumeration of totally complex quartic fields of small discriminant. D. Ford

129

Class number computation by cyclotomic or elliptic units. K. Nakamula

139

Computing fundamental units from independent units. B. Arenz A note on index divisors. M. Pohst Computation of independent units in number fields by a combination of the methods of Buchmann / Pethö and Pohst / Zassenhaus.

163

173

U. Schröter

183

Hecke actions on classes of ternary quadratic forms. B.J. Birch

191

Computation of singular moduli by multi-valued modular equations. H. Cohn

213

Congruent numbers and elliptic curves. P. Serf

227

The rank of elliptic curves upon quadratic extension. U. Schneiders and H.G. Zimmer

239

On the resolution of some diophantine equations. I. Gadl

261

Index form equations in cubic number fields. N. Schulte

281

On the practical solution of the Thue-Mahler equation. N. B.M.M. de Weger Some results and on equations and Thue-Mahler equations. J.H.Tzanakis Evertse andThue K. Györy

289 295

xiii

On the solution of the diophantine equation Gn = P(x) with sieve algorithm. I. Nemes

303

On Thue equations associated with certain quartic number fields. R.J. Stroeker

313

KANT - a tool for computations in algebraic number fields. J. Graf v. Schmettow

321

SIMATH - a computer algebra system. C. Hollinger and P. Serf

331

On the construction of primitive elements and primitive normal bases in a finite field S.A. Stepanov and I.E.

Shparlinskiy

A b s t r a c t . The purpose of this survey is to present some recent, results of the authors on finding fast algorithms for the construction of primitive elements, normal bases and primitive normal bases in a finite field. It should be noted that, our results can be used for the development of other fast algorithms in finite fields.

1 Introduction Let τη, η be positive integers, q = pm a power of a prime number p, IFg a finite field of q elements, and I F ^ a finite extension of JFq of degree n. It is well known (see [1], [2]) that each field IF^n has a normal basis over IF^, i.e. a basis of the form n - l

a, a , . . . , ag Further, IFgn has a primitive element ϋ whose powers

ΰ, ΰ2, . . . , ϋ