Computational and Algorithmic Problems in Finite Fields

Mathematics and Its Applications (Soviet Series)

Managing Editor:

M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board:

A. A. KIRILLOV, MGU, Moscow, Russia, Cl.S. Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, Russia, C1.S. N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, Russia, Cl.S. S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, Russia, Cl.S. Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, Russia, Cl.S.

Volume 88

Computational and Algorithmic Problems in Finite Fields by

Igor E. Shparlinski School of Mathematics , Physics, Computing and Electronics, Macquarie University, Sydney, New South Wales, Australia

SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

ISBN 978-94-010-4796-8 ISBN 978-94-011-1806-4 (eBook) DOI 10.1007/978-94-011-1806-4

Printed on acid-free pa per

AII Rights Reserved © 1992 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1992 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

SERIES EDITOR'S PREFACE

'Et moi, ...• si j'avait su comment en revenir. je n'y semis point a1J6.'

JulesVeme

The series is divergent; therefore we may be able to do something with il

O. Heaviside

One service mathematics bas rendemI !be human race. It bas put common sense back where it belongs. on tile topmost sbelf next to tile dusty canister labelled 'discarded nonsense'.

Eric T.BeIl

Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlineari­ties abound. Similarly, all kinds of pans of mathematics serve as tools for other pans and for other sci­ences.

Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser­vice topology has rendered mathematical physics ... '; 'One service logic has rendered computer science ... '; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way fonn pan of the raison d' 8tre of this series.

This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote

"Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sci­ences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experi­mental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large­scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. "

By and large, all this still applies today. It is still true that at first sight mathematics seems rather frag­mented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available.

If anything, the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no pan of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combina­torics, algebra, probability, and so on.

In addition. the applied scientist needs to cope increasingly with the nonlinear world and the extra

vi

mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics were linear we would have no fun with transistors and computers; we would have no TV; in fact you would not be reading these lines.

There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of 'wormhole' paths. There is no telling where all this is leading - fortunately.

Thus the original scope of the series, which for various (sound) reasons now comprises five subseries: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are used in others. Thus the series still aims at books dealing with:

a central concept which plays an important role in several different mathematical and/or scientific specialization areas; new applications of the results and ideas from one area of scientific endeavour into another; influences which the results, problems and concepts of one field of enquiry have, and have had, on the development of another.

Mathematics is about many things, e.g. concepts, structures and computation. The last aspect is very important if all the elegant theory is also to be applied. And then efficiency of computation, complexity of algorithms is very important. This book is about computation and algorithms in finite fields. It is an exhaustive and unique treatise on the topic that surveys the results of some papers. Many of these were published in the former USSR and the many important results in them were and are insufficiently known in the West. This is by far the most complete treatment of computation and algorithms for finite fields that I know of. The amount of information in this volume is staggering.

1bc shortest path between two truths in !be rcaJ

domain passes through the complex domain.

J.Hadamard

La physique ne nous donne pas seulement

I'occasion de r~udre des probl~es ... elle

IIOUS fait presseotir Ia solution.

H. P0incar6

Bussum, September 1992

Never lend books, for no one ever returns them;

the only books I have in my library are books

that other folk have lent me.

Anatole France

1bc function of an expert is not to be more right

than other people, but to be wrong for more

sophisticated reasons. David Butler

Michiel Hazewinkel

CONTENTS

Series Editor's Preface iI Preface _

Acknowledgements iii

Notations IX

Introduction 1

Chapter 1. Polynomial Factorization 7 1. Univariate factorization 7 2. Multivariate factorization 16 3. Other polynomial decompositions 20

Chapter 2. Finding irreducible and primitive polynomials 21

1. Construction of irreducible polynomials 21 2. Construction of primitive polynomials 27

Chapter 3. The distribution of irreducible and primitive polynomials 30

1. Distribution of irreducible and primitive polynomials 30 2. Irreducible and primitive polynomials of a given height and

weight 42 3. Sparse polynomials 46 4. Applications to algebraic number fields 47

Chapter 4. Bases and computation in finite fields 49 1. Construction of some special bases for finite fields 49 2. Discrete logarithm and Zech's logarithm 54 3. Polynomial multiplication and multiplicative complexity in

finite fields 56 4. Other algorithms in finite fields 64

Chapter 5. Coding theory and algebraic curves 72

1. Codes and points on algebraic curves 72 2. Codes and exponential sums 86 3. Codes and lattice packings and coverings 92

Chapter 6. Elliptic curves 99 1. Some general properties 99 2. Distribution of primitive points on elliptic curves 105

Chapter 7. Recurrent sequences in finite fields and cyclic linear codes 109

1. Distribution of values of recurrent sequences 109 2. Applications of recurrent sequences 113 3. Cyclic codes and recurrent sequences 116

Chapter 8. Finite fields and discrete mathematics 122 1. Cryptography and permutation polynomials 122 2. Graph theory, combinatorics, Boolean functions 129 3. Enumeration problems in finite fields 136 Chapter 9. Congruences 139 1. Optimal coefficients and pseudo-random numbers 139 2. Residues of exponential functions 143 3. Modular arithmetic 148 4. Other applications 150

Chapter 10. Some related problems 153 1. Integer factorization, primality testing and the greatest

common divisor 153 2. Computational algebraic number theory 155 3. Algebraic complexity theory 156 4. Polynomials with integer coefficients 158

Appendix 1 161

Appendix 2 164

Appendix 3 165

Addendum 166

References 191

Index 238

PREFACE

This book is mainly devoted to some computational and algorithmic problems in finite fields such as, for example, polynomial factorization, finding irreducible and primitive polynomials, distribution of these primitive polynomials and of primitive points on elliptic curves, constructing bases of various types, and new applications of finite fields to other areas of mathematics. For completeness we include two special chapters on some recent advances and applications of the theory of congru­ences (optimal coefficients, congruential pseudo-random number generators, modu­lar arithmetic etc.), and computational number theory (primality testing, factoring integers, computing in algebraic number theory, and etc.).

The problems considered here have many applications in Computer Science, Coding Theory, Cryptography, Numerical Methods and so on.

There are a few books devoted to more general questions, but these results have not been collected under one cover. In the present work the author has attempted to point out new links among different areas oftheory of finite fields. Moreover, during the last years a lot of important results have appeared in our area, which previously could be found only in widely scattered conference proceedings and journals. In particular, we extensively review results which originally appeared only in Russian, and are not well known to mathematicians outside the former USSR.

This book may be used for graduate level courses as well as for undergraduate students, who are oriented towards (future) research in various areas of Computer Science, Coding Theory, Cryptography, Number Theory, Discrete Mathematics. The required background for this book is essentially limited to knowledge of the basic facts on finite fields such as one can readily find in the excellent book by R. Lidl and H. Niederreiter, Finite fields, Addison-Wesley, 1983 (and in many other texts).

ix

ACKNOWLEDGEMENTS

The author is very grateful to Steve D. Cohen, Joachim von zur Gathen, Dima Grigoriev, Marek Karpinski, Rudolf Lidl, Oscar Moreno, Gary L. Mullen, Harald Niederreiter, Gregori Perel'muter, Alf J. van der Poorten, Claus P. Schnorr, Victor Shoup, Alexei Skorobogatov, Sergei Stepanov, Sergei Tarasov, Michael Tsfasman, Sergei Vladuts for fruitful discussions of the problems considered here, critical re­marks, and for many additional references. Some open questions stated below were suggested by them.

The author would like to thank them all, as well as many other mathematicians, for sending reprints (a lot of them before they were published).

The author is also thankful to Alexander Polupanov for drawing his attention to some problems in physics that are related to computational number theory.

x

NOTATION

W, iZ, Q, ~, C are the sets of natural, integer, rational, real, complex numbers, respectively;

IF is the set of prime numbers; IFq is a finite field of q = pr elements, pElF, r E W; iF q is the algebraic closure of IF q;

IF; is the multiplicative group of IF q; IFp is identified with the set {O, 1, ... ,p - I}; the input-size of a E IF q is O( r log p) bits (e.g. the description of its r coordinates

ai, 0 ~ aj ~ p - 1, i = 1, ... , r, in some basis IFq over IFp), hence, the input-size of a polynomial f(x) E IFq[x] of degree n is O(nrlogp) bits;

Mn(q), In(q), and Gn(q) are the set of all monic polynomials of degree n over IFq ,

the subset of all irreducible polynomials from Mn(q), and the subset of all primitive polynomials from Mn(q), respectively;

H(f) is the height of f E iZ[XI, ... , xm], i.e. the greatest absolute value of its coefficients;

W(f) is the weight of f E iZ[xl, ... , xm], i.e. the number of its nonzero coeffi­cients;

Tn = {t = (t l , ... ,tn) E iZ I tl,· .. ,tn 2: O,tl +2t2+ .. ·+ntn = n}; f E Mn(q) has a factorization pattern t = (tl,'" ,tn) E Tn, if its factorization

in irreducible polynomials has exactly til polynomials of degree 1I, 1I = 1, ... , n; € denotes any fixed positive number (the implied constants in the symbol "0"

may depend on €); ERH is the Extended Riemann Hypothesis, i.e. the hypothesis that all nontrivial

zeros s of all L-functions have Re s = 1/2; J.l(k) is the Mobius function; <p( k) is the Euler function; 1I( k) is the number of all prime divisors of the non zero integer k; 71'(x) is the number of prime numbers not exceeding x; (a, b) is the greatest common divisor of a and b (which are either integers or

polynomials) ; C~ = n!/k!(n - k)! denote the binomial coefficients; IMI is the size (cardinality) of a set 9Jl; log x = log2 x, In x = loge x, for x 2: 0 and we set log x = In x = 1 for x ~ O. The term "a given finite field IFq" implies that we have given a basis WI, ... , Wr

over the ground field IFp and the multiplication table

r

WiWj = LaijkWk,

k=l

i,j,k = 1, ... ,r.

xi

xii

The term "computing time of an algorithm" denotes the number of elementary bit operations. The term "polynomial algorithm" denotes an algorithm with a computing time which is bounded by a polynomial in the input-size.

The symbol I means the end of a proof.