Character Sums with Exponential Functions and their Applications

173 Pages·2006·2.06 MB·english

by  Sergei V.Konyagin and Igor E.Shparlinski

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CAMBRIDGE TRACTS IN MATHEMATICS
GeneralEditors
B.BOLLOBAS, F.KIRWAN, P.SARNAK, C.T.C.WALL
132 Character Sums with Exponential
Functions and their Applications

SergeiV.Konyagin IgorE.Shparlinski
MoscowStateUniversity MacquarieUniversity
Character Sums with Exponential
Functions and their Applications
         
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
  
The Edinburgh Building, Cambridge CB2 2RU, UK
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http://www.cambridge.org
©Cambridge University Press 2004
First published in printed format 1999
ISBN 0-511-04036-9 eBook (netLibrary)
ISBN 0-521-64263-9 hardback
Contents
Preface pagevii
Acknowledgement viii
Part one: Preliminaries 1
1 Introduction 3
2 NotationandAuxiliaryResults 8
Part two: Bounds of Character Sums 11
3 BoundsofLongCharacterSums 13
4 BoundsofShortCharacterSums 26
5 BoundsofCharacterSumsforAlmostAllModuli 31
6 BoundsofGaussianSums 37
Part three: Multiplicative Translations of Sets 47
7 MultiplicativeTranslationsofSubgroupsofF∗ 49
p
8 MultiplicativeTranslationsofArbitrarySetsModulo p 59
Part four: Applications to Algebraic Number Fields 63
9 RepresentativesofResidueClasses 65
10 CyclotomicFieldsandGaussianPeriods 76
Part five: Applications to Pseudo-Random Number Generators 89
11 PredictionofPseudo-RandomNumberGenerators 91
12 CongruentialPseudo-RandomNumberGenerators 99
Part six: Applications to Finite Fields 107
13 SmallmthRootsModulo p 109
14 SupersingularHyperellipticCurves 115
15 DistributionofPowersofPrimitiveRoots 131
Part seven: Applications to Coding Theory and Combinatorics 141
16 DifferenceSetsinVp 143
17 DimensionofBCHCodes 148
18 AnEnumerationProbleminFiniteFields 154
Bibliography 157
Index 163
v

Preface
In this book, we consider various questions related to the distribution of
integer powers gx of some integer g > 1 modulo a prime number p with
gcd(g,p) = 1. Possible applications where such results play a central role
include, but are not limited to, linear congruential pseudo-random number
generators,algebraicnumbertheory,thetheoryoffunctionfieldsoverafinite
field,complexitytheory,cryptography,andcodingtheory.
We also consider similar questions in a more general setting related to the
distribution of elements of a finitely generated multiplicative group V withr
generatorsinanalgebraicnumberfieldKofdegreenoverQmoduloaninteger
idealq.
vii
Acknowledgement
The authors are grateful to Alf van der Poorten for his help, support and
valuableadvice.
The authors also wish to thank Hugh Montgomery and Martin Tompa for
ahelpfuldiscussionoftheproblemsconsideredinChapter15andFrancesco
Pappalardiforinformationabouthisresult[72]andmanyfruitfuldiscussions
of some problems. The authors are thankful to Roger Heath-Brown for
providing us with some of his unpublished results. Corey Powell kindly
providedapreliminaryversionof[73].
The authors would like to thank Gang Yu who helped them to find some
mistakesinapreliminaryversionofChapter6.
The research of the first author has been supported by the Grant DMS
9304580fromtheNSF.
The research of the second author has been supported in part by the
GrantA69700294oftheAustralianResearchCouncil.
viii