Harald Niederreiter

Introduction to finite fields and their applications

The first part of this book presents an introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications. The second part is devoted to a discussion of the most important applications of finite fields especially information theory, algebraic coding theory and cryptology (including some very recent material that has never before appeared in book form). There is also a chapter on applications within mathematics, such as finite geometries. combinatorics. and pseudorandom sequences. Worked-out examples and list of exercises found throughout the book make it useful as a textbook.

Quasi-Monte Carlo methods and pseudo-random numbers

CONTENTS

Point Sets and Sequences with Small Discrepancy.

A systematic theory of a class of point sets called (t, m, s)-nets and of a class of sequences called (t, s)-sequences is developed. On the basis of this theory, point sets and sequences in thes-dimensional unit cube with the smallest discrepancy that is currently known are constructed. Various connections with other areas arise in this work, e.g. with orthogonal latin squares, finite projective planes, finite fields, and algebraic coding theory. Applications of the theory of (t, m, s)-nets to two methods for pseudorandom number generation, namely the digital multistep method and the GFSR method, are presented. Several open problems, mostly of a combinatorial nature, are stated.

Implementation and tests of low-discrepancy sequences

Low-discrepancy sequences are used for numerical integration, in simulation, and in related applications. Techniques for producing such sequences have been proposed by, among others, Halton, Sobol´, Faure, and Niederreiter. Niederreiters sequences have the best theoretical asymptotic properties. The paper describes two ways to implement the latter sequences on a computer and discusses the results obtained in various practical tests on particular integrals.

Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing

loCal orGanIzerS • Piotr krzyżanowski • Marek kwas • leszek Plaskota • Henryk Woźniakowski (chair) PROGRAM COMMITTEE • William Chen (australia) • ronald Cools (Belgium) • Josef dick (australia) • Henri Faure (France) • alan Genz (USa) • Paul Glasserman (USa) • Stefan Heinrich (Germany) • Fred J. Hickernell (USa) • Stephen Joe (new zealand) • aneta karaivanova (Bulgaria) • alexander keller (Germany) • Frances kuo (australia) • Gerhard larcher (austria) • Pierre l’ecuyer (Canada) • Christiane lemieux (Canada) • Makoto Matsumoto (Japan) • Peter Mathé (Germany) • thomas Müller-Gronbach (Germany) • Harald niederreiter (austria) • erich novak (Germany) • art B. owen (USa) • Friedrich Pillichshammer (austria) • leszek Plaskota (Poland) • klaus ritter (Germany) • Wolfgang Ch. Schmid (austria) • nikolai Simonov (russia) • Ian H. Sloan (australia) • Ilya M. Sobol’ (russia) • Jerome Spanier (USa) • Shu tezuka (Japan) • Xiaoqun Wang (China) • Grzegorz Wasilkowski (USa) • Henryk Woźniakowski (chair) (Poland/USa)

Complete mappings of finite fields

We discuss complete mapping polynomials of finite fields, which are a special class of permutation polynomials. Complete mapping polynomials of small degree are classified. Results are obtained on a class of complete mapping binomials and on permutation binomials.

New Developments in Uniform Pseudorandom Number and Vector Generation

A survey of recent and new developments in the areas of uniform pseudorandom number and uniform pseudorandom vector generation is presented. The emphasis is on generators for which a detailed theoretical analysis is available.

Monte Carlo and Quasi-Monte Carlo Methods 2006

This book represents the refereed proceedings of the Seventh International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, held in Ulm (Germany) in August 2006. The proceedings include carefully selected papers on many aspects of Monte Carlo and quasi-Monte Carlo methods and their applications, as well as providing information on current research in these very active areas. Besides covering theory, the book is an excellent resource work for practitioners as well.

Recent trends in random number and random vector generation

A survey of recent work in the areas of uniform pseudorandom number and uniform pseudorandom vector generation is presented. The emphasis is on methods for which a detailed theory is available. A progress report on the construction of quasirandom points for efficient multidimensional numerical integration is also given.

Linear Complexity and Related Complexity Measures for Sequences

We present a survey of recent work on the linear complexity, the linear complexity profile, and the k-error linear complexity of sequences and on the joint linear complexity of multisequences. We also establish a new enumeration theorem on multisequences and state several open problems. The material is of relevance for the assessment of keystreams in stream ciphers.