Arne Winterhof

Alexander Pott

Pascale Charpin

Character Sums and Polynomials

Finite Fields and Their Applications

This book is based on the invited talks of the "RICAM-Workshop on Finite Fields and Their Applications: Character Sums and Polynomials" held at the Federal Institute for Adult Education (BIfEB) in Strobl, Austria, from September 2-7, 2012. Finite fields play important roles in many application areas such as coding theory, cryptography, Monte Carlo and quasi-Monte Carlo methods, pseudorandom number generation, quantum computing, and wireless communication. In this book we will focus on sequences, character sums, and polynomials over finite fields in view of the above mentioned application areas: Chapters 1 and 2 deal with sequences mainly constructed via characters and analyzed using bounds on character sums. Chapters 3, 5, and 6 deal with polynomials over finite fields. Chapters 4 and 9 consider problems related to coding theory studied via finite geometry and additive combinatorics, respectively. Chapter 7 deals with quasirandom points in view of applications to numerical integration using quasi-Monte Carlo methods and simulation. Chapter 8 studies aspects of iterations of rational functions from which pseudorandom numbers for Monte Carlo methods can be derived. The goal of this book is giving an overview of several recent research directions as well as stimulating research in sequences and polynomials under the unified framework of character theory.

Kai-Uwe Schmidt

Combinatorics and finite fields are of great importance in modern applications such as in the analysis of algorithms, in information and communication theory, and in signal processing and coding theory. This book contains survey articles on topics such as difference sets, polynomials, and pseudorandomness.

Difference Sets, Polynomials, Pseudorandomness and Applications

Combinatorics and Finite Fields

Harald Niederreiter

This textbook effectively builds a bridge from basic number theory to recent advances in applied number theory. It presents the first unified account of the four major areas of application where number theory plays a fundamental role, namely cryptography, coding theory, quasi-Monte Carlo methods, and pseudorandom number generation, allowing the authors to delineate the manifold links and interrelations between these areas. Number theory, which Carl-Friedrich Gauss famously dubbed the queen of mathematics, has always been considered a very beautiful field of mathematics, producing lovely results and elegant proofs. While only very few real-life applications were known in the past, today number theory can be found in everyday life: in supermarket bar code scanners, in our cars’ GPS systems, in online banking, etc. Starting with a brief introductory course on number theory in Chapter 1, which makes the book more accessible for undergraduates, the authors describe the four main application areas in Chapters 2-5 and offer a glimpse of advanced results that are presented without proofs and require more advanced mathematical skills. In the last chapter they review several further applications of number theory, ranging from check-digit systems to quantum computation and the organization of raster-graphics memory. Upper-level undergraduates, graduates and researchers in the field of number theory will find this book to be a valuable resource.

Applied Number Theory

Solomon W. Golomb

Matthew G. Parker

This volume contains the refereed proceedings papers of the Fifth International ConferenceonSequencesandtheirApplications(SETA2008), heldinLexington, Kentucky (USA), September 14-18, 2008. The conference SETA is well est- lished in the mathematics and computer science community. Topics of "SETA" include - Randomness of sequences - Correlation (periodic and aperiodic types) and combinatorial aspects of - quences (di?erence sets) - Sequences with applications in coding theory and cryptography - Sequences over ?nite ?elds/rings/function ?elds - Linear and nonlinear feedback shift register sequences - Sequences for radar distance ranging, synchronization, identi?cation, and hardware testing - Sequences for wireless communication - Pseudorandom sequence generators - Correlation and transformation of Boolean functions - Multidimensional sequences and their correlation properties - Linear and nonlinear complexity of sequences. This year's proceedings contain 32 contributed papers. All papers have been thoroughly refereed by at least two referees. Most of the refereeing was done by members of the program committee. We thank all of them for their help. We arealsogratefultoSerdarBozta, s, Nina Brandst] atter, Nicolas Courtois, Fa- wei Fu, Honggang Hu, Alexander Kholosha, Emmanuel Prou?, Martin Ro ]tteler, Ayineedi Venkateswarlu and Bo-Yin Yang for their assistance in the reviewing process. In addition to the contributed papers, we had four invited lectures given by Claude Carlet (University of Paris 8, France), Pierre L'Ecuyer (Universit ́ede Montr ́ eal, Canada), Guang Gong (University of Waterloo, Canada) and Robert McEliece (California Institute of Technology, USA). You ?nd papers based on these invited talks in this volume.

5th International Conference Lexington, KY, USA, September 14-18, 2008, Proceedings

Sequences and Their Applications - SETA 2008

This book constitutes the refereed proceedings of the 8th International Conference on Sequences and Their Applications, SETA 2014, held in Melbourne, VIC, Australia, in November 2014. The 24 full papers presented together with 2 invited papers were carefully reviewed and selected from 36 submissions. The papers have been organized in topical sections on Boolean functions, perfect sequences, correlation of arrays, relative difference sets, aperiodic correlation, pseudorandom sequences and stream ciphers, crosscorrelation of sequences, prime numbers in sequences, OFDM and CDMA, and frequency-hopping sequences.

8th International Conference, Melbourne, VIC, Australia, November 24-28, 2014, Proceedings

Sequences and Their Applications - SETA 2014

Chaoping Xing

Gerhard Larcher

Friedrich Pillichshammer

Harald Niederreiter's pioneering research in the field of applied algebra and number theory has led to important and substantial breakthroughs in many areas. This collection of survey articles has been authored by close colleagues and leading experts to mark the occasion of his 70th birthday. The book provides a modern overview of different research areas, covering uniform distribution and quasi-Monte Carlo methods as well as finite fields and their applications, in particular, cryptography and pseudorandom number generation. Many results are published here for the first time. The book serves as a useful starting point for graduate students new to these areas or as a refresher for researchers wanting to follow recent trends.

Arne Winterhof

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