Computational and Algorithmic Problems in Finite Fields

Computational and Algorithmic Problems in Finite Fields
Author: Igor Shparlinski
Publisher: Springer Science & Business Media
Total Pages: 253
Release: 2012-12-06
Genre: Mathematics
ISBN: 940111806X

This volume presents an exhaustive treatment of computation and algorithms for finite fields. Topics covered include 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 araes of mathematics. For completeness, also included are 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, etc.) The problems considered here have many applications in computer science, coding theory, cryptography, number theory and discrete mathematics. The level of discussion presuppose only a knowledge of the basic facts on finite fields, and the book can be recommended as supplementary graduate text. For researchers and students interested in computational and algorithmic problems in finite fields.

Finite Fields: Theory and Computation

Finite Fields: Theory and Computation
Author: Igor Shparlinski
Publisher: Springer Science & Business Media
Total Pages: 532
Release: 2013-03-09
Genre: Mathematics
ISBN: 940159239X

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, the 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 in clude two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number gener ators, modular arithmetic, etc.) and computational number theory (primality testing, factoring integers, computation in algebraic number theory, etc.). The problems considered here have many applications in Computer Science, Cod ing Theory, Cryptography, Numerical Methods, and so on. There are a few books devoted to more general questions, but the results contained in this book have not till now been collected under one cover. In the present work the author has attempted to point out new links among different areas of the theory of finite fields. It contains many very important results which previously could be found only in widely scattered and hardly available 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.

Finite Fields: Theory, Applications, and Algorithms

Finite Fields: Theory, Applications, and Algorithms
Author: Gary L. Mullen
Publisher: American Mathematical Soc.
Total Pages: 434
Release: 1994
Genre: Mathematics
ISBN: 0821851837

Because of their applications in so many diverse areas, finite fields continue to play increasingly important roles in various branches of modern mathematics, including number theory, algebra, and algebraic geometry, as well as in computer science, information theory, statistics, and engineering. Computational and algorithmic aspects of finite field problems also continue to grow in importance. This volume contains the refereed proceedings of a conference entitled Finite Fields: Theory, Applications and Algorithms, held in August 1993 at the University of Nevada at Las Vegas. Among the topics treated are theoretical aspects of finite fields, coding theory, cryptology, combinatorial design theory, and algorithms related to finite fields. Also included is a list of open problems and conjectures. This volume is an excellent reference for applied and research mathematicians as well as specialists and graduate students in information theory, computer science, and electrical engineering.

Algorithms and Computation

Algorithms and Computation
Author: Ding-Zhu Du
Publisher: Springer Science & Business Media
Total Pages: 708
Release: 1994-07-27
Genre: Computers
ISBN: 9783540583257

This volume is the proceedings of the fifth International Symposium on Algorithms and Computation, ISAAC '94, held in Beijing, China in August 1994. The 79 papers accepted for inclusion in the volume after a careful reviewing process were selected from a total of almost 200 submissions. Besides many internationally renowned experts, a number of excellent Chinese researchers present their results to the international scientific community for the first time here. The volume covers all relevant theoretical and many applicational aspects of algorithms and computation.

Finite Fields and Their Applications

Finite Fields and Their Applications
Author: Pascale Charpin
Publisher: Walter de Gruyter
Total Pages: 288
Release: 2013-05-28
Genre: Mathematics
ISBN: 3110283603

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.

Algorithms and Computations

Algorithms and Computations
Author: John Staples
Publisher: Springer Science & Business Media
Total Pages: 470
Release: 1995-11-15
Genre: Computers
ISBN: 9783540605737

This book presents the refereed proceedings of the 6th International Symposium on Algorithms and Computation, ISAAC '95, held in Cairns, Australia, in December 1995. The 45 revised full papers presented together with the abstracts of three invited talks were selected from a total of 130 submissions. The papers address many current aspects of research and advanced applications of algorithms and computations; among the topics covered are graph theory and graph algorithms, computational geometry, computational logics, searching and sorting, approximation and optimization, algebraic manipulation, and coding.

Cryptography and Computational Number Theory

Cryptography and Computational Number Theory
Author: Kwok Y. Lam
Publisher: Birkhäuser
Total Pages: 376
Release: 2013-03-07
Genre: Computers
ISBN: 3034882955

This volume contains the refereed proceedings of the Workshop on Cryptography and Computational Number Theory, CCNT'99, which has been held in Singapore during the week of November 22-26, 1999. The workshop was organized by the Centre for Systems Security of the Na tional University of Singapore. We gratefully acknowledge the financial support from the Singapore National Science and Technology Board under the grant num ber RP960668/M. The idea for this workshop grew out of the recognition of the recent, rapid development in various areas of cryptography and computational number the ory. The event followed the concept of the research programs at such well-known research institutions as the Newton Institute (UK), Oberwolfach and Dagstuhl (Germany), and Luminy (France). Accordingly, there were only invited lectures at the workshop with plenty of time for informal discussions. It was hoped and successfully achieved that the meeting would encourage and stimulate further research in information and computer security as well as in the design and implementation of number theoretic cryptosystems and other related areas. Another goal of the meeting was to stimulate collaboration and more active interaction between mathematicians, computer scientists, practical cryptographers and engineers in academia, industry and government.

Handbook of Algebra

Handbook of Algebra
Author:
Publisher: Elsevier
Total Pages: 936
Release: 1995-12-18
Genre: Mathematics
ISBN: 0080532950

Handbook of Algebra defines algebra as consisting of many different ideas, concepts and results. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. Each chapter of the book combines some of the features of both a graduate-level textbook and a research-level survey. This book is divided into eight sections. Section 1A focuses on linear algebra and discusses such concepts as matrix functions and equations and random matrices. Section 1B cover linear dependence and discusses matroids. Section 1D focuses on fields, Galois Theory, and algebraic number theory. Section 1F tackles generalizations of fields and related objects. Section 2A focuses on category theory, including the topos theory and categorical structures. Section 2B discusses homological algebra, cohomology, and cohomological methods in algebra. Section 3A focuses on commutative rings and algebras. Finally, Section 3B focuses on associative rings and algebras. This book will be of interest to mathematicians, logicians, and computer scientists.

Modern Computer Algebra

Modern Computer Algebra
Author: Joachim von zur Gathen
Publisher: Cambridge University Press
Total Pages: 811
Release: 2013-04-25
Genre: Computers
ISBN: 1107039037

Now in its third edition, this highly successful textbook is widely regarded as the 'bible of computer algebra'.