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The areas of communications, computer networks, and signal
processing have undergone rapid development over the past several
years. The advent of VLSI circuitry and increasingly sophisticated
computer hardware and software techniques have made possible the
construction of systems and signal proces sors for. communications
applications not contemplated only a short time ago. The increasing
complexity of communication systems, both by themselves and in
land-based or satellite networks, has created a greater need for
finding use ful mathematical techniques for their analysis. The
rapidly evolving technolo gies involved continue to find exciting
new areas for application, and it remains a challenge for
researchers to keep abreast of developments. In this volume
researchers from a broad cross section of the areas of
communications, signal processing, and computer networks have been
invited to contribute articles to assist readers in learning about
the current state of research and future research directions in
their area. The authors were not given tight guidelines for their
contributions and thus the character and emphasis of each chapter
differs. Although the scope of the areas considered is necessarily
limited in a volume of this size, the coverage here is quite broad
and it is hoped that the reader will find the contents of this
volume to be interesting, useful, and informative."
The theory of finite fields, whose origins can be traced back to
the works of Gauss and Galois, has played a part in various
branches of mathematics, in recent years there has been a
resurgence of interest in finite fields, and this is partly due to
important applications in coding theory and cryptography.
Applications of Finite Fields introduces some of these recent
developments. This book focuses attention on some specific recent
developments in the theory and applications of finite fields. While
the topics selected are treated in some depth, Applications of
Finite Fields does not attempt to be encyclopedic. Among the topics
studied are different methods of representing the elements of a
finite field (including normal bases and optimal normal bases),
algorithms for factoring polynomials over finite fields, methods
for constructing irreducible polynomials, the discrete logarithm
problem and its implications to cryptography, the use of elliptic
curves in constructing public key cryptosystems, and the uses of
algebraic geometry in constructing good error-correcting codes.
This book is developed from a seminar held at the University of
Waterloo. The purpose of the seminar was to bridge the knowledge of
the participants whose expertise and interests ranged from the
purely theoretical to the applied. As a result, this book will be
of interest to a wide range of students, researchers and
practitioners in the disciplines of computer science, engineering
and mathematics. Applications of Finite Fields is an excellent
reference and may be used as a text for a course on the subject.
Since the appearance of the authors' first volume on elliptic curve
cryptography in 1999 there has been tremendous progress in the
field. In some topics, particularly point counting, the progress
has been spectacular. Other topics such as the Weil and Tate
pairings have been applied in new and important ways to
cryptographic protocols that hold great promise. Notions such as
provable security, side channel analysis and the Weil descent
technique have also grown in importance. This second volume
addresses these advances and brings the reader up to date.
Prominent contributors to the research literature in these areas
have provided articles that reflect the current state of these
important topics. They are divided into the areas of protocols,
implementation techniques, mathematical foundations and pairing
based cryptography. Each of the topics is presented in an
accessible, coherent and consistent manner for a wide audience that
will include mathematicians, computer scientists and engineers.
In the past few years elliptic curve cryptography has moved from a
fringe activity to a major system in the commercial world. This
timely work summarizes knowledge gathered at Hewlett-Packard over a
number of years and explains the mathematics behind practical
implementations of elliptic curve systems. Since the mathematics is
advanced, a high barrier to entry exists for individuals and
companies new to this technology. Hence, this book will be
invaluable not only to mathematicians but also to engineers and
computer scientists who want to actually implement such systems.
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