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.

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."

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.