Enhanced descriptions from Syndetics:

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 in mathematics. Inrecent years we have witnessed a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and cryptography. The purpose of this book is to introduce the reader to some of these recent developments. It should be of interest to a wide range of students, researchers and practitioners in the disciplines of computer science, engineering and mathematics. We shall focus our attention on some specific recent developments in the theory and applications of finite fields. While the topics selected are treated in some depth, we have not attempted 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. To limit the size of the volume we have been forced to omit some important applications of finite fields. Some of these missing applications are briefly mentioned in the Appendix along with some key references.

Reviews provided by Syndetics

CHOICE Review

Lately, the theory of finite fields, once the province of algebraists and number theorists, now attracts considerable attention from computer scientists motivated by such applications as public key cryptosystems, error-correcting codes, primality testing, and factorization algorithms. This has led to an explosion of new results, some of a classical nature, others emphasizing issues of computational complexity. Rudolf Lidl and Harold Niederreiter's Finite Fields (1983) is a basic reference that would serve as a prerequisite for both the books under review. One central notion in Applications of Finite Fields is that of a base for one finite field over another, particularly a normal base. Bases have theoretical significance, and constructions of bases with good properties have great practical value because they facilitate the implementation of efficient finite field arithmetic in software or hardware. Another central notion, taken from algebraic geometry, is that of a projective curve over a finite field. Elliptic curves are used here to construct families of public key cryptosystems. The final chapter is devoted to efficient algebraic-geometric error-correcting codes attached to various projective curves. In Shparlinski's book, a central role is played by the estimation of exponential sums and the number of rational points of algebraic varieties over finite fields. These delicate estimates, which often depend on the unproven Extended Riemann Hypothesis, are crucial to many algorithms. The two books cover similar subjects, but where the former is an advanced work treating a small number of topics in considerable detail, the latter is a terse, encyclopedic survey of recent results and open problems, with a comprehensive bibliography of 1,306 items. They complement each other nicely, and both are recommended. Advanced undergraduate through faculty. D. V. Feldman University of New Hampshire