Computational and algorithmic problems in finite fields /

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