This chapter is both a survey of the theory of finite fields as well as an updating of current progress in the important directions. The first section includes an historical review of the subject which traces its development and reviews basic properties. Some brief comments on recent books in the area are given as well as on software packages that include some capability in finite fields. The next two sections deal with various types of bases, polynomials, primitive elements and orthogonal matrices, as well as their construction and enumeration. A survey of recent work on Costas arrays is included. Section 4 gives a survey of existence, construction and characterization of permutation polynomials, with a brief extension into the theory of such polynomials in several indeterminates. A brief description of the discrete logarithm problem includes a discussion of the few cases where formulas for efficient computation are available. The index calculus method is described. Section 6 gives a quite detailed survey of linear recurring sequences.
The final three sections deal with the application of finite fields to cryptography, combinatorics and a more general treatment of pseudo-random sequences, including nonlinear congruential methods and multidimensional analogs of pseudo-random numbers. While the chapter is largely an update on recent research in the theory of finite fields, it can also be read as a comprehensive introduction as well as a companion to the encyclopedic treatment of the subject in the 1983 book of the authors [Finite fields, Cambridge Univ. Press (1984; Zbl 0554.12010)].
For the entire collection see [Zbl 0859.00011].