Permutation polynomials over finite fields. (English) Zbl 0808.11069

Mullen, Gary L.

Mullen, Gary L. (ed.) et al., Finite fields, coding theory, and advances in communications and computing. Proceedings of the international conference on finite fields, coding theory, and advances in communications and computing, held at the University of Nevada, Las Vegas, USA, August 7-10, 1991. New York: Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. 141, 131-151 (1993).

A polynomial \(f\) over a commutative ring \(R\) with identity is called a permutation polynomial PP of \(R\) if \(f\) induces a permutation of \(R\). The paper gives an excellent survey of results obtained on PP over finite fields \(\mathbb{F}_ q\) since the comprehensive treatise of PPs in the book “Finite fields” by R. Lidl and H. Niederreiter in 1983. It expands and continues the paper “When does a polynomial over a finite field permute the elements of the field?” by R. Lidl and G. L. Mullen [Am. Math. Mon. 95, No. 3, 243-246 (1988; Zbl 0653.12010)]. Known results on PP are indicated and the main classes of PPs are discussed. Furthermore, a review of the literature on PPs since 1983 is given.
A polynomial \(f\) in several indeterminates over \(\mathbb{F}_ q\) is called a permutation polynomial of \(\mathbb{F}_ q\) if the equation \(f(x_ 1, \ldots,x_ n)\) \(=\alpha\) has \(q^{n-1}\) solutions in \(\mathbb{F}_ q^ n\) for each \(\alpha\in \mathbb{F}_ q\). Properties and problems on such polynomials are stated. Another section is devoted to a discussion of polynomials \(f\in \mathbb{F}_ q\) which induce permutations on the set of all \(m\times m\)-matrices over \(\mathbb{F}_ q\).
A brief discussion of applications of PP in cryptography and combinatorics and the enumeration of papers concerning PP over residue class rings of the integers, Galois rings etc. concludes the paper. The extensive list of references will be a great help for all researchers in the field of PPs.
For the entire collection see [Zbl 0771.00039].

Reviewer: W.B.Müller (Klagenfurt)

MSC:

11T06	Polynomials over finite fields
11-02	Research exposition (monographs, survey articles) pertaining to number theory

Keywords:

permutations of finite fields; polynomial functions; bibliography; permutation polynomial; survey; finite fields; cryptography; combinatorics; residue class rings of the integers; Galois rings

Citations:

Zbl 0653.12010