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On constructing permutations of finite fields. (English) Zbl 1281.11102
The main topic is finding nontrivial classes of permutation polynomials, which induce permutations of the set of elements of a finite group, with finite fields being of particular interest.
The authors quote Lemma 2.1 from [M. E. Zieve, Int. J. Number Theory 4, No. 5, 851–857 (2008; Zbl 1204.11180)] as Theorem 1.1, which reduces the question as to whether a certain type of polynomial induces a permutation of \(\mathbb{F}_{q}\) to whether another, related polynomial permutes a smaller set. In this theorem, endomorphisms of \(\mathbb{F}_{q}\) of the form \(x\mapsto x^e\) for some integer \(e\) play a role. Analogous results with arbitrary endomorphisms are treated in [M. E. Zieve, Additive number theory. Festschrift in honor of the sixtieth birthday of Melvyn B. Nathanson. New York, NY: Springer, 355–361 (2010; Zbl 1261.11081)].
Taking this approach further, the authors prove a fundamental lemma which contains theorem 1.1 and other constructions of permutation polynomials as special cases. This lemma allows constructions of permutation polynomials of a set \(A\) out of a bijection between two subsets of \(A\).
The case of \(A\) being a finite field is of particular interest, but this lemma can also be applied in the more general context of a finite group. Here it is shown that under certain conditions, a permutation of a finite group can be constructed from two endomorphisms of the group.
This is applied to the multiplicative group of a finite field, with endomorphisms of the form \(x\mapsto x^s\), to elliptic curves over finite fields, with endomorphisms of the form multiplication by \(m\) or the Frobenius map, and to the additive group of a finite field, with endomorphisms given by additive polynomials. This last one is considered the most interesting case.

MSC:
11T06 Polynomials over finite fields
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