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The compositional inverse of a class of permutation polynomials over a finite field. (English) Zbl 1023.11061
Let \(q\) be a power of \(2\) and \(n\) be an odd positive integer. Denote the trace mapping from \({\mathbb F}_{q^n}\) to \({\mathbb F}_q\) by \(\text{Tr}(X) = \sum_{i=0}^{n-1}X^{q^i}\). Then as it has been shown in A. Blokhuis, R. Coulter, M. Henderson, and C. O’Keefe [Finite fields and applications V, Augsburg, 1999, 37-42 (2001; Zbl 1009.11064)] that the polynomial \[ f_\alpha(X) = X\operatorname {Tr}(X) + (\alpha+1)X^2 \] induces a permutation of \({\mathbb F}_{q^n}\) for any \(\alpha \in {\mathbb F}_q\setminus\{0,1\}\). The authors provide formulas describing a polynomial \(g_\alpha(X)\) which induces the inverse permutation.

MSC:
11T06 Polynomials over finite fields
12E10 Special polynomials in general fields
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[2] DOI: 10.2307/2323626 · Zbl 0653.12010 · doi:10.2307/2323626
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