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On the permutation behaviour of Dickson polynomials of the second kind. (English) Zbl 1032.11056

Let \(q = p^e\) be a prime power and let \(\mathbb F_q\) denote the finite field of order \(q\). The Dickson polynomials of the first kind and the Dickson polynomials of the second kind (DPSK) are defined by \(g_k(X,a) = \sum_{i=0}^{\lfloor k/2 \rfloor}\frac{k}{k-i}{k-i \choose i}{-a}^iX^{k-2i}\) and \(f_k(X,a) = \sum_{i=0}^{\lfloor k/2 \rfloor}{k-i \choose i}{-a}^iX^{k-2i}\), \(a \neq 0 \in \mathbb F_q\), respectively. \(g_k(X,a)\) is a permutation polynomial (PP) over \(\mathbb F_q\), i.e., it induces a permutation on \(\mathbb F_q\) if and only if \(\gcd(k,q^2-1) = 1\). It seems to be much more difficult to decide whether or not \(f_k(X,a)\) is a PP over \(\mathbb F_q\). For the case that \(q = 3^e\) and \(a \neq 0 \in \mathbb F_q\) is a non-square the authors describe a class of DPSK which are PPs over \(\mathbb F_q\). The result expands the known PPs among DPSK in characteristic \(3\) and simplifies the description of classes given in [M. Henderson and R. Matthews, N. Z. J. Math. 27, 227–244 (1998; Zbl 0976.12002)].

MSC:

11T06 Polynomials over finite fields
12E10 Special polynomials in general fields

Citations:

Zbl 0976.12002
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References:

[1] Cohen, S. D., Dickson polynomials of the second kind that are permutations, Canad. J. Math., 46, 225-238 (1994) · Zbl 0796.11054
[2] Cohen, S. D., Dickson permutations, Number-Theoretic and Algebraic Methods in Computer Science (Moscow, 1993) (1995), World Scientific Publishing: World Scientific Publishing River Edge, p. 29-51 · Zbl 0924.11015
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[6] Henderson, M.; Matthews, R., Dickson polynomials of the second kind which are permutation polynomials over a finite field, New Zealand J. Math., 27, 227-244 (1998) · Zbl 0976.12002
[7] Lidl, R.; Mullen, G. L.; Turnwald, G., Dickson Polynomials (1993), Longman Scientific and Technical: Longman Scientific and Technical Essex
[8] R. Matthews, Permutation Polynomials in One and Several Variables, Ph.D. thesis, University of Tasmania, 1982.; R. Matthews, Permutation Polynomials in One and Several Variables, Ph.D. thesis, University of Tasmania, 1982.
[9] Nobauer, W., Über eine Klasse von Permutationspolynomen und die dadurch dargestellten Gruppen, J. Reine Angew. Math., 231, 215-219 (1968) · Zbl 0159.05402
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