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Two classes of permutation polynomials over finite fields. (English) Zbl 1230.11146
Summary: Two new classes of permutation polynomials over finite fields are presented:
(i)
$$f(x)= (1- x- x^2)x^{{3^e+1\over 2}}- 1- x+ x^2$$ over $$\mathbb{F}_{3^e}$$, where $$e$$ is a positive even integer;
(ii)
$$g_{n,p}(x)= \displaystyle\sum_{\frac{n}{p} \leq l \leq \frac{n}{p-1}} \frac{n}{l} \binom{l}{n-l(p-1)} \times x^{n-l(p-1)}$$ over $$\mathbb{F}_{p^e}$$, where $$e$$ is a positive integer such that $$e \equiv 0 \pmod 2$$ if $$p=2$$, and $$n = (p-1) p^m + p^{0e}+ p^{1e}+\dots+ p^{(p-1)e}$$, $$(m-1,e) = 1$$.
The permutation polynomial in (i) answers an open question about reversed Dickson polynomials.

##### MSC:
 11T06 Polynomials over finite fields
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##### References:
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