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A new approach to permutation polynomials over finite fields. (English) Zbl 1273.11169
Summary: Let $$p$$ be a prime and $$q=p^{\kappa }$$. We study the permutation properties of the polynomial $$g_{n,q}\in \mathbb F_{p}[x]$$ defined by the functional equation $$\sum _{a\in \mathbb F_{q}}(x+a)^{n} = g_{n,q}(x^{q} - x)$$. The polynomial $$g_{n,q}$$ is a q-ary version of the reversed Dickson polynomial in characteristic 2. We are interested in the parameters $$(n,e;q)$$ for which $$g_{n,q}$$ is a permutation polynomial (PP) of $$\mathbb F_{q^{e}}$$. We find several families of such parameters and obtain various necessary conditions on such parameters. Initial results, both theoretical and numerical, indicate that the class $$g_{n,q}$$ contains an abundance of PPs over finite fields, many of which are yet to be explained and understood.

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