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Motivated by several constructions of permutation polynomials done by several authors (most notably by Zieve), we propose a unified treatment for a large set of classes of permutation polynomials of F q . Our approach yields a recipe for constructing several new and old classes of permutation polynomials of F q .

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We count permutation polynomials of F q which are sums of m + 1 ( 2) monomials of prescribed degrees. This allows us to prove certain results about existence of permutation polynomials of prescribed shape.

We use cyclotomy to design new classes of permutation polynomials over finite fields. This allows us to generate many classes of permutation polynomials in an algorithmic way. Many of them are permutation polynomials of large indices.2000 Mathematics Subject Classification. 11T06.