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Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials D n (x, a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on F q with q < 200, with only one exception, all belong to the RDPP families established in this paper.

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A theorem of Kneser states that in an abelian group G; if A and B are finite subsets in G and AB ¼ fab : a 2 A; b 2 Bg; then jABj5jAj þ jBj À jHðABÞj where HðABÞ ¼ fg 2 G : gðABÞ ¼ ABg: Motivated by the study of a problem in finite fields, we prove an analogous result for vector spaces over a field E in an extension field K of E: Our proof is algebraic and it gives an immediate proof of Kneser's Theorem. # 2002 Elsevier Science (USA)