Polynomials Let p be a prime and q = pe. Let Fq denote the nite eld of order q and F

q represent the set of non-zero elements of Fq. The ring of polynomials in the indeterminate X with coecients from Fq will be represented by Fq[X]. A polynomial f 2 Fq[X] which permutes Fq under evaluation is called a permutation polynomial of Fq. Permutation polynomials have important ap-plications in cryptography. This is because one of the basic requirements of a mapping used to encrypt a message is that it be invertible so that the origi-nal message can be recovered. In particular, Dembowski-Ostrom polynomials have been used for a cryptographic application in the public key cryptosys-tem HFE, see [7]. There the author states that \it seems dicult to choose f (a DO polynomial) such that it is a permutation". It is the purpose of this article to provide some examples of Dembowski-Ostrom permutations. We consider this problem in the purely theoretical spirit of problem P2 of [5]. We do not claim that any of the classes identied in this article could be used to provide a \secure " cryptosystem when implemented in HFE. The class of polynomials of primary interest in this article is now dened. De nition 1. The polynomial f 2 Fq [X] is called a Dembowski-Ostrom polynomial (DO polynomial) if f has the shape f(X) = nX i;j=0 aijX pi+pj: This class of polynomials was described by Dembowski and Ostrom in [4]. In that article, the authors considered projective planes of order n which admitted a collineation group of order n2. They introduced the notion of a planar function as an aid for describing these planes. A polynomial g 2 Fq[X