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Planar functions were introduced by Dembowski and Ostrom ([4]) to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a(More)
We consider the implications of the equivalence of commutative semifields of odd order and planar Dembowski-Ostrom polynomials. This equivalence was outlined recently by Coulter and Henderson. In particular, following a more general statement concerning semifields we identify a form of planar Dembowski-Ostrom polynomial which must define a commutative(More)
The development of Public Key Infrastructures (PKIs) is highly desirable to support secure digital transactions and communications throughout existing networks. It is important to adopt a particular trust structure or PKI model at an early stage as this forms a basis for the PKI's development. Many PKI models have been proposed but use only natural language(More)
We give an alternative proof of the fact that a planar function cannot exist on groups of even order. The argument involved leads us to define a class of functions which we call semi-planar. Through the introduction of an incidence structure we construct semi-biplanes using semi-planar functions. The method involved represents a new approach to constructing(More)
Let F q be a finite field of characteristic p and F q [X] denote the ring of poly-nomials in X over F q. A polynomial f ∈ F q [X] is called a permutation polynomial over F q if f induces a bijection of F q under substitution. A polynomial f ∈ F q [X] is said to be planar over F q if for every non-zero a ∈ F q , the polynomial f (X + a) − f (X) is a(More)
For non-negative integers n we determine the roots of the trinomial X p n − aX − b, with a = 0, over a finite field of characteristic p. Throughout q = p k where p is a prime and k is a positive integer. Let F q be the finite field of order q, F * q be the set of non-zero elements of F q and F q [X] be the ring of polynomials in the indeterminate X over F(More)
Several authors have recently shown that a planar function over a finite field of order q must have at least (q + 1)/2 distinct values. In this note this result is extended by weakening the hypothesis significantly and strengthening the conclusion. We also give an algorithm for determining whether a given bivariate polynomial φ(X, Y) can be written as f (X(More)
Motivated by several recent results, we determine precisely when F k (X d , a) − F k (0, a) is a Dembowski-Ostrom polynomial, where F k (X, a) is a Dickson polynomial of the first or second kind. As a consequence, we obtain a classification of all such polynomials which are also planar; all examples found are equivalent to previously known examples.