Planar functions were introduced by Dembowski and Ostrom () 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)
Let H be a subgroup of the multiplicative group of a finite field. In this note we give a method for constructing permutation polynomials over the field using a bi-jective map from H to a coset of H. A similar, but inequivalent, method for lifting permutation behaviour of a polynomial to an extension field is also given.
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.