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1 Dembowski-Ostrom Polynomials and Linearised Polynomials Let p be a prime and q = p e. Let F q denote the finite field of order q and F * q represent the set of non-zero elements of F q. The ring of polynomials in the indeterminate X with coefficients from F q will be represented by F q [X]. A polynomial f ∈ F q [X] which permutes F q under evaluation is… (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)

We report on a recent implementation of Giesbrecht's algorithm for factoring poly-nomials in a skew-polynomial ring. We also discuss the equivalence between factor-ing polynomials in a skew-polynomial ring and decomposing p s-polynomials over a finite field, and how Giesbrecht's algorithm is outlined in some detail by Ore in the 1930's. We end with some… (More)

Note: This is a personal preprint; for correct page numbering and references please see the original paper, the proper citation for which is: Abstract In a recent paper, Kyureghyan and¨Ozbudak proved that u ∈ {1, 2} was a sufficient condition for the polynomial X(X q 2 + X q + (1 − u)X) to be planar over F q 3 , and conjectured the condition was also… (More)