Marie Henderson

Permutations amongst the Dembowski-Ostrom polynomials

We note that certain Dembowski-Ostrom polynomials can be obtained from the product of two linearised polynomials. We examine this subclass for permutation behaviour over finite fields. In particular, a new infinite class of permutation polynomials is identified.

Planar polynomials for commutative semifields with specified nuclei

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 semifield with the nuclei specified. Since any strong isotopy class of commutative semifields must contain at least one example of a commutative semifield described by such a planar polynomial, to classify commutative semifields it is enough to classify planar Dembowski-Ostrom polynomials of this form and determine when they describe non-isotopic commutative semifields. We prove several results along these lines. We end by introducing a new commutative semifield of order 38 with left nucleus of order 3 and middle nucleus of order 32.

A note on constructing permutation polynomials

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 bijective 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.

The compositional inverse of a class of permutation polynomials over a finite field

A new class of bilinear permutation polynomials was recently identified. In this note we determine the class of permutation polynomials which represents the functional inverse of the bilinear class.

A class of functions and their application in constructing semi-biplanes and association schemes

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 semi-biplanes and provides infinite classes of semi-biplanes unlike any known to the authors. For a particular class of semi-planar functions, we provide a method to construct association schemes with two associate classes. Such an association scheme is equivalent to a strongly regular graph

On decomposition of sub-linearised-polynomials

We give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that there is a one-to-one correspondence between decompositions of linearised polynomials and sub-linearised polynomials. This correspondence leads to a formula for the number of indecomposable sub-linearised polynomials of given degree over a finite field. We also show how to extend existing factorisation algorithms over skew-polynomial rings to decompose sub-linearised polynomials without asymptotic cost.

A note on the roots of trinomials over a finite field

where n is a positive integer. Throughout we assume a ∈ F∗q as otherwise f is a binomial and the factorisation is known, see [3]. The trinomial (1) has been considered in [2] for the case a = 1. The article [4] mainly considers the case where n divides k. There is one result in [4] concerning the general case which we include below (see Lemma 2). We determine all roots of the trinomial (1) in Theorem 3 below and then cast these against the previous results described above. We make use of the following lemma. This is essentially [1, Theorem 57].

Giesbrecht's Algorithm, The HFE Cryptosystem and ORE's ps-Polynomials

We report on a recent implementation of Giesbrechts algorithm for factoring polynomials in a skew-polynomial ring. We also discuss the equivalence between factoring polynomials in a skew-polynomial ring and decomposing

On a conjecture on planar polynomials of the form X(Trn(X)−uX)

p^s

Applications of Trusted Review to Information Security

-polynomials over a finite field, and how Giesbrechts algorithm is outlined in some detail by Ore in the 1930s. We end with some observations on the security of the Hidden Field Equation (HFE) cryptosystem, where