In this thesis we study the relationship between secret sharing schemes and various discrete structures. Chapter 1 contains the mathematical background necessary for the rest of the work. In Chapter 2 t-affine designs are studied. Such designs with block class size two and the maximum number of parallel classes are classified and it is shown that a (3,1, q, q + 1) affine design with q even can always be extended to a (3,1, q, q + 2) affine design. Chapter 3 introduces secret sharing schemes and a new model for secret sharing is proposed in Chapter 4. This model is designed to facilitate the comparison with existing secret sharing models in the literature. We introduce the notion of equivalence of secret sharing schemes and propose a new definition of the information rate of a scheme. In Chapter 5 the problem of constructing schemes for general monotone access structures is considered by using manipulation of logical expressions. We use this method to construct geometrical secret sharing schemes for any monotone access structure. Chapter 6 considers ideal secret sharing schemes. It is shown that any ideal monotone access structure can be associated with a unique matroid. We enumerate the number of distinct rows of an ideal scheme and classify ideal threshold schemes as transversal TD1(t,k,n) designs. In Chapter 7 we look at several new constructions of new secret sharing schemes from existing ones. We use these constructions to show that the distinct rows of an ideal scheme occur with equal frequency and to find a lower bound for the maximum information rate of a monotone access structure. 2 Chapter 8 presents a solution to the problem of how to cope with a participant in a secret sharing scheme becoming untrustworthy after the scheme has been initiated. Finally in Chapter 9 we present a model for a different type of secret sharing scheme and construct such schemes for a family of two-level access structures.

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We start by giving a description of some generalised braids. For the cases of singular and virtual braids we give the algebraic definition and the geometric interpretation. Then we use combinatorial methods to give a proof that the singular braid group is torsion free. Motivated by the formation of labels for the edges of diagrams of virtual knots we give the abstract definiton for biracks and biquandles. Examples of these new algebraic objects are given and some properties are proved as well. We show how biracks provide representations of virtual braids and biquandles provide invariants of virtual knots. Finally, we give some applications of this theory using elements from the Alexander birack as labels for diagrams of virtual knots.

This thesis is organised and presented in nine chapters. The first chapter, the introduction, is in two broad sections and begins by discussing the origins of matched data and the reasons for matching. The general problems of censored data are mentioned and brief descriptions of the past attempts to analyse matched censored data are given, together with their shortcomings. The second section de-fines the notation used and presents the background to the failure time distributions and the types of censoring considered. Chapter 2 is concerned with the analysis of data from the proportional hazards model. The two existing methods are reviewed then a new solution, the integrated method, is proposed and the theory developed. These methods are com-pared in the following chapter, Chapter 3. Chapter 4 concentrates on data arising from the normal theory accelerated failure model. The previous solution is discussed and the results are derived for a new solution based upon the EM algorithm. This is extended to allow for right and interval censored data. The existing solution and the new solution are compared in Chapter 5. Chapter 6 provides analyses of some data sets to compare the results arising from the new methods and the existing solutions, in a practical framework. Chapter 7 discusses the relative merits of the new methods as compared with the previous solutions in the analysis of matched censored data and concludes with an outline of other areas in this field which require further research and the way in which the problems might be tackled. Chapter 8 comprises four appendices whilst Chapter 9 lists the references cited in the text.

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