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Title: Discrete structures in the theory of secret sharing.
Author: Martin, Keith Murray.
ISNI:       0000 0001 2445 8655
Awarding Body: Royal Holloway, University of London
Current Institution: Royal Holloway, University of London
Date of Award: 1991
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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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Pure mathematics Mathematics