Discrete structures in the theory of secret sharing.
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
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
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.
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
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.