The combinatorics of generalised cumulative arrays.

In this paper we present a combinatorial analysis of generalised cumulative arrays. These are structures that are associated with a monotone collections of subsets of a base set and have properties that find application in areas of information security. We propose a number of basic measures of efficiency of a generalised cumulative array and then study fundamental bounds on their parameters. We then look at a number of construction techniques and show that the problem of finding good generalised cumulative arrays is closely related to the problem of finding boolean expressions with special properties

Anonymity in Shared Symmetric Key Primitives

We provide a stronger definition of anonymity in the context of shared symmetric key primitives, and show that existing schemes do not provide this level of anonymity. A new scheme is presented to share symmetric key operations amongst a set of participants according to a (t, n)-threshold access structure. We quantify the amount of information the output of the shared operation provides about the group of participants which collaborated to produce it.

Hash Families and Cover-Free Families with Cryptographic Applications

This thesis is focused on hash families and cover-free families and their application to problems in cryptography. We present new necessary conditions for generalized separating hash families, and provide new explicit constructions. We then consider three cryptographic applications of hash families and cover-free families. We provide a stronger de nition of anonymity in the context of shared symmetric key primitives and give a new scheme with improved anonymity properties. Second, we observe that nding the invalid signatures in a set of digital signatures that fails batch veri cation is a group testing problem, then apply and compare many group testing algorithms to solve this problem e ciently. In particular, we apply group testing algorithms based on cover-free families. Finally, we construct a one-time signature scheme based on cover-free families with short signatures