| Popis: |
Given a network of users, with certain secure communication requirements, we examine the mathematics that underpins the distribution of the necessary secret information, to enable the secure communications within that network. More precisely, we let f!lJ be a network of users and ~, § be some prede- termined families of subsets of those users. The secret information (keys or subkeys) must be distributed in such a way that for any G E ~, the members of G can communicate securely among themselves without fear of the members of some F E § (that have no users in common with G), colluding together to either eavesdrop on what is being said (and understand the content of the message) or tamper with the message, undetected., In the case when ~ and § comprise of all the subsets of f!lJ that have some fixed cardinality t and w respectively, we have a well-known and much studied problem. However, in this thesis we remove these rigid cardinality constraints and make ~ and § as unrestricted as possible. This allows for situations where the members of ~ and § are completely irregular, giving a much less well-known and less studied problem. Without any regularity emanating from cardinality constraints, the best approach to the study of these general structures is unclear. It is unreason- able to expect that highly regular objects (such as designs or finite geometries) play any significant role in the analysis of such potentially irregular structures. Thus, we require some new techniques and a more general approach. In this thesis we use methods from set theory and ideas from convex analysis in order to construct these general structures and provide some mathematical insight into their behaviour. Furthermore, we analyse these general structures by ex- ploiting the proof techniques of other authors in new ways, tightening existing inequalities and generalising results from the literature. |
