Distinct Difference Configurations in Groups

A distinct difference configuration is a set of elements contained in a (finite or infinite) group such that the differences between those elements are pairwise distinct. Such configurations may be used to create key predistribution schemes for wireless sensor networks. After detailing preliminary results and describing the applications, we consider distinct difference configurations in the free group. This has applications to networks distributed in a tree-like structure in addition to being an extreme case combinatorially speaking and therefore mathematically interesting in its own right. Furthermore, our results on the free group inform our results on other groups. We provide upper bounds on the number of elements contained in a distinct difference configuration in the free group, in addition to constructions which provide lower bounds. We then consider distinct difference configurations in all groups before looking at the group Zn, rather than restricting ourselves to the group Z2 as much of the existing literature does. Next, we consider a natural generalisation of a distinct difference configuration which we call a difference from unique pair configuration. We describe the relation between these two objects and their appropriateness for use in key predistribution in wireless sensor networks. Finally, we outline some open problems which are worthy of further study.