Kathleen A. S. Quinn

Bounds for Key Distribution Patterns

Abstract. This paper is concerned with the problem of distributing pieces of information to nodes in a network in such a way that any pair of nodes can compute a secure common key but the amount of information stored at each node is small. It has been proposed that a special type of finite incidence structure, called a key distribution pattern (KDP) , might provide a good solution to this problem. We give various lower bounds on the information storage of KDPs. Our main result shows that in general KDP schemes necessarily have greater information storage at the nodes than the minimum possible. This minimum is achieved by a scheme not based on KDPs.

Existence and embeddings of partial Steiner triple systems of order ten with cubic leaves

Denote the set of 21 non-isomorphic cubic graphs of order 10 by L. We first determine precisely which L@?L occur as the leave of a partial Steiner triple system, thus settling the existence problem for partial Steiner triple systems of order 10 with cubic leaves. Then we settle the embedding problem for partial Steiner triple systems with leaves L@?L. This second result is obtained as a corollary of a more general result which gives, for each integer v>=10 and each L@?L, necessary and sufficient conditions for the existence of a partial Steiner triple system of order v with leave consisting of the complement of L and v-10 isolated vertices.

Mendelsohn directed triple systems

Abstract We introduce a class of ordered triple systems which are both Mendelsohn triple systems and directed triple systems. We call these Mendelsohn directed triple systems (MDTS(v,λ)), characterise them, and prove that they exist if and only if λ(v−1)≡0 ( mod 3) . This is the same spectrum as that of regular directed triple systems, of which they are a special case. We also prove that cyclic MDTS(v,λ) exist if and only if λ(v−1)≡0 ( mod 6) . In so doing we simplify a known proof of the existence of cyclic directed triple systems. Finally, we enumerate some ‘small’ MDTS.

On directed designs with block size five

In at-(v, k,λ) directed design the blocks are orderedk-tuples and every orderedt-tuple of distinct points occurs in exactly λ blocks (as a subsequence). We studyt−(v, 5, 1) directed designs witht=3 andt=4. In particular, we construct the first known examples, and an infinite class, of 3-(v, 5, 1) directed designs, and the first known infinite class of 4-(v, 5, 1) directed designs.