On Key and Multiple Space Key Pre-Distribution Schemes in Wireless Sensor Networks with a Combinatorial Block Design Structure

Abstract

The aim of this work is to present, along with performing their comparative analysis, the structural characteristics and security parameters of the Key Pre-Distribution Schemes (KPDS) and Multiple Space Key Pre-Distribution Schemes (MSKPDS) in wireless sensor networks (WSN) with combinatorial block design (CBD) and combined combinatorial block design (CCBD) structure. A KPDS is determined by a totality of the sets of WSN nodes, in the key memory of which the same keys are pre-recorded for confidential exchange between the nodes of these sets. An MSKPDS differs in that instead of keys, identical pre-keys are written to the specified nodes, which make it possible to calculate, using node identifiers, different keys for confidential communication between the nodes of these sets. As a rule, the specified sets of nodes correspond to blocks or dual blocks of a certain CBD or blocks and dual blocks of a certain CCBD. If direct links between the nodes of these sets are provided in the WSN, and there are no direct links between other nodes, then, by definition, the KPDS, MSKPDS and WSN are built with the structure of this CBD or CCBD. The CBD structure is determined by a graph on the set of its blocks or dual blocks. Vertices that have common elements are regarded as adjacent ones. The nodes and links of the WSN with the CBD structure correspond to the vertices and edges of this graph. In such WSN, the sets of the KPDS or MSKPDS nodes, to which common keys or common pre-keys are assigned, are composed of nodes corresponding to the vertices of the CBD graph containing the element corresponding to this common key or pre-key. The CCBD structure is determined by a graph on a set of pairs (a CBD block, an element of this block). Vertices that have the same first or the same second elements are adjacent ones. The WSN with the CCBD structure is determined by this graph according to the same rule. The KPDS and MSKPDS in it are defined on the sets of all vertices of this graph that have the same first or the same second components (constituting the CCBD blocks and dual blocks). As a result, two keys are assigned to each node. Once key identifiers are assigned to the vertices, the above-mentioned CBD and CCBD graphs become key sharing graphs (KSG). The objective of this work is to perform a comparative analysis of the structural characteristics (the longest length of the shortest path in the KSG, the size of the node key memory), deterministic and probabilistic security characteristics of the MSKPDS with the CBD structure and the MSPDS with the CCBD structure. Two CBDs were chosen for considering specific cases: symmetric (a projective plane) and asymmetric (a unital combinatorial block design). It is shown that the advantages of MSKPDS with the CCBD structure are a smaller share of compromised direct links when one node is compromised and a small amount of key memory (equal to 2 or linearly dependent on the security parameter), and the shortcoming is that there are three-step routes in the key agreement graph. By limiting the security parameter of deterministic MSKPDS, it is possible to improve the security parameter of probabilistic schemes with a significant reduction of key memory. The results of the work can be used in searching for compromise solutions in choosing the structure and security parameters of the KPDS and MSKPDS in the WSN.