Joint work with Jerzy Jaworski and Katarzyna Rybarczyk Sensor networks, that is networks of nodes with sensing ability, and wireless communication capacity are proving very useful in fields of environment and industrial monitoring, as well as security. Since the nodes are assumed to be as cheap as possible (with the so far unattained ideal being "smart dust"), and have limited energy capacity (batteries), they can not perform many operations nor standard protocols used to ensure security in more capable devices. Of particular interest is the problem of key distribution in wireless sensor networks, made difficult by lack of, or very limited, ability to perform public-key operations as well as vulnerability to node compromise -the nodes can not be tamper-resistant, so physical compromise of a node compromises all of its key material. Previous work in this area has mostly concentrated on the so-called random key predistribution methods, introduced by Eschenauer and Gligor [10].Numerous enhancements to the random key predistribution methods have been proposed, chief among them the utilization of Blom's scheme [1] to decrease the memory consumption and communication overhead as well as ensuring that as long as the number of compromised nodes remains under a certain threshold, the scheme remains secure [8], or using polynomial-based threshold schemes [5] to achieve the same improvements [12].An interesting class of approaches to the problem has emerged, which aims to improve the properties of key distribution schemes by utilizing deployment knowledge, that is knowledge of the physical location of the nodes. The most commonly considered schemes make use of threshold key predistribution schemes based on Blom's scheme, and use a square deployment grid [9]. Only a few proposals exist for non-square deployment grid models with no analytical results but rather simulation ones, some utilizing Blom's scheme [14], and some utilizing polynomial threshold schemes [11,15].
Figure 1. Arrangement of hexagonal clustersWe consider the following model of the sensor grid. The area on which sensor nodes are deployed consists of hexagons (see fig. 1), with sensor deployment points corresponding to centers of the hexagons. It is assumed that the communication range (both for sending and receiving) of sensor nodes is equal to the circumradius of the hexagon. More precisely, lest s be communication range, let H be the division of the deployment area into hexagons, each with edge length of s 2 . We will call H a hexagon grid. Let H + be H with added hexagons, adjacent to the border hexagons of H. To each of the hexagons in H and H + we may assign coordinates (i, j) ∈ I and (i, j) ∈ I + , respectively (see fig. 1). To each hexagon H i,j ∈ H + we will assign a set S i,j of m keys (for different heksagons those sets will be disjoint) and we will attribute a set V i,j of N sensors to each hexagon H i,j ∈ H. Moreover, each sensor v ∈ V i,j will be deployed over a middle point of