In this paper, some combinatorial constructions are presented for
deterministic optimal and nearly optimal KPSs. The present construction
is useful when an optimal configuration is not available for the given . A recent survey on the
constructions of KPSs for DSNs using some combinatorial designs may be
found in [3].
Further a resolvable design is said to be affine if any two blocks
belonging to different resolution classes intersect in constant number
of elements. Some examples of resolvable and affine resolvable designs
are given in subsections 2.2–2.4.
Example 1. Consider the following resolvable
solution of an SRGD design SR9 with parameters as given in [6]:
RI: [(1 5) (2 6) (3 7) (4 8)];
RII: [(2 7) (1 8) (4 5) (3
6)];
RIII: [(4 6) (3 5) (2 8) (1
7)];
RIV: [(3 8) (4 7) (1 6) (2
5)].
The arrangement of
elements in array is
given as:
2.3 type
design
Let elements be
arranged in an array. An
type (Latin square) design
is an arrangement of the
elements in blocks each of size
such that:
Every element occurs at most once in a block;
Every element occurs in r blocks;
Every pair of elements, which are in the same row or in the
same column of the
array, occur together in blocks whereas every other
pair of elements occur together in blocks.
The non–negative integers and are known as parameters of
the L2 – type design and they satisfy the relations:
Some recent constructions of
these designs may be found in [9].
Example 2. Consider an type design as given in [6] with parameters whose blocks are given as:
The arrangement of
elements in array is
given as,
2.4 Nearly Optimal KPS and
Efficiency
A block design is
said to be a configuration if any two blocks intersect in at most
one element. A configuration is a common intersection design ( – CID) if , whenever where are blocks of
the configuration.
This implies that any two disjoint blocks intersect with at least
blocks in common or any two
disjoint blocks can be connected through at least blocks. In general, it is
desirable to construct a configuration with
as large as possible for a given . This maximum value of is denoted by .
Such a configuration is called optimal [1,2]. The KPSs obtained from optimal configurations
are optimal in the sense of local connectivity.
The efficiency of a KPS obtained from a configuration is defined here
as:
Clearly for an optimal
configuration. A KPS will be called nearly optimal if its efficiency
when number of nodes is
very large. If an optimal KPS for a given is not available, we go for
nearly optimal KPS. Some examples of such KPSs are given in Section
3.
A block design may be
used to obtain a KPS having
sensor nodes, is the number of
keys per node with efficiency where is determined using the
corresponding series of designs and if the KPS is
optimal. A nearly optimal KPS will be denoted as
Example 3. Consider an affine resolvable SRGD
design SR23 as listed in [6] with parameters which is a configuration. The resolution classes are given
as:
RI: [(1 2 3) (4 5 6) (7 8 9)];
RII: [(1 5 9) (2 6 7) (3 4
8)];
RIII: [(1 6 8) (2 4 9) (3 5
7)].
It is easy to verify that any two disjoint blocks of this design
intersect with six blocks in common, i.e., for where . Hence SR23 is
an optimal
configuration which is a 6 – CID. The arrangement of elements in array is given as,
3. The Constructions
3.1 Optimal
KPS from Affine Resolvable type
designs
Let
and be
two matrices. Then their
Hadamard product
is also an matrix given
by
The following Series I of type designs may be found in [9]:
Series I: There exists an type design with parameters: The incidence matrix of
above design is given as: where is an
matrix whose ith column contains only +1’s and 0
elsewhere. Now it is shown below that the design with parameters (1) is
affine resolvable.
Since each row sum of and
is one, the design is
resolvable. Clearly the number of resolution classes is and each class contains blocks. The blocks
corresponding to the
columns of form one
resolution class and the blocks corresponding to the columns of form another resolution class . Further let and be two blocks from and respectively and let and be columns of which represent the blocks and respectively. Since the Hadamard
product
contains one exactly once and remaining entry zero in the resultant
column, any two blocks from different resolution classes intersect in
exactly one element. Hence the design is affine resolvable.
The above Series I of type designs can be mapped to an optimal KPS having number of
nodes , number of keys per node,
and
efficiency [vide
Lemma 1 and [3], Section
4.1.2].
Example 4. For we obtain an type design with parameters which can be mapped to an optimal KPS having number of nodes
, keys per node, and
efficiency
3.2 Nearly Optimal KPS
from Resolvable GD Design
Example 5. A GD design with parameters: is a
configuration and the configuration is optimal if the GD design is
affine resolvable and
Proof. Consider a GD design with parameters: . Let and be any two distinct blocks of the
GD with above mentioned parameters. Since any pair of distinct elements
occur together in either one block or no block, we have
and hence the above GD design is a configuration. The proof of second part follows from
Lemma 1 of [3].
The following Series of resolvable GD designs using generalized
Hadamard matrices may be found in [7]:
Series II: There exists a GD design with parameters:
where is a prime and .
First, we show that any two disjoint blocks of Series II intersect
with blocks in common.
Let and be any two disjoint blocks
and be their
cartesian product if we consider them as sets. It is sufficient to count
the blocks in which the elements (or unordered pairs) of occur. In there are total
pairs out of which pairs of elements belong to the rows of array on which the GD design
is based and thus these pairs cannot be part of any block of the GD
design as . Hence
The Series I may be mapped to a nearly optimal KPS having sensor
nodes, keys
per node, and efficiency given as:
which tends to 1 as 
Example 6. Consider the following resolvable
solution of an SRGD design SR26 with parameters: as given in [6],
RI: [(1 2 3) (7 8 12) (5 9 10) (4 6
11)];
RII: [(1 11 12) (3 5 7) (6 8 10) (2 4
9)];
RIII: [(1 5 6) (3 4 8) (7 9 11) (2 10
12)];
RIV: [(1 8 9) (2 6 7) (3 10 11) (4 5
12)].
The arrangement of
elements in array is
given as,
Consider disjoint blocks and of RI and consider the cartesian product . We
count the number of blocks in which these pairs occur. Since the pairs
and of elements belong to first,
second and third rows respectively of the array, these pairs cannot be part of any block of SR26
as . Clearly the
disjoint blocks intersect with six blocks: (1 11 12), (3 5 7), (3 4 8),
(2 10 12), (1 8 9), (2 6 7) in common. It can be also verified that any
pair of remaining disjoint blocks also intersects with six blocks in
common. Hence and SR26
is a CID.
The following Table 1 lists nearly optimal KPSs using Series II
having number of sensor nodes 1000 for different values of and with . The
design numbers 16 listed below may
be found in [6].
Table 1 Nearly Optimal KPSs from SRGD Designs
No.
|
SRGD Parameters
|
KPSs
|
1
|
|
|
2
|
|
|
3
|
|
|
4
|
|
|
5
|
|
|
6
|
|
|
7
|
|
|
8
|
|
|
9
|
|
|
10
|
|
|
11
|
|
|
12
|
|
|
13
|
|
|
14
|
|
|
15
|
|
|
16
|
|
|
4. Resiliency
In a configuration, the compromise of random nodes affect a given
link with probability roughly equal to: Clearly for a larger resiliency, fail should have a smaller value [1].
Example 7. For Series II yields nearly optimal configuration.
This may be mapped to a KPS with 961 sensor nodes. Suppose 10 nodes are
compromised, then . This implies that any given link is affected with a
probability of when 10 nodes
are compromised.
Furthermore, suppose that and are two neighboring nodes then the
probability that and share a common key is: . Clearly
for a symmetric
balanced incomplete block design having and for another block
designs.
Let denote the number
of nodes in the intersection of the neighborhoods of the two nodes and where depends on the size of the physical
area where the nodes are deployed, the distance between nodes and on the
total number of sensor nodes in the DSN. The probability that and do not share a common key, but
there exists a node such that
shares a key with both and , is given as follows: . Then the probability
that is connected to via a path of length one or two is
approximately: [1,3].
Example 8. The values of and for the optimal KPS obtained from
Series I are:
Clearly the value of
is the minimum which is desirable for a KPS.
Earlier Lee and Stinson [1,2] and Saurabh and Sinha [3] obtained optimal KPSs from transversal designs,
affine and partially affine resolvable designs. Here nearly optimal KPS
has been introduced and a class of such KPS has also been proposed using
resolvable group divisible designs. An optimal KPS has also been
obtained using affine resolvable type design. A metric for efficiency of KPSs is given. Some
other resolvable combinatorial designs (which are not affine) such as
rectangular designs, balanced incomplete block designs, type designs and triangular
designs may also be used to obtain nearly optimal KPSs under certain
conditions. The constructions of these schemes would be taken up as a
future work.
Declaration of Competing
Interest
The author declares that they have no known competing financial
interests or personal relationships that could have appeared to
influence the work reported in this paper.