sensor network
kei pool
kei predistribut
hierarch hypercub model
secur
pairwis kei establish algorithm
cluster-base distribut model
polynomi kei
encrypt
node code
high fault-toler
pairwis kei

Researches on Scheme of Pairwise Key Establishment for DistributedSensor Networks Wang Lei Fujian University Technology Fuzhou,Funjian, PR.China (+)86-591-8755-9001, 350014 wanglei_hn@hn165.com Chen Zhi-ping Fujian University Technology Fuzhou,Funjian, PR.China (+)86-591-8755-9001, 350014 jt_zpchen@hnu.cn Jiang Xin-hua Fujian University Technology Fuzhou,Funjian, PR.China (+)86-591-8755-9001, 350014 xhj@csu.edu.cn ABSTRACT Security schemes of pairwise key establishment, which enable sensors to communicate with each other securely, play a fundamental role in research on security issue in wireless sensor networks.
A new kind of cluster deployed sensor networks distribution model is presented, and based on which, an innovative Hierarchical Hypercube model - H(k,u,m,v,n) and the mapping relationship between cluster deployed sensor networks and the H(k,u,m,v,n) are proposed.
By utilizing nice properties of H(k,u,m,v,n) model, a new general framework for pairwise key predistribution and a new pairwise key establishment algorithm are designed, which combines the idea of KDC(Key Distribution Center) and polynomial pool schemes.
Furthermore, the working performance of the newly proposed pairwise key establishment algorithm is seriously inspected.
Theoretic analysis and experimental figures show that the new algorithm has better performance and provides higher possibilities for sensor to establish pairwise key, compared with previous related works.
Categories and Subject Descriptors C.2.4 [Computer-Communication-Networks]: Distributed Systems-Distributed applications.
General Terms: Security.
1.
INTRODUCTION Security communication is an important requirement in many sensor network applications, so shared secret keys are used between communicating nodes to encrypt data.
As one of the most fundamental security services, pairwise key establishment enables the sensor nodes to communicate securely with each other using cryptographic techniques.
However, due to the sensor nodes' limited computational capabilities, battery energy, and available memory, it is not feasible for them to use traditional pairwise key establishment techniques such as public key cryptography and key distribution center (KDC).
Several alternative approaches have been developed recently to perform pairwise key establishment on resource-constrained sensor networks without involving the use of traditional cryptography [14].
Eschenauer and Gligor proposed a basic probabilistic key predistribution scheme for pairwise key establishment [1].
In the scheme, each sensor node randomly picks a set of keys from a key pool before the deployment so that any two of the sensor nodes have a certain probability to share at least one common key.
Chan et al. further extended this idea and presented two key predistribution schemes: a q-composite key pre-distribution scheme and a random pairwise keys scheme.
The q-composite scheme requires any two sensors share at least q pre-distributed keys.
The random scheme randomly picks pair of sensors and assigns each pair a unique random key [2].
Inspired by the studies above and the polynomial-based key pre-distribution protocol [3], Liu et al. further developed the idea addressed in the previous works and proposed a general framework of polynomial pool-based key predistribution [4].
The basic idea can be considered as the combination of the polynomial-based key pre-distribution and the key pool idea used in [1]] and [2].
Based on such a framework, they presented two pairwise key pre-distribution schemes: a random subset assignment scheme and a grid-based scheme.
A polynomial pool is used in those schemes, instead of using a key pool in the previous techniques.
The random subset assignment scheme assigns each sensor node the secrets generated from a random subset of polynomials in the polynomial pool.
The gridbased scheme associates polynomials with the rows and the columns of an artificial grid, assigns each sensor node to a unique coordinate in the grid, and gives the node the secrets generated from the corresponding row and column polynomials.
Based on this grid, each sensor node can then identify whether it can directly establish a pairwise key with another node, and if not, what intermediate nodes it can contact to indirectly establish the pairwise key.
A similar approach to those schemes described by Liu et al was independently developed by Du et a. [5].
Rather than on Blundo's scheme their approach is based on Blom's scheme [6].
In some cases, it is essentially equivalent to the one in [4].
All of those schemes above improve the security over the basic probabilistic key pre-distribution scheme.
However, the pairwise key establishment problem in sensor networks is still not well solved.
For the basic probabilistic and the q-composite key predistribution schemes, as the number of compromised nodes increases, the fraction of affected pairwise keys increases quickly.
As a result, a small number of compromised nodes may affect a large fraction of pairwise keys [3].
Though the random pairwise keys scheme doses not suffer from the above security problem, it incurs a high memory overhead, which increases linearly with the number of nodes in the network if the level of security is kept constant [2][4].
For the random subset assignment scheme, it suffers higher communication and computation overheads.
In 2004, Liu proposed a new hypercube-based pairwise key predistribution scheme [7], which extends the grid-based scheme from a two dimensional grid to a multi-dimensional hypercube.
The analysis shows that hypercube-based scheme keeps some attractive properties of the grid-based scheme, including the guarantee of establishing pairwise keys and the resilience to node compromises.
Also, when perfect security against node compromise is required, the hypercube-based scheme can support a larger network by adding more dimensions instead of increasing the storage overhead on sensor nodes.
Though hypercube-based scheme (we consider the grid-based scheme is a special case of hypercube-based scheme) has many attractive properties, it requires any two nodes in sensor networks can communication directly with each other.
This strong assumption is impractical in most of the actual applications of the sensor networks.
In this paper, we present a kind of new cluster-based distribution model of sensor networks, and for which, we propose a new pairwise key pre-distribution scheme.
The main contributions of this paper are as follows: Combining the deployment knowledge of sensor networks and the polynomial pool-based key pre-distribution, we setup a clusterbased topology that is practical with the real deployment of sensor networks.
Based on the topology, we propose a novel cluster distribution based hierarchical hypercube model to establish the pairwise key.
The key contribution is that our scheme dose not require the assumption of all nodes can directly communicate with each other as the previous schemes do, and it still maintains high probability of key establishment, low memory overhead and good security performance.
We develop a kind of new pairwise key establishment algorithm with our hierarchical hypercube model.
The structure of this paper is arranged as follows: In section 3, a new distribution model of cluster deployed sensor networks is presented.
In section 4, a new Hierarchical Hypercube model is proposed.
In section 5, the mapping relationship between the clusters deployed sensor network and Hierarchical Hypercube model is discussed.
In section 6 and section 7, new pairwise key establishment algorithm are designed based on the Hierarchical Hypercube model and detailed analyses are described.
Finally, section 8 presents a conclusion.
2.
PRELIMINARY Definition 1 (Key Predistribution): The procedure, which is used to encode the corresponding encryption and decryption algorithms in sensor nodes before distribution, is called Key Predistribution.
Definition 2 (Pairwise Key): For any two nodes A and B, if they have a common key E, then the key E is called a pairwise key between them.
Definition 3 (Key Path): For any two nodes A0 and Ak, when there has not a pairwise key between them, if there exists a path A0,A1,A2,......,Ak-1,Ak, and there exists at least one pairwise key between the nodes Ai and Aj for 0≤i≤k-1 and 1≤j≤k, then the path consisted of A0,A1,A2,......,Ak-1,Ak is called a Key Path between A0 and Ak.
Definition 4 (n-dimensional Hypercube): An n-dimensional Hypercube (or n−cube) H(v,n) is a topology with the following properties: (1) It is consisted of n·vn-1 edges, (2) Each node can be coded as a string with n positions such as b1b2...bn, where 0≤b1,b2,...,bn≤v-1, (3) Any two nodes are called neighbors, which means that there is an edge between them, iff there is just one position different between their node codes.
3.
MODEL OF CLUSTERS DEPLOYED SENSOR NETWORKS In some actual applications of sensor networks, sensors can be deployed through airplanes.
Supposing that the deployment rounds of sensors are k, and the communication radius of any sensors is r, then the sensors deployed in the same round can be regarded as belonging to a same Cluster.
We assign a unique cluster number l (1 ≤ l ≤ k) for each cluster.
Supposing that the sensors form a connected graph in any cluster after deployment through airplanes, and then the Fig.1 presents an actual model of clusters deployed sensor networks.
Figure.1 An actual model of clusters deployed sensor networks.
From Figure.1, it is easy to know that, for a given node A, there exist lots of nodes in the same cluster of A, which can be communicated directly with A, since the nodes are deployed densely in a cluster.
But there exist much less nodes in a cluster neighboring to the cluster of A, which can be communicated directly with A. since the two clusters are not deployed at the same time.
4.
HIERARCHICAL HYPERCUBE MODEL Definition 5 (k-levels Hierarchical Hypercube): Let there are N nodes totally, then a k-levels Hierarchical Hypercube named H(k,u,m,v,n) can be constructed as follows: 1) The N nodes are divided into k clusters averagely, and the [N/k] nodes in any cluster are connected into an n-dimensional Hypercube: In the n-dimensional Hypercube, any node is encoded 55 as i1i2...in, which are called In-Cluster-Hypercube-Node-Codes, where 0 ≤ i1,i2,...in ≤ v-1,v=[ n kN / ],[j] equals to an integer not less than j.
So we can obtain k such kind of different hypercubes.
2) The k different hypercubes obtained above are encoded as j1j2...jm, which are called Out-Cluster-Hypercube-Node-Codes, where 0 ≤ j1,j2,...jm ≤ u-1,u=[ m k ].
And the nodes in the k different hypercubes are connected into m-dimensional hypercubes according to the following rules: The nodes with same In-Cluster-Hypercube-Node-Codes and different Out-ClusterHypercube-Node-Codes are connected into an m-dimensional hypercube.
(The graph constructed through above steps is called a k-levels Hierarchical Hypercube abbreviated as H(k,u,m,v,n).)
3) Any node A in H(k,u,m,v,n) can be encoded as (i, j), where i(i=i1i2...in, 0 ≤ i1,i2,...in ≤ v-1) is the In-Cluster-HypercubeNode-Code of node A, and j(j=j1j2...jm, 0 ≤ j1,j2,...jm ≤ u-1) is the Out-Cluster-Hypercube-Node-Code of node A. Obviously, the H(k,u,m,v,n) model has the following good properties: Property 1: The diameter of H(k,u,m,v,n) model is m+n. Proof: Since the diameter of n-dimensional hypercube is n, and the diameter of m-dimensional hypercube is m, so it is easy to know that the diameter of H(k,u,m,v,n) model is m+n from the definition 5.
Property 2: The distance between any two nodes A(i1, j1) and B(i2, j2) in H(k,u,m,v,n) model is d(A,B)= dh(i1, i2)+dh(j1, j2), where dh represents the Hamming distance.
Proof: Since the distance between any two nodes in hypercube equals to the Hamming distance between them, so it is obvious that the theorem 2``s conclusion stands from definition 5.
5.
MAPPING CLUSTERS DEPLOYED SENSOR NETWORKS TO H(K,U,M,V,N) Obviously, from the description in section 3 and 4, we can know that the clusters deployed sensor network can be mapped into a klevels- hierarchical hypercube model as follows: At first, the k clusters in the sensor network can be mapped into k different levels (or hypercubes) in the k-levels- hierarchical hypercube model.
Then, the sensor nodes in each cluster can be encoded with the In-Cluster-Hypercube-Node-Codes, and the sensor nodes in the k different clusters with the same In-ClusterHypercube-Node-Codes can be encoded with the Out-ClusterHypercube-Node-Codes according to the definition 5 respectively.
Consequently, the whole sensor network has been mapped into a k-levels- hierarchical hypercube model.
6.
H(K,U,M,V,N) MODEL-BASED PAIRWISE KEY PREDISTRIBUTION ALGORITHM FOR SENSOR NETWORKS In order to overcome the drawbacks of polynomial-based and polynomial pool-based key predistribution algorithms, this paper proposed an innovative H(k,u,m,v,n) model-based key predistribution scheme and pairwise key establishment algorithm, which combines the advantages of polynomial-based and key pool-based encryption schemes, and is based on the KDC and polynomials pool-based key predistribution models.
The new H(k,u,m,v,n) model-based pairwise key establishment algorithm includes three main steps: (1) Generation of the polynomials pool and key predistribution, (2) Direct pairwise key discovery, (3) Path key discovery.
6.1 Generation of Polynomials Pool and Key Predistribution Supposing that, the sensor network includes N nodes, and is deployed through k different rounds.
Then we can predistribute keys for each sensor node on the basis of the H(k,u,m,v,n) model as follows: Step 1: Key setup server randomly generates a bivariate polynomials pool such as the following: F={ f i iiil n >< −121 ,...,,, (x,y), f j jjjinii m >< −121 ,...,,,,...,2,1 (x,y) | 0 ≤ iii n 121 ... −≤≤≤ ≤ v-1, 1 ≤ i ≤ n, 1 ≤ l ≤ k; 0 ≤ jjj m 121 ... −≤≤≤ ≤ u-1 , 1 ≤ j ≤ m} with vn *m*um-1 +[N/vn ]*n*vn-1 different t-degree bivariate polynomials over a finite field Fq, and then assigns a unique polynomial ID to each bivariate polynomial in F. Step 2: In each round, key setup server assigns a unique node ID: (i1i2...in,j1j2...jm) to each sensor node from small to big, where 0 ≤ i1,i2,...in ≤ v-1, 0 ≤ j1,j2,...jm ≤ u-1.
Step 3: key setup server assigns a unique cluster ID: l to all the sensor nodes deployed in the same round, where 1 ≤ l ≤ k. Step 4: key setup server predistributes m+n bivariate polynomials { f iiil n 1 ,...,,, 32 >< (i1,y),..., f n iiil n >< −121 ,...,,, (in,y); f jjinii m 1 ,...,,,...,2,1 2 >< ( j1,y),..., f m jjinii m >< −11 ,...,,,...,2,1 ( jm,y) } and the corresponding polynomial IDs to the sensor node deployed in the lth round and with ID (i1i2...in, j1j2...jm).
6.2 Direct Pairwise Key Discovery If the node A(i1i2...in,j1j2...jm) in the sensor network wants to establish pairwise key with a node B (i'1i'2...i`n,j'1j'2...j'm), then node A can establish pairwise key with the node B trough the following methods.
Firstly, node A computes out the distance between itself and node B: d= d1+ d2, where d1=dh(i1i2...in, i'1i'2...i`n) and d2=dh(j1j2...jm, j'1j'2...j'm).
If d=1, then node A obtains the direct pairwise key between itself and node B according to the following theorem 1: Theorem 1: For any two sensor nodes A(i1i2...in,j1j2...jm) and B (i'1i'2...i`n,j'1j'2...j'm) in the sensor network, supposing that the 56 distance between nodes A and B is d= d1+ d2, where d1=dh(i1i2...in, i'1i'2...i`n) and d2=dh(j1j2...jm, j'1j'2...j'm).
If d=1, then there exists a direct pairwise key between nodes A and B. Poof: Since d=1, then there is d1=1, d2=0, or d1=0, d2=1.
1) If d1=1, d2=0: From d2=0, there is nodes A, B belong to the same cluster.
Supposing that nodes A, B belong to the same cluster l, then from d1=1 ⇒ There is only one position different between i1i2...in and i``1i``2...i``n. Let it=i``t, when 1 ≤ t ≤ n-1, and in ≠ i``n ⇒ f n iiil n >< −121 ,...,,, (in,i``n)= f n iiil n >′′′< −121 ,...,,, (i``n,in).
So, there exists a direct pairwise key f n iiil n >< −121 ,...,,, (in,i``n) between nodes A and B. 2) If d1=0, d2=1: From d2=1 ⇒ There is only one position different between j1j2...jm and j``1j``2...j``m. Let jt=j``t, when 1 ≤ t ≤ m1, and jm ≠ j``m.
Since d1=0 ⇒ i1i2...in equals to i``1i``2...i``n ⇒ f m jjjinii m >< −121 ,...,,,,...,2,1 (jm, j``m)= f m jjji nii m >′′′′′′< −121 ,...,,,,...,2,1 (j``m,jm).
So, there exists a direct pairwise key f m jjjinii m >< −121 ,...,,,,...,2,1 (jm, j``m) between nodes A and B.
According to theorem 1, we present the detailed description of the direct pairwise key discovery algorithm as follows: Step 1: Obtain the node IDs and cluster IDs of the source node A and destination node B; Step 2: Compute out the distance between nodes A and B: d= d1+ d2; Step 3: If d1=1, d2=0, then select out a common polynomial share of nodes A and B from { f iiil n 1 ,...,,, 32 >< ,..., f n iiil n >< −121 ,...,,, } to establish direct pairwise key; Step 4: If d1=0, d2=1, then select out a common polynomial share of nodes A and B from { f jjinii m 1 ,...,,,...,2,1 2 >< ,..., f m jjjinii m >< −121 ,...,,,,...,2,1 } to establish direct pairwise key; Step 5: Otherwise, there exists no direct pairwise key between nodes A and B. And then turn to the following path key discovery process.
6.3 Path Key Discovery If d>1, then node A can establish path key with node B according to the following theorem 2: Theorem 2: For any two sensor nodes A(i1i2...in,j1j2...jm) and B (i'1i'2...i`n,j'1j'2...j'm) in the sensor network, supposing that the distance between nodes A and B is d= d1+ d2, where d1=dh(i1i2...in, i'1i'2...i`n) and d2=dh(j1j2...jm, j'1j'2...j'm).
If d>1, then there exists a path key between nodes A and B. Proof: Let d1=a, d2=b, then we can think that it ≠ i`t, when 1 ≤ t ≤ a; but it=i`t, when t>a; and jt ≠ j't, when 1 ≤ t ≤ b; but jt=j't, when t>b. Obviously, nodes A(i1i2...in, j1j2...jm) ,(i'1i2 i3...in, j1j2...jm),(i'1i'2 i3...in, j1j2...jm),...,(i'1i'2...i`n, j1j2...jm) belong to the same cluster.
So, according to the supposing condition of The nodes in the same cluster form a connected graph, there is a route among those nodes.
In addition, in those nodes, the distance between any two neighboring nodes is 1, so from theorem 1, it is easy to know that there exists direct pairwise key between any two neighboring nodes among those nodes.
For nodes (i'1i'2...i`n,j1j2...jm), (i'1i'2...i`n,j'1 j2 j3...jm), (i'1i'2...i`n,j'1j'2 j3...jm-1 jm),..., (i'1i'2...i`n,j'1j'2...j'm-1jm), since they have the same Out-Cluster-Hypercube-Node-Codes with the node B(i'1i'2...i`n,j'1j'2...j'm), so nodes (i'1i'2...i`n,j1j2...jm), (i'1i'2...i`n,j'1 j2 j3...jm), (i'1i'2...i`n,j'1j'2 j3...jm-1 jm),..., (i'1i'2...i`n,j'1j'2...j'm-1 jm) and node B belong to a same logical hypercube.
Obviously, from the supposing condition of The whole sensor network forms a connected graph, there is a route among those nodes.
In addition, in those nodes, the distance between any two neighboring nodes is 1, so from theorem 1, it is easy to know that there exists direct pairwise key between any two neighboring nodes among those nodes.
So, it is obvious that there exists a path key between nodes A and B.
According to theorem 2, we present the detailed description of the path key discovery algorithm as follows: Step 1: Compute out the intermediate nodes (i'1i2 i3...in,j1j2...jm), (i'1i'2 i3...in,j1j2...jm),...,(i'1i'2...i`n, j1j2...jm) and (i'1i'2...i`n,j1`j2 j3...jm), (i'1i'2...i`n,j'1j'2 j3...j'm-1 jm),...,(i'1i'2...i`n,j'1j'2...j'm-1 jm) from the source node A(i1i2...in,j1j2...jm) and the destination node B (i'1i'2...i`n,j'1j'2...j'm).
Step 2: In those nodes series A(i1i2...in,j1j2...jm), (i'1i2 i3...in,j1j2...jm), (i'1i'2 i3...in,j1j2...jm),...,(i'1i'2...i`n,j1j2...jm) and (i'1i'2...i`n,j1`j2 j3...jm), (i'1i'2...i`n,j'1j'2 j3...j'm-1 jm),..., (i'1i'2...i`n, j'1j'2...j'm-1 jm), B (i'1i'2...i`n,j'1j'2...j'm), the neighboring nodes select their common polynomial share to establish direct pairwise key.
From theorem 2, it is easy to know that any source node A can compute out a key path P to the destination node B according to the above algorithm, when there are no compromised nodes in the sensor network.
Once the key path P is computed out, then node A can send messages to B along the path P to establish indirect pairwise key with node B. Fig.2 presents a example of key path establishment.
Figure.2 Key path establishment example.
For example: In the above Figure.2, node A((012),(1234)) can establish pairwise key with node B((121),(2334)) through the following key path: A((012),(1234)) → C((112),(1234)) → D((122),(1234)) → E((121),(1234)) → F((121),(2234)) → B((121),(2334)), where node F shall route through nodes G, H, I, J to establish direct pairwise key with node B. 57 According to the properties of H(k,u,m,v,n) model, we can prove that the following theorem by combing the proof of theorem 2: Theorem 3: Supposing that there exist no compromised nodes in the sensor network, and the distance between node A and B, then there exists a shortest key path with k distance between node A and B logically.
That is to say, node A can establish indirect pairwise key with node B through t-1 intermediate nodes.
Proof: Supposing that the distance between node A(i1i2...in, j1j2...jm) and B (i'1i'2...i`n, j'1j'2...j'm) is d=d1+ d2, where d1=dh(i1i2...in, i'1i'2...i`n), d2=dh(j1j2...jm, j'1j'2...j'm).
Since d=t, according to the construction properties of H(k,u,m,v,n), it is easy to know that there exist t-1 intermediate nodes I1,...,It-1, in the logical space H(k,u,m,v,n), which satisfy that the distance between any two neighboring nodes in the nodes series A, I1,...,It1, B equals to 1.
So according to the theorem 1, we can know that nodes A, I1,...,It-1, B form a correct key path between node A and B.
If any two neighboring nodes in the nodes series A, I1,...,It-1, B can communicate directly, then node A can establish indirect pairwise key with node B through those t-1 intermediate nodes.
6.4 Dynamic Path Key Discovery The path key discovery algorithm proposed in the above section can establish a key path correctly, only when there exist no compromised nodes in the whole sensor network, since the key path is computed out beforehand.
And the proposed algorithm cannot find an alternative key path when there exist some compromised nodes or some intermediate nodes not in the communication radius, even that there exists other alternative key paths in the sensor network.
From the following example we can know that there are many parallel paths in the H(k,u,m,v,n) model for any two given source and destination nodes, since the H(k,u,m,v,n) model is high fault-tolerant[9,10] .
Figure.3 Alternative key path establishment example.
For example: Considering the key path establishment example given in the above section based on Figure.2: A((012),(1234)) → C((112),(1234)) → D((122),(1234)) → E((121),(1234)) → F((121),(2234)) → B((121),(2334)), supposing that node F((121),(2234)) has compromised, then from Figure.3, we can know that there exists another alternative key path as A((012),(1234)) → C((112),(1234)) → D((122),(1234)) →E((121),(1234)) → M((121),(1334)) → B((121),(2334)), which can be used to establish the indirect pairwise key between node A and B, where node E shall route through nodes D and K to establish direct pairwise key with node M, and node M shall route through nodes N, O, G, H, I, J to establish direct pairwise key with node B.
Since the sensors are source limited, so they are easy to die or out of the communication radius, therefore the algorithm proposed in the above section cannot guarantee to establish correct key path efficiently.
In this section, we will propose a dynamic path key discovery algorithm as follows, which can improve the probability of key path effectively: Algorithm I: Dynamic key path establishment algorithm based on H(k,u,m,v,n) model for cluster deployed sensor networks.
Input: Sub-sensor network H(k,u,m,v,n), which has some compromised /fault sensors and fault links, And two reachable nodes A(a1...an,a``1...a``m) and B(b1...bn,b``1...b``m) in H(k,u,m,v,n), where a``t ≠ b``t, t∈[1,s], a``t=b``t, t >s. Output: A correct key path from node A to B in H(k,u,m,v,n).
Step 1: Obtain the code strings of node A and B: A ← (a1...an,a``1...a``m), B ← (b1...bn,b``1...b``m), where aj, bj [0,∈ u-1], a``j, b``j [0,∈ v-1].
Step 2: If a``1...a``m = b``1...b``m, then node A can find a route to B according to the routing algorithms of hypercube [9-10].
Step 3: Otherwise, node A can find a route to C(b1...bn, a``1...a``m) according to the Algorithm I or Algorithm II.
Then let I0=C(b1...bn,a``1...a``m), I1=(b1...bn,b``1 a``2...a``m),..., Is=B(b1...bn,b``1 b``2...b``s a``s+1...a``m), and each node It in the above nodes series find a route to its neighboring node It+1 on the basis of the location information (Detailed routing algorithms based on location information can see the references[11-14]).
Step 4: Algorithm exits.
If such kind of a correct key path exists, then through which node A can establish an indirect pairwise key with node B. Otherwise, node A fails to establish an indirect pairwise key with node B. And node A will tries again to establish an indirect pairwise key with node B some time later.
7.
ALGORITHM ANALYSES 7.1 Practical Analyses According to the former description and analyses, it is easy to know that the above newly proposed algorithm has the following properties: Property 3: When there exist no fault and compromised nodes, by using new pairwise key predistribution scheme based on H(k,u,m,v,n) model, the probability of direct pairwise key establishment between any two nodes can be estimated as P=(m(u-1)+n(v-1))/(N-1), where N is the total number of nodes in the sensor network, and N=um * vn .
Proof: Since the predistributed pairwise keys for any node FA ={ f iiil n 1 ,...,,, 32 >< (i1,y),..., f n iiil n >< −121 ,...,,, (in,y); f jjinii m 1 ,...,,,...,2,1 2 >< (j1 ,y),..., f m jjinii m >< −11 ,...,,,...,2,1 ( jm,y) } in the newly proposed algorithm.
Obviously, in the logical hypercube formed by the nodes in the same cluster of node A, there are n(v-1) nodes, which 58 have direct pairwise key with node A. And in the logical hypercube formed by the nodes in different clusters from that of node A, there are m(u-1) nodes, which have direct pairwise key with node A. Therefore, there are totally m(u-1)+n(v-1) nodes, which have direct pairwise key with node A. So, the probability of pairwise key establishment between any two nodes can be estimated as P=(m(u-1)+n(v-1))/(N-1), since the whole sensor network has N sensor nodes in all.
Figure.4 presents the comparision between the probability of direct pairwise key establishment between any two nodes and the dimension n, when the sensor network has different total nodes, and use the new pairwise key predistribution scheme based on H(8,2,3,v,n) model.
2 3 4 5 6 7 8 9 10 0 0.002 0.004 0.006 0.008 0.01 Number of Dimension ProbabilitytoEstablishDirectKey N = 8000 N=10000 N=20000 N=30000 Figure.4 Comparision between the probability of direct pairwise key establishment between any two nodes and the dimension n, when the sensor network has different total nodes, and use the new pairwise key predistribution scheme based on H(8,2,3,v,n) model.
From Figure.4, it is easy to know that by using new pairwise key predistribution scheme based on H(k,u,m,v,n) model, the probability of direct pairwise key establishment between any two nodes decreases with the increasing of the scale of the sensor networks, and in addition, the probability of direct pairwise key establishment between any two nodes decreases with the increasing of the dimension n, when the scale of the sensor network is fixed.
Theorem 4: Supposing that the total sensors is N in the sensor network, then when u ≥ v2 , the probability of direct pairwise key establishment between any two nodes, when using the key distribution scheme based on the hypercube model H(v,p), is smaller than that when using the key distribution scheme based on the H(k,u,m,v,n) model.
Proof: Since u ≥ v, then we can let u=vt , where t ≥ 2.
Since the total number of nodes in H(v,p) is vp =N, the total number of nodes in H(k,u,m,v,n) is um * vn =N. Let p=x+n, then there is um *vn = vx * vn ⇒ um =vx ⇒ x=tm.
From the property 3, it is easy to know that the probability of direct pairwise key establishment between any two nodes can be estimated as P=(m(u-1)+n(v-1))/(N-1).
According to the description in [7], it is well know that the probability of direct pairwise key establishment between any two nodes can be estimated as P''= p(v-1)/(N-1)= (x(v-1)+n(v-1))/(N-1).
Next, we will prove that m(u-1) ≥ x(v-1): m(u-1)= m(vt -1), x(v-1)= tm(v-1).
Construct a function as f(t)= vt -1- t(v-1), where t ≥ 2.
When t=2, it is obvious that there is f(t)= vt -2v+1=-LRB- v-1)2 ≥ 0 and f''(t)=t vt-1 - v+1 ≥ 2v- v+1= v+1>0.
So, there is f(t) ≥ 0 ⇒ vt -1 ≥ t(v-1) ⇒ m(vt -1) ≥ tm(v-1) ⇒ m(u1) ≥ x(v-1).
Therefore, the conclusion of the theorem stands.
As for the conclusion of theorem 4, we give an example to illustrate.
Supposing that the total number of nodes in the sensor network is N=214 , and H(k,u,m,v,n)=H(16,4,2,2,10), H(v,p)= H(10,14), then the probability of direct pairwise key establishment between any two nodes based on the H(k,u,m,v,n) model is P= (m(u-1)+n(v1))/(N-1)= (2(4-1)+10(2-1))/(214 -1)=16/(214 -1), but the probability of direct pairwise key establishment between any two nodes based on the H(v,p) model is P''= p(v-1)/(N-1)=14(2-1)/(214 -1)= 14/(214 1).
Supposing that the total number of nodes in the sensor network is N, Figure.5 illustrates the comparison between the probability of direct pairwise key establishment between any two nodes based on the H(k,u,m,v,n) model and the probability of direct pairwise key establishment between any two nodes based on the H(v,p) model, when u=4 and v=2.
1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 x 10 -3 scaleofthesensornetwork ProbabilitytoEstablishDirectKey H(k,u,m,v,n)model-based H(v,p)model-based Figure.5 Comparison between the probability of direct pairwise key establishment between H(v,n) and H(k,u,m,v,n) models.
From Figure.5, it is easy to know that the theorem 5 stands.
Theorem 5: Supposing that the total sensors is N in the sensor network, then the pairwise key distribution scheme based on the hypercube model H(v,p), is only a special case of the pairwise key distribution scheme based on the H(k,u,m,v,n) model.
Proof: As for the pairwise key distribution scheme based on the H(k,u,m,v,n) model, let k=1 (u=1, m=0), which means that the total sensor network includes only one cluster.
Then obviously, the H(k,u,m,v,n) model will degrade into the H(v,n) model.
According to the former anayses in this paper and the definition of the pairwise key distribution scheme based on the hypercube model H(v,p) in [7], it is easy to know that the conclusion of the theorem stands.
59 7.2 Security Analyses By using the pairwise key establishment algorithm based on the H(k,u,m,v,n) model, the intruders can launch two kinds of attacks: 1) The attackers may target the pairwise key between two particular sensor node, in order to compromise the pairwise key between them, or prevent them to establish pairwise key.
2) The attackers may attack against the whole sensor network, inorder to decrease the probability of the pairwise key establishment, or increase the cost of the pairwise key establishment.
Attacks against a Pair of sensor nodes 1.
Supposing that the intruders want to attack two particular sensor nodes u,v, where u,v are all not compromised nodes, but the intruders want to compromise the pairwise key between them.
1) If u,v can establish direct pairwise key, then the only way to compromise the key is to compromise the common bivariate polynomial f(x,y) between u,v.
Since the degree of the bivariate polynomial f(x,y) is t, so the intruders need to compromise at least t+1 sensor nodes that have a share of the bivariate polynomial f(x,y).
2) If u,v can establish indirect pairwise key through intermediate nodes, then the intruders need to compromise at least one intermediate node, or compromise the common bivariate polynomial f(x,y) between two neighboring intermediate nodes.
But even if the intruders succeed to do that, node u and v can still reestablish indirect pairwise key through alternative intermediate nodes.
2.
Supposing that the intruders want to attack two particular sensor nodes u,v, where u,v are all not compromised nodes, but the intruders want to prevent them to establish the pairwise key.
Then, the intruders need to compromise all of the m+n bivariate polynomials of node u or v.
Since the degree of the bivariate polynomial f(x,y) is t, so for bivariate polynomial, the intruders need to compromise at least t+1 sensor nodes that have a share of the given bivariate polynomial.
Therefore, the intruders need to compromise (m+n)(t+1) sensor nodes altogether to prevent u,v to establish the pairwise key.
Attacks against the sensor network Supposing that the Attackers know the distribution of the polynomials over sensor nodes, it may systematically attack the network by compromising the polynomials in F one by one in order to compromise the entire network.
Assume the fraction of the compromised polynomials is pc, then there are up to N''=pc × { vn v N umv n n mn ××+×× ][ }= pc ××N (m+n) Sensor nodes that have at least one compromised polynomial share.
Among all of the remaining N- N'' sensor nodes, none of them includes a compromised polynomial share.
So, the remaining N- N'' sensor nodes can establish direct pairwise key by using any one of their polynomial shares.
However, the indirect pairwise keys in the remaining N- N'' sensor nodes may be affected.
And they may need to re-establish a new indirect pairwise key between them by select alternative intermediate nodes that do not belong to the N'' compromised nodes.
Supposing that the scale of the sensor network is N=10000, Figure.6 presents the comparison between pc and the number of sensor nodes with at least one compromised polynomial share in sensor networks based on different H(k,u,m,v,n) distribution models.
From Figure.6, it is easy to know that, when the scale of the sensor network is fixed, the number of the affected sensor nodes in the sensor network increases with the increasing of the number of compromised nodes.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 1000 2000 3000 4000 5000 6000 7000 8000 F rac tion of C om prom is ed B ivariate P oly nom ialsSensorNodeswithatleastoneCompromisedPolynomialShare H (1,0,0,100,2) H (2,2,1,71,2) H (4,2,2,50,2) H (8,2,3,36,2) Figure.6 the comparison between pc and the number of sensor nodes with at least one compromised polynomial share in sensor networks based on different H(k,u,m,v,n) distribution models.
Theorem 6: Supposing that the total sensors is N in the sensor network, and the fraction of compromised nodes is pc, then when u>v, the number of affected nodes of the H(v,p) model based key predistribution scheme, is bigger than that of the H(k,u,m,v,n) model based key predistribution scheme.
Proof: Since the number of affected nodes of the H(k,u,m,v,n) model based key predistribution scheme is pc ××N (m+n), and it is proved in [7] that the number of affected nodes of the H(v,p) model based key predistribution scheme is pc ××N p. Let p=x+n, then there is um * vn = vx * vn ⇒ um =vx .
Since u>v ⇒ x>m ⇒ pc ××N (m+n)< pc ××N (x+n)= pc ××N p. Supposing that the scale of the sensor network is N=10000, Figure.7 presents the comparison between pc and the number of sensor nodes with at least one compromised polynomial share in sensor networks based on H(9,3,2,2,n) and H(2,p) distribution models.
From Figure.7, it is easy to know that the conclusion of theorem 9 is correct, and the number of the affected sensor nodes in the sensor network increases with the increasing of the number of compromised nodes, when the scale of the sensor network is fixed.
60 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Fraction of Compromised Bivariate Polynomials SensorNodeswithatleastoneCompromisedPolynomialShare H(9,3,2,34,2) H(16,4,2,25,2) H(225,15,2,7,2) H(1296,36,2,3,2) H(2,14) Figure.7 the comparison between pc and the number of sensor nodes with at least one compromised polynomial share in sensor networks based on H(9,3,2,2,n) and H(2,p) distribution models.
8.
CONCLUSION A new hierarchical hypercube model named H(k,u,m,v,n) is proposed, which can be used for pairwise key predistribution for cluster deployed sensor networks.
And Based on the H(k,u,m,v,n) model, an innovative pairwise key predistribution scheme and algorithm are designed respectively, by combing the good properties of the Polynomial Key and Key Pool encryption schemes.
The new algorithm uses the good characteristics of node codes and high fault-tolerance of H(k,u,m,v,n) model to route and predistribute pairwise keys, in which nodes are not needed to be able to communicate with each other directly such as that the algorithms proposed by [7] shall need.
So, the traditional pairwise key predistribution algorithm based on hypercube model [7] is only a special case of the new algorithm proposed in this paper.
Theoretical and experimental analyses show that the newly proposed algorithm is an efficient pairwise key establishment algorithm that is suitable for the cluster deployed sensor networks.
9.
ACKNOWLEDGMENTS Our thanks to ACM SIGCHI for allowing us to modify templates they had developed, and to nature science fund of Fujian province of PR.China under grant No.A0510024.
10.
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A key-management scheme for distribute sensor networks.
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Random key predistribution schemes for sensor networks.
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Establishing pairwise keys in distributed sensor networks.
In Proceedings of the 10th ACM Conference on Computer and Communications Security.
ACM Press, Washingtion, DC, USA, 2003, 52-61.
[5] W. Du, J. Deng, Y. Han, and P. Varshney.
A pairwise key pre-distribution scheme for wireless sensor networks.
In Proceedings of the Tenth ACM Conference on Computer and Communications Security.
Washingtion, DC, USA,2003, 4251.
[6] R. Blom.
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[7] Donggang Liu, Peng Ning, Rongfang Li, Establishing Pairwise Keys in Distributed Sensor Networks.
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[8] L. Fang, W. Du, and N. Peng.
A Beacon-Less Location Discovery Scheme for Wireless Sensor Networks, INFOCOM 2005.
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[14] B. Krishnamachari, D. Estrin, and S. Wicker.
Modelling Data-Centric Routing in Wireless Sensor Networks.
In Proceedings of IEEE Infocom, 2002.
61

Researches on Scheme of Pairwise Key Establishment for Distributed Sensor Networks
ABSTRACT
Security schemes of pairwise key establishment , which enable sensors to communicate with each other securely , play a fundamental role in research on security issue in wireless sensor networks .
A new kind of cluster deployed sensor networks distribution model is presented , and based on which , an innovative Hierarchical Hypercube model - H ( k , u , m , v , n ) and the mapping relationship between cluster deployed sensor networks and the H ( k , u , m , v , n ) are proposed .
By utilizing nice properties of H ( k , u , m , v , n ) model , a new general framework for pairwise key predistribution and a new pairwise key establishment algorithm are designed , which combines the idea of KDC ( Key Distribution Center ) and polynomial pool schemes .
Furthermore , the working performance of the newly proposed pairwise key establishment algorithm is seriously inspected .
Theoretic analysis and experimental figures show that the new algorithm has better performance and provides higher possibilities for sensor to establish pairwise key , compared with previous related works .
1 .
INTRODUCTION
Security communication is an important requirement in many sensor network applications , so shared secret keys are used between communicating nodes to encrypt data .
As one of the most fundamental security services , pairwise key establishment enables the sensor nodes to communicate securely with each other using cryptographic techniques .
However , due to the sensor nodes ' limited computational capabilities , battery energy , and available memory , it is not feasible for them to use traditional pairwise key establishment techniques such as public key cryptography and key distribution center ( KDC ) .
Several
alternative approaches have been developed recently to perform pairwise key establishment on resource-constrained sensor networks without involving the use of traditional cryptography [ 14 ] .
Eschenauer and Gligor proposed a basic probabilistic key predistribution scheme for pairwise key establishment [ 1 ] .
In the scheme , each sensor node randomly picks a set of keys from a key pool before the deployment so that any two of the sensor nodes have a certain probability to share at least one common key .
Chan et al. further extended this idea and presented two key predistribution schemes : a q-composite key pre-distribution scheme and a random pairwise keys scheme .
The q-composite scheme requires any two sensors share at least q pre-distributed keys .
The random scheme randomly picks pair of sensors and assigns each pair a unique random key [ 2 ] .
Inspired by the studies above and the polynomial-based key pre-distribution protocol [ 3 ] , Liu et al. further developed the idea addressed in the previous works and proposed a general framework of polynomial pool-based key predistribution [ 4 ] .
The basic idea can be considered as the combination of the polynomial-based key pre-distribution and the key pool idea used in [ 1 ] ] and [ 2 ] .
Based on such a framework , they presented two pairwise key pre-distribution schemes : a random subset assignment scheme and a grid-based scheme .
A polynomial pool is used in those schemes , instead of using a key pool in the previous techniques .
The random subset assignment scheme assigns each sensor node the secrets generated from a random subset of polynomials in the polynomial pool .
The gridbased scheme associates polynomials with the rows and the columns of an artificial grid , assigns each sensor node to a unique coordinate in the grid , and gives the node the secrets generated from the corresponding row and column polynomials .
Based on this grid , each sensor node can then identify whether it can directly establish a pairwise key with another node , and if not , what intermediate nodes it can contact to indirectly establish the pairwise key .
A similar approach to those schemes described by Liu et al was independently developed by Du et a. [ 5 ] .
Rather than on Blundo 's scheme their approach is based on Blom 's scheme [ 6 ] .
In some cases , it is essentially equivalent to the one in [ 4 ] .
All of those schemes above improve the security over the basic probabilistic key pre-distribution scheme .
However , the pairwise key establishment problem in sensor networks is still not well solved .
For the basic probabilistic and the q-composite key predistribution schemes , as the number of compromised nodes increases , the fraction of affected pairwise keys increases quickly .
As a result , a small number of compromised nodes may affect a
large fraction of pairwise keys [ 3 ] .
Though the random pairwise keys scheme doses not suffer from the above security problem , it incurs a high memory overhead , which increases linearly with the number of nodes in the network if the level of security is kept constant [ 2 ] [ 4 ] .
For the random subset assignment scheme , it suffers higher communication and computation overheads .
In 2004 , Liu proposed a new hypercube-based pairwise key predistribution scheme [ 7 ] , which extends the grid-based scheme from a two dimensional grid to a multi-dimensional hypercube .
The analysis shows that hypercube-based scheme keeps some attractive properties of the grid-based scheme , including the guarantee of establishing pairwise keys and the resilience to node compromises .
Also , when perfect security against node compromise is required , the hypercube-based scheme can support a larger network by adding more dimensions instead of increasing the storage overhead on sensor nodes .
Though hypercube-based scheme ( we consider the grid-based scheme is a special case of hypercube-based scheme ) has many attractive properties , it requires any two nodes in sensor networks can communication directly with each other .
This strong assumption is impractical in most of the actual applications of the sensor networks .
In this paper , we present a kind of new cluster-based distribution model of sensor networks , and for which , we propose a new pairwise key pre-distribution scheme .
The main contributions of this paper are as follows : Combining the deployment knowledge of sensor networks and the polynomial pool-based key pre-distribution , we setup a clusterbased topology that is practical with the real deployment of sensor networks .
Based on the topology , we propose a novel cluster distribution based hierarchical hypercube model to establish the pairwise key .
The key contribution is that our scheme dose not require the assumption of all nodes can directly communicate with each other as the previous schemes do , and it still maintains high probability of key establishment , low memory overhead and good security performance .
We develop a kind of new pairwise key establishment algorithm with our hierarchical hypercube model .
The structure of this paper is arranged as follows : In section 3 , a new distribution model of cluster deployed sensor networks is presented .
In section 4 , a new Hierarchical Hypercube model is proposed .
In section 5 , the mapping relationship between the clusters deployed sensor network and Hierarchical Hypercube model is discussed .
In section 6 and section 7 , new pairwise key establishment algorithm are designed based on the Hierarchical Hypercube model and detailed analyses are described .
Finally , section 8 presents a conclusion .
2 .
PRELIMINARY
Definition 1 ( Key Predistribution ) : The procedure , which is used to encode the corresponding encryption and decryption algorithms in sensor nodes before distribution , is called Key Predistribution .
Definition 2 ( Pairwise Key ) : For any two nodes A and B , if they have a common key E , then the key E is called a pairwise key between them .
Definition 3 ( Key Path ) : For any two nodes A0 and Ak , when there has not a pairwise key between them , if there exists a path A0,A1,A2,...,Ak-1 , Ak , and there exists at least one pairwise key between the nodes Ai and Aj for 05i5k-1 and 15j5k , then the path consisted of A0 , A1 , A2 , ... , Ak-1 , Ak is called a Key Path between A0 and Ak .
Definition 4 ( n-dimensional Hypercube ) : An n-dimensional Hypercube ( or n − cube ) H ( v , n ) is a topology with the following properties : ( 1 ) It is consisted of n · vn-1 edges , ( 2 ) Each node can be coded as a string with n positions such as b1b2 ... bn , where 05b1,b2,...,bn5v-1 , ( 3 ) Any two nodes are called neighbors , which means that there is an edge between them , iff there is just one position different between their node codes .
3 .
MODEL OF CLUSTERS DEPLOYED SENSOR NETWORKS
4 .
HIERARCHICAL HYPERCUBE MODEL
5 .
MAPPING CLUSTERS DEPLOYED SENSOR NETWORKS TO H ( K , U , M , V , N )
6 .
H ( K , U , M , V , N ) MODEL-BASED PAIRWISE KEY PREDISTRIBUTION ALGORITHM FOR SENSOR NETWORKS
6.1 Generation of Polynomials Pool and Key Predistribution
6.2 Direct Pairwise Key Discovery
6.3 Path Key Discovery
Figure .2 Key path establishment example .
6.4 Dynamic Path Key Discovery
7 .
ALGORITHM ANALYSES
7.1 Practical Analyses
7.2 Security Analyses
Attacks against a Pair of sensor nodes
Attacks against the sensor network
Fraction of Compromised B ivariate Polynomials
Fraction of Compromised Bivariate Polynomials
8 .
CONCLUSION
A new hierarchical hypercube model named H ( k , u , m , v , n ) is proposed , which can be used for pairwise key predistribution for cluster deployed sensor networks .
And Based on the H ( k , u , m , v , n ) model , an innovative pairwise key predistribution scheme and algorithm are designed respectively , by combing the good properties of the Polynomial Key and Key Pool encryption schemes .
The new algorithm uses the good characteristics of node codes and high fault-tolerance of H ( k , u , m , v , n ) model to route and predistribute pairwise keys , in which nodes are not needed to be able to communicate with each other directly such as that the algorithms proposed by [ 7 ] shall need .
So , the traditional pairwise key predistribution algorithm based on hypercube model [ 7 ] is only a special case of the new algorithm proposed in this paper .
Theoretical and experimental analyses show that the newly proposed algorithm is an efficient pairwise key establishment algorithm that is suitable for the cluster deployed sensor networks .

Researches on Scheme of Pairwise Key Establishment for Distributed Sensor Networks
ABSTRACT
Security schemes of pairwise key establishment , which enable sensors to communicate with each other securely , play a fundamental role in research on security issue in wireless sensor networks .
A new kind of cluster deployed sensor networks distribution model is presented , and based on which , an innovative Hierarchical Hypercube model - H ( k , u , m , v , n ) and the mapping relationship between cluster deployed sensor networks and the H ( k , u , m , v , n ) are proposed .
By utilizing nice properties of H ( k , u , m , v , n ) model , a new general framework for pairwise key predistribution and a new pairwise key establishment algorithm are designed , which combines the idea of KDC ( Key Distribution Center ) and polynomial pool schemes .
Furthermore , the working performance of the newly proposed pairwise key establishment algorithm is seriously inspected .
Theoretic analysis and experimental figures show that the new algorithm has better performance and provides higher possibilities for sensor to establish pairwise key , compared with previous related works .
1 .
INTRODUCTION
Security communication is an important requirement in many sensor network applications , so shared secret keys are used between communicating nodes to encrypt data .
As one of the most fundamental security services , pairwise key establishment enables the sensor nodes to communicate securely with each other using cryptographic techniques .
However , due to the sensor nodes ' limited computational capabilities , battery energy , and available memory , it is not feasible for them to use traditional pairwise key establishment techniques such as public key cryptography and key distribution center ( KDC ) .
Several
alternative approaches have been developed recently to perform pairwise key establishment on resource-constrained sensor networks without involving the use of traditional cryptography [ 14 ] .
Eschenauer and Gligor proposed a basic probabilistic key predistribution scheme for pairwise key establishment [ 1 ] .
In the scheme , each sensor node randomly picks a set of keys from a key pool before the deployment so that any two of the sensor nodes have a certain probability to share at least one common key .
Chan et al. further extended this idea and presented two key predistribution schemes : a q-composite key pre-distribution scheme and a random pairwise keys scheme .
The q-composite scheme requires any two sensors share at least q pre-distributed keys .
The random scheme randomly picks pair of sensors and assigns each pair a unique random key [ 2 ] .
Based on such a framework , they presented two pairwise key pre-distribution schemes : a random subset assignment scheme and a grid-based scheme .
A polynomial pool is used in those schemes , instead of using a key pool in the previous techniques .
The random subset assignment scheme assigns each sensor node the secrets generated from a random subset of polynomials in the polynomial pool .
The gridbased scheme associates polynomials with the rows and the columns of an artificial grid , assigns each sensor node to a unique coordinate in the grid , and gives the node the secrets generated from the corresponding row and column polynomials .
Based on this grid , each sensor node can then identify whether it can directly establish a pairwise key with another node , and if not , what intermediate nodes it can contact to indirectly establish the pairwise key .
A similar approach to those schemes described by Liu et al was independently developed by Du et a. [ 5 ] .
Rather than on Blundo 's scheme their approach is based on Blom 's scheme [ 6 ] .
All of those schemes above improve the security over the basic probabilistic key pre-distribution scheme .
However , the pairwise key establishment problem in sensor networks is still not well solved .
For the basic probabilistic and the q-composite key predistribution schemes , as the number of compromised nodes increases , the fraction of affected pairwise keys increases quickly .
As a result , a small number of compromised nodes may affect a
large fraction of pairwise keys [ 3 ] .
Though the random pairwise keys scheme doses not suffer from the above security problem , it incurs a high memory overhead , which increases linearly with the number of nodes in the network if the level of security is kept constant [ 2 ] [ 4 ] .
For the random subset assignment scheme , it suffers higher communication and computation overheads .
In 2004 , Liu proposed a new hypercube-based pairwise key predistribution scheme [ 7 ] , which extends the grid-based scheme from a two dimensional grid to a multi-dimensional hypercube .
The analysis shows that hypercube-based scheme keeps some attractive properties of the grid-based scheme , including the guarantee of establishing pairwise keys and the resilience to node compromises .
Also , when perfect security against node compromise is required , the hypercube-based scheme can support a larger network by adding more dimensions instead of increasing the storage overhead on sensor nodes .
Though hypercube-based scheme ( we consider the grid-based scheme is a special case of hypercube-based scheme ) has many attractive properties , it requires any two nodes in sensor networks can communication directly with each other .
This strong assumption is impractical in most of the actual applications of the sensor networks .
In this paper , we present a kind of new cluster-based distribution model of sensor networks , and for which , we propose a new pairwise key pre-distribution scheme .
Based on the topology , we propose a novel cluster distribution based hierarchical hypercube model to establish the pairwise key .
We develop a kind of new pairwise key establishment algorithm with our hierarchical hypercube model .
The structure of this paper is arranged as follows : In section 3 , a new distribution model of cluster deployed sensor networks is presented .
In section 4 , a new Hierarchical Hypercube model is proposed .
In section 5 , the mapping relationship between the clusters deployed sensor network and Hierarchical Hypercube model is discussed .
In section 6 and section 7 , new pairwise key establishment algorithm are designed based on the Hierarchical Hypercube model and detailed analyses are described .
Finally , section 8 presents a conclusion .
2 .
PRELIMINARY
Definition 1 ( Key Predistribution ) : The procedure , which is used to encode the corresponding encryption and decryption algorithms in sensor nodes before distribution , is called Key Predistribution .
Definition 2 ( Pairwise Key ) : For any two nodes A and B , if they have a common key E , then the key E is called a pairwise key between them .
8 .
CONCLUSION
A new hierarchical hypercube model named H ( k , u , m , v , n ) is proposed , which can be used for pairwise key predistribution for cluster deployed sensor networks .
And Based on the H ( k , u , m , v , n ) model , an innovative pairwise key predistribution scheme and algorithm are designed respectively , by combing the good properties of the Polynomial Key and Key Pool encryption schemes .
So , the traditional pairwise key predistribution algorithm based on hypercube model [ 7 ] is only a special case of the new algorithm proposed in this paper .
Theoretical and experimental analyses show that the newly proposed algorithm is an efficient pairwise key establishment algorithm that is suitable for the cluster deployed sensor networks .

Researches on Scheme of Pairwise Key Establishment for Distributed Sensor Networks
ABSTRACT
Security schemes of pairwise key establishment , which enable sensors to communicate with each other securely , play a fundamental role in research on security issue in wireless sensor networks .
A new kind of cluster deployed sensor networks distribution model is presented , and based on which , an innovative Hierarchical Hypercube model - H ( k , u , m , v , n ) and the mapping relationship between cluster deployed sensor networks and the H ( k , u , m , v , n ) are proposed .
By utilizing nice properties of H ( k , u , m , v , n ) model , a new general framework for pairwise key predistribution and a new pairwise key establishment algorithm are designed , which combines the idea of KDC ( Key Distribution Center ) and polynomial pool schemes .
Furthermore , the working performance of the newly proposed pairwise key establishment algorithm is seriously inspected .
Theoretic analysis and experimental figures show that the new algorithm has better performance and provides higher possibilities for sensor to establish pairwise key , compared with previous related works .
1 .
INTRODUCTION
Security communication is an important requirement in many sensor network applications , so shared secret keys are used between communicating nodes to encrypt data .
As one of the most fundamental security services , pairwise key establishment enables the sensor nodes to communicate securely with each other using cryptographic techniques .
However , due to the sensor nodes ' limited computational capabilities , battery energy , and available memory , it is not feasible for them to use traditional pairwise key establishment techniques such as public key cryptography and key distribution center ( KDC ) .
Several
alternative approaches have been developed recently to perform pairwise key establishment on resource-constrained sensor networks without involving the use of traditional cryptography [ 14 ] .
Eschenauer and Gligor proposed a basic probabilistic key predistribution scheme for pairwise key establishment [ 1 ] .
In the scheme , each sensor node randomly picks a set of keys from a key pool before the deployment so that any two of the sensor nodes have a certain probability to share at least one common key .
Chan et al. further extended this idea and presented two key predistribution schemes : a q-composite key pre-distribution scheme and a random pairwise keys scheme .
The q-composite scheme requires any two sensors share at least q pre-distributed keys .
The random scheme randomly picks pair of sensors and assigns each pair a unique random key [ 2 ] .
Inspired by the studies above and the polynomial-based key pre-distribution protocol [ 3 ] , Liu et al. further developed the idea addressed in the previous works and proposed a general framework of polynomial pool-based key predistribution [ 4 ] .
The basic idea can be considered as the combination of the polynomial-based key pre-distribution and the key pool idea used in [ 1 ] ] and [ 2 ] .
Based on such a framework , they presented two pairwise key pre-distribution schemes : a random subset assignment scheme and a grid-based scheme .
A polynomial pool is used in those schemes , instead of using a key pool in the previous techniques .
The random subset assignment scheme assigns each sensor node the secrets generated from a random subset of polynomials in the polynomial pool .
The gridbased scheme associates polynomials with the rows and the columns of an artificial grid , assigns each sensor node to a unique coordinate in the grid , and gives the node the secrets generated from the corresponding row and column polynomials .
Based on this grid , each sensor node can then identify whether it can directly establish a pairwise key with another node , and if not , what intermediate nodes it can contact to indirectly establish the pairwise key .
A similar approach to those schemes described by Liu et al was independently developed by Du et a. [ 5 ] .
Rather than on Blundo 's scheme their approach is based on Blom 's scheme [ 6 ] .
In some cases , it is essentially equivalent to the one in [ 4 ] .
All of those schemes above improve the security over the basic probabilistic key pre-distribution scheme .
However , the pairwise key establishment problem in sensor networks is still not well solved .
For the basic probabilistic and the q-composite key predistribution schemes , as the number of compromised nodes increases , the fraction of affected pairwise keys increases quickly .
As a result , a small number of compromised nodes may affect a
large fraction of pairwise keys [ 3 ] .
Though the random pairwise keys scheme doses not suffer from the above security problem , it incurs a high memory overhead , which increases linearly with the number of nodes in the network if the level of security is kept constant [ 2 ] [ 4 ] .
For the random subset assignment scheme , it suffers higher communication and computation overheads .
In 2004 , Liu proposed a new hypercube-based pairwise key predistribution scheme [ 7 ] , which extends the grid-based scheme from a two dimensional grid to a multi-dimensional hypercube .
The analysis shows that hypercube-based scheme keeps some attractive properties of the grid-based scheme , including the guarantee of establishing pairwise keys and the resilience to node compromises .
Also , when perfect security against node compromise is required , the hypercube-based scheme can support a larger network by adding more dimensions instead of increasing the storage overhead on sensor nodes .
Though hypercube-based scheme ( we consider the grid-based scheme is a special case of hypercube-based scheme ) has many attractive properties , it requires any two nodes in sensor networks can communication directly with each other .
This strong assumption is impractical in most of the actual applications of the sensor networks .
In this paper , we present a kind of new cluster-based distribution model of sensor networks , and for which , we propose a new pairwise key pre-distribution scheme .
The main contributions of this paper are as follows : Combining the deployment knowledge of sensor networks and the polynomial pool-based key pre-distribution , we setup a clusterbased topology that is practical with the real deployment of sensor networks .
Based on the topology , we propose a novel cluster distribution based hierarchical hypercube model to establish the pairwise key .
The key contribution is that our scheme dose not require the assumption of all nodes can directly communicate with each other as the previous schemes do , and it still maintains high probability of key establishment , low memory overhead and good security performance .
We develop a kind of new pairwise key establishment algorithm with our hierarchical hypercube model .
The structure of this paper is arranged as follows : In section 3 , a new distribution model of cluster deployed sensor networks is presented .
In section 4 , a new Hierarchical Hypercube model is proposed .
In section 5 , the mapping relationship between the clusters deployed sensor network and Hierarchical Hypercube model is discussed .
In section 6 and section 7 , new pairwise key establishment algorithm are designed based on the Hierarchical Hypercube model and detailed analyses are described .
Finally , section 8 presents a conclusion .
2 .
PRELIMINARY
Definition 1 ( Key Predistribution ) : The procedure , which is used to encode the corresponding encryption and decryption algorithms in sensor nodes before distribution , is called Key Predistribution .
Definition 2 ( Pairwise Key ) : For any two nodes A and B , if they have a common key E , then the key E is called a pairwise key between them .
Definition 3 ( Key Path ) : For any two nodes A0 and Ak , when there has not a pairwise key between them , if there exists a path A0,A1,A2,...,Ak-1 , Ak , and there exists at least one pairwise key between the nodes Ai and Aj for 05i5k-1 and 15j5k , then the path consisted of A0 , A1 , A2 , ... , Ak-1 , Ak is called a Key Path between A0 and Ak .
Definition 4 ( n-dimensional Hypercube ) : An n-dimensional Hypercube ( or n − cube ) H ( v , n ) is a topology with the following properties : ( 1 ) It is consisted of n · vn-1 edges , ( 2 ) Each node can be coded as a string with n positions such as b1b2 ... bn , where 05b1,b2,...,bn5v-1 , ( 3 ) Any two nodes are called neighbors , which means that there is an edge between them , iff there is just one position different between their node codes .
3 .
MODEL OF CLUSTERS DEPLOYED SENSOR NETWORKS
In some actual applications of sensor networks , sensors can be deployed through airplanes .
Supposing that the deployment rounds of sensors are k , and the communication radius of any sensors is r , then the sensors deployed in the same round can be regarded as belonging to a same Cluster .
We assign a unique cluster number l ( 1 < _ l < _ k ) for each cluster .
Supposing that the sensors form a connected graph in any cluster after deployment through airplanes , and then the Fig. 1 presents an actual model of clusters deployed sensor networks .
Figure .1 An actual model of clusters deployed sensor networks .
From Figure .1 , it is easy to know that , for a given node A , there exist lots of nodes in the same cluster of A , which can be communicated directly with A , since the nodes are deployed densely in a cluster .
But there exist much less nodes in a cluster neighboring to the cluster of A , which can be communicated directly with A. since the two clusters are not deployed at the same time .
4 .
HIERARCHICAL HYPERCUBE MODEL
Definition 5 ( k-levels Hierarchical Hypercube ) : Let there are N nodes totally , then a k-levels Hierarchical Hypercube named H ( k , u , m , v , n ) can be constructed as follows :
as i1i2 ... in , which are called In-Cluster-Hypercube-Node-Codes , where 0 ≤ i1 , i2 , ... in ≤ v-1 , v =[ n N/k ] , [ j ] equals to an integer not less than j .
So we can obtain k such kind of different hypercubes .
2 ) The k different hypercubes obtained above are encoded as j1j2 ... jm , which are called Out-Cluster-Hypercube-Node-Codes , where 0 ≤ j1 , j2 , ... jm ≤ u-1 , u =[ m k ] .
And the nodes in the k different hypercubes are connected into m-dimensional hypercubes according to the following rules : The nodes with same In-Cluster-Hypercube-Node-Codes and different Out-ClusterHypercube-Node-Codes are connected into an m-dimensional hypercube .
( The graph constructed through above steps is called a k-levels Hierarchical Hypercube abbreviated as H ( k , u , m , v , n ) . )
3 ) Any node A in H ( k , u , m , v , n ) can be encoded as ( i , j ) , where i ( i = i1i2 ... in , 0 ≤ i1 , i2 , ... in ≤ v-1 ) is the In-Cluster-HypercubeNode-Code of node A , and j ( j = j1j2 ... jm , 0 ≤ j1 , j2 , ... jm ≤ u-1 ) is the Out-Cluster-Hypercube-Node-Code of node A. Obviously , the H ( k , u , m , v , n ) model has the following good properties :
Property 1 : The diameter of H ( k , u , m , v , n ) model is m + n.
Proof : Since the diameter of n-dimensional hypercube is n , and the diameter of m-dimensional hypercube is m , so it is easy to know that the diameter of H ( k , u , m , v , n ) model is m + n from the definition 5 .
Property 2 : The distance between any two nodes A ( i1 , j1 ) and B ( i2 , j2 ) in H ( k , u , m , v , n ) model is d ( A , B ) = dh ( i1 , i2 ) + dh ( j1 , j2 ) , where dh represents the Hamming distance .
Proof : Since the distance between any two nodes in hypercube equals to the Hamming distance between them , so it is obvious that the theorem 2 's conclusion stands from definition 5 .
5 .
MAPPING CLUSTERS DEPLOYED SENSOR NETWORKS TO H ( K , U , M , V , N )
Obviously , from the description in section 3 and 4 , we can know that the clusters deployed sensor network can be mapped into a klevels - hierarchical hypercube model as follows : At first , the k clusters in the sensor network can be mapped into k different levels ( or hypercubes ) in the k-levels - hierarchical hypercube model .
Then , the sensor nodes in each cluster can be encoded with the In-Cluster-Hypercube-Node-Codes , and the sensor nodes in the k different clusters with the same In-ClusterHypercube-Node-Codes can be encoded with the Out-ClusterHypercube-Node-Codes according to the definition 5 respectively .
Consequently , the whole sensor network has been mapped into a k-levels - hierarchical hypercube model .
6 .
H ( K , U , M , V , N ) MODEL-BASED PAIRWISE KEY PREDISTRIBUTION ALGORITHM FOR SENSOR NETWORKS
In order to overcome the drawbacks of polynomial-based and polynomial pool-based key predistribution algorithms , this paper proposed an innovative H ( k , u , m , v , n ) model-based key predistribution scheme and pairwise key establishment algorithm , which combines the advantages of polynomial-based and key pool-based encryption schemes , and is based on the KDC and polynomials pool-based key predistribution models .
The new H ( k , u , m , v , n ) model-based pairwise key establishment algorithm includes three main steps : ( 1 ) Generation of the polynomials pool and key predistribution , ( 2 ) Direct pairwise key discovery , ( 3 ) Path key discovery .
6.1 Generation of Polynomials Pool and Key Predistribution
Supposing that , the sensor network includes N nodes , and is deployed through k different rounds .
Then we can predistribute keys for each sensor node on the basis of the H ( k , u , m , v , n ) model as follows : Step 1 : Key setup server randomly generates a bivariate polynomials pool such as the following : F ={ f i l , < i 1 , i 2 , ... , i n − 1 > ( x , y ) , f < ji1 , i2 , ... , in , j1 , j2 , ... , jm − 1 >
+ [ N/vn ] * n * vn-1 different t-degree bivariate polynomials over a finite field Fq , and then assigns a unique polynomial ID to each bivariate polynomial in F. Step 2 : In each round , key setup server assigns a unique node ID : ( i1i2 ... in , j1j2 ... jm ) to each sensor node from small to big , where 0 ≤ i1 , i2 , ... in ≤ v-1 , 0 ≤ j1 , j2 , ... jm ≤ u-1 .
Step 3 : key setup server assigns a unique cluster ID : l to all the sensor nodes deployed in the same round , where 1 ≤ l ≤ k. Step 4 : key setup server predistributes m + n bivariate polynomials { f l i i in
corresponding polynomial IDs to the sensor node deployed in the lth round and with ID ( i1i2 ... in , j1j2 ... jm ) .
6.2 Direct Pairwise Key Discovery
If the node A ( i1i2 ... in , j1j2 ... jm ) in the sensor network wants to establish pairwise key with a node B ( i ' 1i ' 2 ... i 'n , j' 1j ' 2 ... j 'm ) , then node A can establish pairwise key with the node B trough the following methods .
Firstly , node A computes out the distance between itself and node B : d = d1 + d2 , where d1 = dh ( i1i2 ... in , i ' 1i ' 2 ... i 'n ) and d2 = dh ( j1j2 ... jm , j' 1j ' 2 ... j 'm ) .
If d = 1 , then node A obtains the direct pairwise key between itself and node B according to the following theorem 1 : Theorem 1 : For any two sensor nodes A ( i1i2 ... in , j1j2 ... jm ) and B ( i ' 1i ' 2 ... i 'n , j' 1j ' 2 ... j 'm ) in the sensor network , supposing that the
distance between nodes A and B is d = d1 + d2 , where d1 = dh ( i1i2 ... in , i ' 1i ' 2 ... i 'n ) and d2 = dh ( j1j2 ... jm , j' 1j ' 2 ... j 'm ) .
If d = 1 , then there exists a direct pairwise key between nodes A and B. Poof : Since d = 1 , then there is d1 = 1 , d2 = 0 , or d1 = 0 , d2 = 1 .
1 ) If d1 = 1 , d2 = 0 : From d2 = 0 , there is nodes A , B belong to the same cluster .
Supposing that nodes A , B belong to the same cluster l , then from d1 = 1 ⇒ There is only one position different between i1i2 ... in and i ' 1i ' 2 ... i 'n .
Let it = i ' t , when 1 ≤ t ≤ n-1 , and in ≠ i 'n ⇒ f n l , < i 1 , i 2 , ... , i n − 1 > ( in , i 'n ) = f n l , < i ′ 1 , i ′ 2 , ... , i ′ n − 1 > ( i 'n , in ) .
So , there exists a direct pairwise key f nl , < i1 , i2 , ... , in − 1 > ( in , i 'n ) between nodes A and B.
2 ) If d1 = 0 , d2 = 1 : From d2 = 1 ⇒ There is only one position different between j1j2 ... jm and j' 1j ' 2 ... j 'm .
Let jt = j' t , when 1 ≤ t ≤ m1 , and jm ≠ j 'm .
Since d1 = 0 ⇒ i1i2 ... in equals to
i ' 1i ' 2 ... i 'n ⇒ f < mi 1 , i 2 , ... , in , j1 , j2 , ... , jm − 1 > ( jm , j 'm ) = f < m i ′ 1 , i ′ 2 , ... , i ′ n , j ′ 1 , j ′ 2 , ... , j ′ m − 1 > ( j 'm , jm ) .
So , there exists a direct pairwise key f < m i 1 , i2 , ... , in , j 1 , j2 , ... , j m − 1 > ( jm , j 'm ) between nodes A and B .
According to theorem 1 , we present the detailed description of the direct pairwise key discovery algorithm as follows : Step 1 : Obtain the node IDs and cluster IDs of the source node A and destination node B ; Step 2 : Compute out the distance between nodes A and B : d = d1 + d2 ; Step 3 : If d1 = 1 , d2 = 0 , then select out a common polynomial share of nodes A and B from { f l i i in
establish direct pairwise key ; Step 4 : If d1 = 0 , d2 = 1 , then select out a common polynomial share of nodes A and B from { f i i in j jm 1 < 1,2 , ... , ,2 , ... , > , ... , f < m i1 , i2 , ... , in , j1 , j2 , ... , jm − 1 > } to establish direct pairwise key ; Step 5 : Otherwise , there exists no direct pairwise key between nodes A and B. And then turn to the following path key discovery process .
6.3 Path Key Discovery
If d > 1 , then node A can establish path key with node B according to the following theorem 2 : Theorem 2 : For any two sensor nodes A ( i1i2 ... in , j1j2 ... jm ) and B ( i ' 1i ' 2 ... i 'n , j' 1j ' 2 ... j 'm ) in the sensor network , supposing that the distance between nodes A and B is d = d1 + d2 , where d1 = dh ( i1i2 ... in , i ' 1i ' 2 ... i 'n ) and d2 = dh ( j1j2 ... jm , j' 1j ' 2 ... j 'm ) .
If d > 1 , then there exists a path key between nodes A and B. Proof : Let d1 = a , d2 = b , then we can think that it ≠ i ' t , when 1 ≤ t ≤ a ; but it = i ' t , when t > a ; and jt ≠ j' t , when 1 ≤ t ≤ b ; but jt = j' t , when t > b. Obviously , nodes A ( i1i2 ... in , j1j2 ... jm ) , ( i ' 1i2 i3 ... in , j1j2 ... jm ) , ( i ' 1i ' 2 i3 ... in , j1j2 ... jm ) , ... , ( i ' 1i ' 2 ... i 'n , j1j2 ... jm ) belong to the same cluster .
So , according to the supposing condition of `` The nodes in the same cluster form a connected graph '' , there is a route among those nodes .
In addition , in those nodes , the distance between any two neighboring nodes is 1 , so from theorem 1 , it is easy to know that there exists direct pairwise key between any two neighboring nodes among those nodes .
For nodes ( i ' 1i ' 2 ... i 'n , j1j2 ... jm ) , ( i ' 1i ' 2 ... i 'n , j' 1 j2 j3 ... jm ) , ( i ' 1i ' 2 ... i 'n , j' 1j ' 2 j3...jm-1 jm ) , ... , ( i ' 1i ' 2 ... i 'n , j' 1j ' 2 ... j 'm -1 jm ) , since they have the same Out-Cluster-Hypercube-Node-Codes with the node B ( i ' 1i ' 2 ... i 'n , j' 1j ' 2 ... j 'm ) , so nodes ( i ' 1i ' 2 ... i 'n , j1j2 ... jm ) , ( i ' 1i ' 2 ... i 'n , j' 1 j2 j3 ... jm ) , ( i ' 1i ' 2 ... i 'n , j' 1j ' 2 j3...jm-1 jm ) , ... , ( i ' 1i ' 2 ... i 'n , j' 1j ' 2 ... j 'm -1 jm ) and node B belong to a same logical hypercube .
Obviously , from the supposing condition of `` The whole sensor network forms a connected graph '' , there is a route among those nodes .
In addition , in those nodes , the distance between any two neighboring nodes is 1 , so from theorem 1 , it is easy to know that there exists direct pairwise key between any two neighboring nodes among those nodes .
So , it is obvious that there exists a path key between nodes A and B .
According to theorem 2 , we present the detailed description of the path key discovery algorithm as follows :
their common polynomial share to establish direct pairwise key .
From theorem 2 , it is easy to know that any source node A can compute out a key path P to the destination node B according to the above algorithm , when there are no compromised nodes in the sensor network .
Once the key path P is computed out , then node A can send messages to B along the path P to establish indirect pairwise key with node B. Fig. 2 presents a example of key path establishment .
Figure .2 Key path establishment example .
For example : In the above Figure .2 , node A ( ( 012 ) , ( 1234 ) ) can establish pairwise key with node B ( ( 121 ) , ( 2334 ) ) through the following key path : A ( ( 012 ) , ( 1234 ) ) → C ( ( 112 ) , ( 1234 ) ) → D ( ( 122 ) , ( 1234 ) ) → E ( ( 121 ) , ( 1234 ) ) → F ( ( 121 ) , ( 2234 ) ) → B ( ( 121 ) , ( 2334 ) ) , where node F shall route through nodes G , H , I , J to establish direct pairwise key with node B.
According to the properties of H ( k , u , m , v , n ) model , we can prove that the following theorem by combing the proof of theorem 2 : Theorem 3 : Supposing that there exist no compromised nodes in the sensor network , and the distance between node A and B , then there exists a shortest key path with k distance between node A and B logically .
That is to say , node A can establish indirect pairwise key with node B through t-1 intermediate nodes .
Proof : Supposing that the distance between node A ( i1i2 ... in , j1j2 ... jm ) and B ( i ' 1i ' 2 ... i 'n , j' 1j ' 2 ... j 'm ) is d = d1 + d2 , where d1 = dh ( i1i2 ... in , i ' 1i ' 2 ... i 'n ) , d2 = dh ( j1j2 ... jm , j' 1j ' 2 ... j 'm ) .
Since d = t , according to the construction properties of H ( k , u , m , v , n ) , it is easy to know that there exist t-1 intermediate nodes I1,...,It-1 , in the logical space H ( k , u , m , v , n ) , which satisfy that the distance between any two neighboring nodes in the nodes series A , I1 , ... , It1 , B equals to 1 .
So according to the theorem 1 , we can know that nodes A , I1,...,It-1 , B form a correct key path between node A and B .
If any two neighboring nodes in the nodes series A , I1,...,It-1 , B can communicate directly , then node A can establish indirect pairwise key with node B through those t-1 intermediate nodes .
6.4 Dynamic Path Key Discovery
The path key discovery algorithm proposed in the above section can establish a key path correctly , only when there exist no compromised nodes in the whole sensor network , since the key path is computed out beforehand .
And the proposed algorithm can not find an alternative key path when there exist some compromised nodes or some intermediate nodes not in the communication radius , even that there exists other alternative key paths in the sensor network .
From the following example we can know that there are many parallel paths in the H ( k , u , m , v , n ) model for any two given source and destination nodes , since the H ( k , u , m , v , n ) model is high fault-tolerant [ 9,10 ] .
Figure .3 Alternative key path establishment example .
For example : Considering the key path establishment example given in the above section based on Figure .2 :
supposing that node F ( ( 121 ) , ( 2234 ) ) has compromised , then from Figure .3 , we can know that there exists another alternative key path as A ( ( 012 ) , ( 1234 ) ) → C ( ( 112 ) , ( 1234 ) ) → D ( ( 122 ) , ( 1234 ) ) → E ( ( 121 ) , ( 1234 ) ) → M ( ( 121 ) , ( 1334 ) ) → B ( ( 121 ) , ( 2334 ) ) , which can be used to establish the indirect pairwise key between node A and B , where node E shall route through nodes D and K to establish direct pairwise key with node M , and node M shall route through nodes N , O , G , H , I , J to establish direct pairwise key with node B .
Since the sensors are source limited , so they are easy to die or out of the communication radius , therefore the algorithm proposed in the above section can not guarantee to establish correct key path efficiently .
In this section , we will propose a dynamic path key discovery algorithm as follows , which can improve the probability of key path effectively : Algorithm I : Dynamic key path establishment algorithm based on H ( k , u , m , v , n ) model for cluster deployed sensor networks .
Input : Sub-sensor network H ( k , u , m , v , n ) , which has some compromised / fault sensors and fault links , And two reachable nodes A ( a1 ... an , a ' 1 ... a 'm ) and B ( b1 ... bn , b ' 1 ... b 'm ) in H ( k , u , m , v , n ) , where a ' t # b ' t , tE [ 1 , s ] , a ' t = b ' t , t > s.
Output : A correct key path from node A to B in H ( k , u , m , v , n ) .
Step 1 : Obtain the code strings of node A and B : A F ( a1 ... an , a ' 1 ... a 'm ) , B F ( b1 ... bn , b ' 1 ... b 'm ) , where aj , bj ∈ [ 0,u-1 ] , a ' j , b ' j ∈ [ 0,v-1 ] .
Step 2 : If a ' 1 ... a 'm = b ' 1 ... b 'm , then node A can find a route to B according to the routing algorithms of hypercube [ 9-10 ] .
Step 3 : Otherwise , node A can find a route to C ( b1 ... bn , a ' 1 ... a 'm ) according to the Algorithm I or Algorithm II .
Then let I0 = C ( b1 ... bn , a ' 1 ... a 'm ) , I1 = ( b1 ... bn , b ' 1 a ' 2 ... a 'm ) , ... , Is = B ( b1 ... bn , b ' 1 b ' 2 ... b 's a 's +1 ... a 'm ) , and each node It in the above nodes series find a route to its neighboring node It +1 on the basis of the location information ( Detailed routing algorithms based on location information can see the references [ 11-14 ] ) .
Step 4 : Algorithm exits .
If such kind of a correct key path exists , then through which node A can establish an indirect pairwise key with node B. Otherwise , node A fails to establish an indirect pairwise key with node B. And node A will tries again to establish an indirect pairwise key with node B some time later .
7 .
ALGORITHM ANALYSES
7.1 Practical Analyses
According to the former description and analyses , it is easy to know that the above newly proposed algorithm has the following properties : Property 3 : When there exist no fault and compromised nodes , by using new pairwise key predistribution scheme based on H ( k , u , m , v , n ) model , the probability of direct pairwise key establishment between any two nodes can be estimated as P = ( m ( u-1 ) + n ( v-1 ) ) / ( N-1 ) , where N is the total number of nodes in the sensor network , and N = um * vn .
Proof : Since the predistributed pairwise keys for any node FA
, y ) , ... , f < m i1 , i2 , ... , in , j 1 , ... , jm − 1 > ( jm , y ) } in the newly proposed algorithm .
Obviously , in the logical hypercube formed by the nodes in the same cluster of node A , there are n ( v-1 ) nodes , which
have direct pairwise key with node A. And in the logical hypercube formed by the nodes in different clusters from that of node A , there are m ( u-1 ) nodes , which have direct pairwise key with node A. Therefore , there are totally m ( u-1 ) + n ( v-1 ) nodes , which have direct pairwise key with node A. So , the probability of pairwise key establishment between any two nodes can be estimated as P = ( m ( u-1 ) + n ( v-1 ) ) / ( N-1 ) , since the whole sensor network has N sensor nodes in all .
Figure .4 presents the comparision between the probability of direct pairwise key establishment between any two nodes and the dimension n , when the sensor network has different total nodes , and use the new pairwise key predistribution scheme based on H ( 8,2,3 , v , n ) model .
Figure .4 Comparision between the probability of direct pairwise key establishment between any two nodes and the dimension n , when the sensor network has different total nodes , and use the new pairwise key predistribution scheme based on H ( 8,2,3 , v , n ) model .
From Figure .4 , it is easy to know that by using new pairwise key predistribution scheme based on H ( k , u , m , v , n ) model , the probability of direct pairwise key establishment between any two nodes decreases with the increasing of the scale of the sensor networks , and in addition , the probability of direct pairwise key establishment between any two nodes decreases with the increasing of the dimension n , when the scale of the sensor network is fixed .
Theorem 4 : Supposing that the total sensors is N in the sensor network , then when u > -- v2 , the probability of direct pairwise key establishment between any two nodes , when using the key distribution scheme based on the hypercube model H ( v , p ) , is smaller than that when using the key distribution scheme based on the H ( k , u , m , v , n ) model .
Proof : Since u > -- v , then we can let u = vt , where t > -- 2 .
Since the total number of nodes in H ( v , p ) is vp = N , the total number of nodes in H ( k , u , m , v , n ) is um * vn = N. Let p = x + n , then there is um * vn = vx * vn = > um = vx = > x = tm .
From the property 3 , it is easy to know that the probability of direct pairwise key establishment between any two nodes can be estimated as P = ( m ( u-1 ) + n ( v-1 ) ) / ( N-1 ) .
According to the description in [ 7 ] , it is well know that the probability of direct pairwise key establishment between any two nodes can be estimated as P ' = p ( v-1 ) / ( N-1 ) = ( x ( v-1 ) + n ( v-1 ) ) / ( N-1 ) .
Next , we will prove that m ( u-1 ) > -- x ( v-1 ) :
Therefore , the conclusion of the theorem stands .
As for the conclusion of theorem 4 , we give an example to illustrate .
Supposing that the total number of nodes in the sensor network is N = 214 , and H ( k , u , m , v , n ) = H ( 16,4,2,2,10 ) , H ( v , p ) = H ( 10,14 ) , then the probability of direct pairwise key establishment between any two nodes based on the H ( k , u , m , v , n ) model is P = ( m ( u-1 ) + n ( v1 ) ) / ( N-1 ) = ( 2 ( 4-1 ) +10 ( 2-1 ) ) / ( 214-1 ) = 16 / ( 214-1 ) , but the probability of direct pairwise key establishment between any two nodes based on the H ( v , p ) model is P ' = p ( v-1 ) / ( N-1 ) = 14 ( 2-1 ) / ( 214-1 ) = 14 / ( 2141 ) .
Supposing that the total number of nodes in the sensor network is N , Figure .5 illustrates the comparison between the probability of direct pairwise key establishment between any two nodes based on the H ( k , u , m , v , n ) model and the probability of direct pairwise key establishment between any two nodes based on the H ( v , p ) model , when u = 4 and v = 2 .
01 2 3 4 5 6 7 8 9 10 scale of the sensor network Figure .5 Comparison between the probability of direct pairwise key establishment between H ( v , n ) and H ( k , u , m , v , n ) models .
From Figure .5 , it is easy to know that the theorem 5 stands .
Theorem 5 : Supposing that the total sensors is N in the sensor network , then the pairwise key distribution scheme based on the hypercube model H ( v , p ) , is only a special case of the pairwise key distribution scheme based on the H ( k , u , m , v , n ) model .
Proof : As for the pairwise key distribution scheme based on the H ( k , u , m , v , n ) model , let k = 1 ( u = 1 , m = 0 ) , which means that the total sensor network includes only one cluster .
Then obviously , the H ( k , u , m , v , n ) model will degrade into the H ( v , n ) model .
According to the former anayses in this paper and the definition of the pairwise key distribution scheme based on the hypercube model H ( v , p ) in [ 7 ] , it is easy to know that the conclusion of the theorem stands .
7.2 Security Analyses
By using the pairwise key establishment algorithm based on the H ( k , u , m , v , n ) model , the intruders can launch two kinds of attacks : 1 ) The attackers may target the pairwise key between two particular sensor node , in order to compromise the pairwise key between them , or prevent them to establish pairwise key .
2 ) The attackers may attack against the whole sensor network , inorder to decrease the probability of the pairwise key establishment , or increase the cost of the pairwise key establishment .
Attacks against a Pair of sensor nodes
1 .
Supposing that the intruders want to attack two particular sensor nodes u , v , where u , v are all not compromised nodes , but the intruders want to compromise the pairwise key between them .
1 ) If u , v can establish direct pairwise key , then the only way to compromise the key is to compromise the common bivariate polynomial f ( x , y ) between u , v .
Since the degree of the bivariate polynomial f ( x , y ) is t , so the intruders need to compromise at least t +1 sensor nodes that have a share of the bivariate polynomial f ( x , y ) .
2 ) If u , v can establish indirect pairwise key through intermediate nodes , then the intruders need to compromise at least one intermediate node , or compromise the common bivariate polynomial f ( x , y ) between two neighboring intermediate nodes .
But even if the intruders succeed to do that , node u and v can still reestablish indirect pairwise key through alternative intermediate nodes .
2 .
Supposing that the intruders want to attack two particular sensor nodes u , v , where u , v are all not compromised nodes , but the intruders want to prevent them to establish the pairwise key .
Then , the intruders need to compromise all of the m + n bivariate polynomials of node u or v .
Since the degree of the bivariate polynomial f ( x , y ) is t , so for bivariate polynomial , the intruders need to compromise at least t +1 sensor nodes that have a share of the given bivariate polynomial .
Therefore , the intruders need to compromise ( m + n ) ( t +1 ) sensor nodes altogether to prevent u , v to establish the pairwise key .
Attacks against the sensor network
Supposing that the Attackers know the distribution of the polynomials over sensor nodes , it may systematically attack the network by compromising the polynomials in F one by one in order to compromise the entire network .
Assume the fraction of the compromised polynomials is pc , then there are up to n N ' = pcx { v m u nx x m + [ ] x x } = pcxNx ( m + n ) N n vvn Sensor nodes that have at least one compromised polynomial share .
Among all of the remaining N - N ' sensor nodes , none of them includes a compromised polynomial share .
So , the remaining N - N ' sensor nodes can establish direct pairwise key by using any one of their polynomial shares .
However , the indirect pairwise keys in the remaining N - N ' sensor nodes may be affected .
And they may need to re-establish a new indirect pairwise key between them by select alternative intermediate nodes that do not belong to the N ' compromised nodes .
Supposing that the scale of the sensor network is N = 10000 , Figure .6 presents the comparison between pc and the number of sensor nodes with at least one compromised polynomial share in sensor networks based on different H ( k , u , m , v , n ) distribution models .
Fraction of Compromised B ivariate Polynomials
Figure .6 the comparison between pc and the number of sensor nodes with at least one compromised polynomial share in sensor networks based on different H ( k , u , m , v , n ) distribution models .
Theorem 6 : Supposing that the total sensors is N in the sensor network , and the fraction of compromised nodes is pc , then when u > v , the number of affected nodes of the H ( v , p ) model based key predistribution scheme , is bigger than that of the H ( k , u , m , v , n ) model based key predistribution scheme .
Proof : Since the number of affected nodes of the H ( k , u , m , v , n ) model based key predistribution scheme is pcxNx ( m + n ) , and it is proved in [ 7 ] that the number of affected nodes of the H ( v , p ) model based key predistribution scheme is pcxNx p. Let p = x + n , then there is um * vn = vx * vn = > um = vx .
Since u > v = > x > m = > pcxNx ( m + n ) < pcxNx ( x + n ) = pcxNx p. Supposing that the scale of the sensor network is N = 10000 , Figure .7 presents the comparison between pc and the number of sensor nodes with at least one compromised polynomial share in sensor networks based on H ( 9,3,2,2 , n ) and H ( 2 , p ) distribution models .
From Figure .7 , it is easy to know that the conclusion of theorem 9 is correct , and the number of the affected sensor nodes in the sensor network increases with the increasing of the number of compromised nodes , when the scale of the sensor network is fixed .
From Figure .6 , it is easy to know that , when the scale of the sensor network is fixed , the number of the affected sensor nodes in the sensor network increases with the increasing of the number of compromised nodes .
Fraction of Compromised Bivariate Polynomials
Figure .7 the comparison between pc and the number of sensor nodes with at least one compromised polynomial share in sensor networks based on H ( 9,3,2,2 , n ) and H ( 2 , p ) distribution models .
8 .
CONCLUSION
A new hierarchical hypercube model named H ( k , u , m , v , n ) is proposed , which can be used for pairwise key predistribution for cluster deployed sensor networks .
And Based on the H ( k , u , m , v , n ) model , an innovative pairwise key predistribution scheme and algorithm are designed respectively , by combing the good properties of the Polynomial Key and Key Pool encryption schemes .
The new algorithm uses the good characteristics of node codes and high fault-tolerance of H ( k , u , m , v , n ) model to route and predistribute pairwise keys , in which nodes are not needed to be able to communicate with each other directly such as that the algorithms proposed by [ 7 ] shall need .
So , the traditional pairwise key predistribution algorithm based on hypercube model [ 7 ] is only a special case of the new algorithm proposed in this paper .
Theoretical and experimental analyses show that the newly proposed algorithm is an efficient pairwise key establishment algorithm that is suitable for the cluster deployed sensor networks .