Connected Component in Secure Sensor Network Induced by a Random Key Pre-Distribution Scheme Bhupendra Gupta, and Subir Singh Lamba Abstract—Wireless sensor network (WSN) has a wide range of applications in various areas. Many time the environment in which these sensor were deployed are hostile in nature and sensors have continuous attacks from the adversary, in such environmental conditions we need a secure communication between the sensors. For secure communication, neighbors must posses a secret common key or there must exists a key-path among these nodes. In this paper, the object of study is a random graph induced by the random key pre-distribution scheme of Eschenauer and Gligor under the assumption of full visibility. Here we establish the threshold value of the parameters (Key pool size and key-ring of an individual node) for which the entire network is almost surely a single connected component. We prove that for a network having N nodes, is a single connected component almost surely, if size of the key-ring is m = √ 2 log N and the size of key pool is K = N log N. Index Terms- Secure Sensor Networks, Random Key Pre-distribution Scheme, Secure Connectivity. I. I NTRODUCTION Wireless Sensor Networks (WSNs) is a distributed collection of sensor nodes. These wireless sensor networks have wide applications in various areas like in disaster recovery, military operation, tracking etc. Generally these sensor nodes having limited computational and communication capacities. Also WSNs do not have any pre-deployed network architecture (for example sensors like smart dust usually deployed by aeroplanes), some times it is due to the hostile environment. Thus sensors need to communicate via an ad-hoc wireless network. After deployment an individual sensor need to connect with the other sensors and create a secure connected wireless Manuscript received 21 April 2011. B.Gupta is with the Indian Institute of Information Technol- ogy, Design & Manufacturing Jabalpur, MP, India 482005.E-mail: gupta.bhupendra@gmail.com, http://www.iiitdmj.ac.in/ bhupen S.S.Lamba with Indian Institute of Information Technology, Design & Manufacturing Jabalpur, MP, India 482005.Email: subirs@gmail.com, http://www.iiitdmj.ac.in/ subirs sensor network. Many applications (like in military operation) needs highly secure operation of sensor network, and have serious consequences if the network is compromised or disrupted. For this a secure pairwise communication must be established. Also the sensor network must be robust against the compromise of individual sensor due to failure and manipulation by an adversary. One of the most promising approach for achieving a secure sensor wireless network is the random predistribution of keys introduced in [6]. We are initiated by describing the model introduced in [6]. A secured wireless sensor network is composed by a number (says N ) of sensors. Each sensor is preloaded with a key-ring having m distinct encryption keys, randomly chosen from a pool of K keys. The shared key discovery take place during WSN initialization in the operational environment where every sensor discovers its neighbors and two sensors can form a secure link if they are within the wireless communication range and they have one or more common encryption key in their respective key rings. In [6], authors raise a question that for a secure connected network, what should be the size of key pool and the size of the key-ring of an individual node? When every node have full visibility and mutually independent source link allocation. Here full visibility means that two node can communicate with each other, irrespective of their geographical position in the operational area. It might be a case when the communication range of sense deployed in the area is sufficiently lager than the operational area where they deployed. Few extensions of the above described model has found in [3]. In [3], authors present three new schemes for key establishment using the framework of pre-distribution of a random key set among the node of network. The Schemes suggested in q-composite keys scheme, multipath-reinforcement scheme and International Journal of Machine Learning and Computing, Vol. 1, No. 3, August 2011 247