Security in wireless sensor networks (WSNs) is an upcoming research field which is quite different from traditional network security mechanisms. Many applications are dependent on the secure operation of a WSN, and have serious effects if the network is disrupted. Therefore, it is necessary to protect communication between sensor nodes. Key management plays an essential role in achieving security in WSNs. To achieve security, various key predistribution schemes have been proposed in the literature. A secure key management technique in WSN is a real challenging task. In this paper, a novel approach to the above problem by making use of elliptic curve cryptography (ECC) is presented. In the proposed scheme, a seed key, which is a distinct point in an elliptic curve, is assigned to each sensor node prior to its deployment. The private key ring for each sensor node is generated using the point doubling mathematical operation over the seed key. When two nodes share a common private key, then a link is established between these two nodes. By suitably choosing the value of the prime field and key ring size, the probability of two nodes sharing the same private key could be increased. The performance is evaluated in terms of connectivity and resilience against node capture. The results show that the performance is better for the proposed scheme with ECC compared to the other basic schemes.