Due to the sensitivity of the Wireless Sensor Networks (WSN) applications and resource constraints, authentication and key management emerge as a challenging issue for WSN. In general, various approaches have been developed for the key management in WSN. This paper has come up with a new robust key pre-distribution scheme using random polynomial functions and matrix. This new proposed scheme significantly increases the storage efficiency and provides resilience to network against node capture by using random prime numbers, polynomial functions and matrix properties. The effectiveness of the scheme is demonstrated through a security analysis and comparison with the existing schemes.