Computational grids are believed to be the ultimate framework to meet the growing computational needs of the scientific community. Here, the processing power of geographically distributed resources working under different ownerships, having their own access policy, cost structure and the likes, is logically coupled to make them perform as a unified resource. The continuous increase of availability of high-bandwidth communication as well as powerful computers built of low-cost components further enhance chances of computational grids becoming a reality. However, the question of grid security remains one of the important open research issues. Here, we present some novel ideas about how to implement grid security, without appreciable performance degradation in grids. A suitable alternative to the computationally expensive encryption is suggested, which uses a key for message authentication. Methods of secure transfer and exchange of the required key(s) are also discussed.