Coded caching schemes on broadcast networks with user caches help to offload traffic from peak times to off-peak times by prefetching information from the server to the users during off-peak times and thus serving the users more efficiently during peak times using coded transmissions. We consider the problem of secretive coded caching which was proposed recently, in which a user should not be able to decode any information about any file that the user has not demanded. We propose a new secretive coded caching scheme which has a lower average rate compared to the existing state-of-the-art scheme, for the same memory available at the users. The proposed scheme is based on exploiting the presence of common demands between multiple users.