1. A method executed in a computer system that minimizes eavesdropping by a service provider (SP) on a query graph Q to a database from a client computer or graph data G in the database from a data owner in a structure-preserving subgraph query processing (SPQP), the method comprising: encrypting the graph data G, by the data owner, and delivering the encrypted graph data in the database to the SP that is accessible to the client computer with the query graph Q that execute through the SP; minimizing the eavesdropping on the query graph Q to the database from the client computer or the graph data G from the data owner by: delivering, by the data owner, private keys to the client computer for encryption of the query graph Q and decryption of an encrypted result; encrypting the query graph Q, by the client computer, using the private keys, and submitting the encrypted query to SP, wherein the encrypting of the query graph Q is based on a cyclic group based private key encryption and comprises: enumerating all possible subgraph isomorphism mappings M i s from the query graph Q to the data graph G by matching for creating a structure-preserving structure; verifying if the mapping M i is valid or not; reducing search space of the M i s by degree constraints and neighborhood constraints; performing query evaluation between the query graph Q and the graph data G using adjacency matrices; and using the cyclic group based private key encryption to encrypt M Q and M G as encrypted matrices of query M Qk and encrypted matrices of graph M G k respectively for verifying the validity of M i ; checking the validity of each mapping M i by the SP under the cyclic group based private key encryption; aggregating computational results under the cyclic group based private key encryption by the SP for reducing communication overheads between the client computer and the SP as the query graph Q has a bounded size; and determining, by the SP, subgraph structure with a number of possible mappings reduced.