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Summary form only given. Emerging computing paradigms such as database service outsourcing and utility computing (a.k.a. cloud computing) offer attractive financial and technological advantages. These are drawing interests of enterprises in migrating their computing operations, including DBMS's, to service providers. Nevertheless, many vocal consultants, including Gartner, have issued warnings on the security threats in the cloud computing model. Private information, which includes both customer data and business information, should not be revealed to unauthorized parties. In this work, we address a very important problem of security in services outsourcing: the elements of an encryption scheme and the execution protocol for encrypted query processing. More specifically, we study how sensitive data and queries should be transformed in an encrypted database environment and how a service provider processes encrypted queries on an encrypted database without the plain data revealed. We call our model of secure query processing SCONEDB (for Secure Computation ON an Encrypted DataBase). The conventional way to deal with security threats is to apply encryption on the plain data and to allow only authorized parties to perform decryption. Unauthorized parties, including the service provider, should not be able to recover the plain data even if they can access the encrypted database. Some previous works have studied this encryption problem in the outsourced database (ODB) model. However, these studies are restricted to simple SQL operations, e.g., exact match of attribute value in point query; comparisons between numeric values in range query. In practice, users often interact with a database via applications in which queries are not easily expressible in SQL. Moreover, most of the previous methods were specially engineered to work against one specific attack model. However, the problem should be studied with respect to various security requirements, considering different at- - tacker capabilities. In this work we focus on k-nearest neighbor (kNN) queries and show how various encryption schemes are designed to support secure kNN query processing under different attacker capabilities.