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Maintaining the confidentiality of data sent to clouds is a vital issue that must be satisfactorily resolved as a pre-condition in order for the cloud computing paradigm to survive and thrive. In this paper we explore a completely new paradigm to approximately achieve the functionality implied by "Oblivious Data Processing(ODP)". Our strategy is to partition the data as well as the underlying Residue Number System (RNS) in the Residue Domain (RD) and then distribute and process the partitions independently. We have also outlined some new fundamental limitations to ODP in general and identified application scenarios where ODP can be achieved as well as those wherein ODP appears to be doomed to fail in the long run.