Motivated by applications to maintaining confidentiality and efficiency of encrypted data access in cloud computing, we uncovered an inherent confidentiality weakness in databases outsourced to cloud servers, even when encrypted. To address this weakness, we formulated a new privacy notion for outsourced databases and (variants of) a classical record length optimization problem, whose solutions achieve the new privacy notion. Our algorithmic investigation resulted in a number of exact and approximate algorithms,for arbitrary input distributions, and in the presence of record additions and deletions. Previous work only analyzed an unconstrained variant of our optimization problem for specific input distributions, with no attention to running time or database updates.