HOPE: improved order preserving encryption with the power to homomorphic operations of ciphertexts

Abstract

Database applications that manage and utilize massive data must address the issues of {element comparisons}, the core operations in index accessing, {metric computations} and {metric comparisons}, and the core operations in result ranking. In the cloud era, to avoid private information leakage, encrypted data are subcontracted, and resolutions for the problems that arise in three operations over ciphertexts are urgently required. Indeed, it is possible to handle element comparison through {order preserving encryption/encoding} (OPE) or metric computation through {homomorphic encryption} (HE) directly over ciphertexts. Unfortunately, the simultaneous achievement of both goals (i.e., metric computation and comparison) by directly combining OPE and HE remains intractable. In this work, an improved OPE, named hOPE, is proposed to support homomorphic operations over ciphertexts in addition to comparisons. Based on hOPE, AhOPE and PhOPE are designed to support homomorphic addition and product, respectively. Both schemes are proved to be indistinguishable under operated and ordered chosen-plaintext attack (IND-O\textsuperscript{2}CPA) secure when the adopted HE algorithm provides indistinguishable under ordered chosen-plaintext attack (IND-OCPA secure). hOPE is a general construction that supports arbitrary HE algorithms and achieves consistent security. We deploy AhOPE and PhOPE in practice with a trusted/untrusted third party and compare the result with the state-of-the-art methods. The results show that our presented algorithms need few interactions and fill the gap between OPE and HE.