Order-preserving encryption (OPE) is an encryption scheme with the property that the ordering of the plaintexts carry over to the ciphertexts. This primitive is particularly useful in the setting of encrypted databases because it enables efficient range queries over encrypted data. Given its practicality and usefulness in the design of databases on encrypted data, OPE's popularity is growing. Unfortunately, nearly all computationally efficient OPE constructions are vulnerable against ciphertext frequency-leakage, which allows for inferring the underlying plaintext frequency. To overcome this weakness, Kerschbaum recently proposed a security model, designed a frequency-hiding OPE scheme, and analyzed its security in the programmable random oracle model (CCS 2015). In this work, we demonstrate that Kerschbaum's definition is imprecise and using its natural interpretation, we describe an attack against his scheme. We generalize our attack and show that his definition is, in fact, not satisfiable. The basic idea of our impossibility result is to show that any scheme satisfying his security notion is also IND-CPA-secure, which contradicts the very nature of OPE. As a consequence, no such scheme can exist. To complete the picture, we rule out the imprecision in the security definition and show that a slight adaption of Kerschbaum's tree-based scheme fulfills it.
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