We initiate the cryptographic study of order-preserving symmetric encryption (OPE), a primitive suggested in the database community by Agrawal et al. (SIGMOD '04) for allowing efficient range queries on encrypted data. Interestingly, we first show that a straightforward relaxation of standard security notions for encryption such as indistinguishability against chosen-plaintext attack (IND-CPA) is unachievable by a practical OPE scheme. Instead, we propose a security notion in the spirit of pseudorandom functions (PRFs) and related primitives asking that an OPE scheme look "as-random-as-possible" subject to the order-preserving constraint. We then design an efficient OPE scheme and prove its security under our notion based on pseudorandomness of an underlying blockcipher. Our construction is based on a natural relation we uncover between a random order-preserving function and the hypergeometric probability distribution. In particular, it makes black-box use of an efficient sampling algorithm for the latter.

Abstract.We further the study of order-preserving symmetric encryption (OPE), a primitive for allowing efficient range queries on encrypted data, recently initiated (from a cryptographic perspective) by Boldyreva et al. (Eurocrypt '09). First, we address the open problem of characterizing what encryption via a random order-preserving function (ROPF) leaks about underlying data (ROPF being the "ideal object" in the security definition, POPF, satisfied by their scheme.) In particular, we show that, for a database of randomly distributed plaintexts and appropriate choice of parameters, ROPF encryption leaks neither the precise value of any plaintext nor the precise distance between any two of them. The analysis here is quite technically non-trivial and introduces useful new techniques. On the other hand, we also show that ROPF encryption does leak both the value of any plaintext as well as the distance between any two plaintexts to within a range of possibilities roughly the square root of the domain size. We then study schemes that are not order-preserving, but which nevertheless allow efficient range queries and achieve security notions stronger than POPF. In a setting where the entire database is known in advance of key-generation (considered in several prior works), we show that recent constructions of "monotone minimal perfect hash functions" allow to efficiently achieve (an adaptation of) the notion of IND-O(rdered) CPA also considered by Boldyreva et al., which asks that only the order relations among the plaintexts is leaked. Finally, we introduce modular order-preserving encryption (MOPE), in which the scheme of Boldyreva et al. is prepended with a shift cipher. MOPE improves the security of OPE in a sense, as it does not leak any information about plaintext location. We clarify that our work should not be interpreted as saying the original scheme of Boldyreva et al., or the variants that we introduce, are "secure" or "insecure." Rather, the goal of this line of research is to help practitioners decide whether the options provide a suitable security-functionality tradeoff for a given application.

In an order-preserving encryption scheme, the encryption algorithm produces ciphertexts that preserve the order of their plaintexts. Order-preserving encryption schemes have been studied intensely in the last decade, and yet not much is known about the security of these schemes. Very recently, Boneh et al. (Eurocrypt 2015) introduced a generalization of order-preserving encryption, called order-revealing encryption, and presented a construction which achieves this notion with best-possible security. Because their construction relies on multilinear maps, it is too impractical for most applications and therefore remains a theoretical result.In this work, we build efficiently implementable order-revealing encryption from pseudorandom functions. We present the first efficient order-revealing encryption scheme which achieves a simulation-based security notion with respect to a leakage function that precisely quantifies what is leaked by the scheme. In fact, ciphertexts in our scheme are only about 1.6 times longer than their plaintexts. Moreover, we show how composing our construction with existing order-preserving encryption schemes results in order-revealing encryption that is strictly more secure than all preceding order-preserving encryption schemes.

Order-preserving encryption (OPE) schemes, whose ciphertexts preserve the natural ordering of the plaintexts, allow efficient range query processing over outsourced encrypted databases without giving the server access to the decryption key. Such schemes have recently received increased interest in both the database and the cryptographic communities. In particular, modular order-preserving encryption (MOPE), due to Boldyreva et al. [8], is a promising extension that increases the security of the basic OPE by introducing a secret modular offset to each data value prior to encrypting it. However, executing range queries via MOPE in a naïve way allows the adversary to learn this offset, negating any potential security gains of this approach.In this paper, we systematically address this vulnerability and show that MOPE can be used to build a practical system for executing range queries on encrypted data while providing a significant security improvement over the basic OPE. We introduce two new query execution algorithms for MOPE: our first algorithm is efficient if the user's query distribution is well-spread, while the second scheme is efficient even for skewed query distributions. Interestingly, our second algorithm achieves this efficiency by leaking the leastimportant bits of the data, whereas OPE is known to leak the most-important bits of the data. We also show that our algorithms can be extended to the case where the query distribution is adaptively learned online. We present new, appropriate security models for MOPE and use them to rigorously analyze the security of our proposed schemes. Finally, we design a system prototype that integrates our schemes on top of an existing database system and apply query optimization methods to execute SQL queries with range predicates efficiently. We provide a performance evaluation of our prototype under a number of different database and query distributions, using both synthetic and real datasets.

We exhibit an explicit list of nine graphs such that a graph drawn in the Klein bottle is 5-colorable if and only if it has no subgraph isomorphic to a member of the list.