Private Set Operations from Multi-Query Reverse Private Membership Test
Private set operations allow two parties to perform secure computation on their private sets, including intersection, union and functions of intersection/union. In this paper, we put forth a framework to perform private set operations. The technical core of our framework is the multi-query reverse private membership test (mqRPMT) protocol (Zhang et al., USENIX Security 2023). We present two constructions of mqRPMT from newly introduced cryptographic notions, one is based on commutative weak pseudorandom function (cwPRF), and the other is based on permuted oblivious pseudorandom function (pOPRF). Both cwPRF and pOPRF can be realized from the decisional Diffie-Hellman (DDH)-like assumptions in the random oracle model. We demonstrate the practicality of our framework with implementations. By plugging our cwPRF-based mqRPMT into the framework, we obtain various PSO protocols that are superior or competitive to the state-of-the-art protocols. For intersection functionality, our protocol is faster than the most efficient one for small sets. For cardinality functionality, our protocol achieves a $2.4-10.5\times$ speedup and a $10.9-14.8\times$ reduction in communication cost. For cardinality-with-sum functionality, our protocol achieves a $28.5-76.3\times$ speedup and $7.4\times$ reduction in communication cost. For union functionality, our protocol is the first one that achieves strictly linear complexity, and requires the lowest concrete computation and communication costs in all settings, achieving a $2.7-17\times$ speedup and about $2\times$ reduction in communication cost. Furthermore, our improvement on PSU also translates to related functionality, yielding the most efficient private-ID protocol to date.
The Relationship Between Idealized Models Under Computationally Bounded Adversaries
The random oracle, generic group, and generic bilinear map models (ROM, GGM, GBM, respectively) are fundamental heuristics used to justify new computational assumptions and prove the security of efficient cryptosystems. While known to be invalid in some contrived settings, the heuristics generally seem reasonable for real-world applications. In this work, we ask: which heuristics are closer to reality? Or conversely, which heuristics are a larger leap? We answer this question through the framework of computational indifferentiability, showing that the ROM is a strictly \milder" heuristic than the GGM, which in turn is strictly milder than the GBM. While this may seem like the expected outcome, we explain why it does not follow from prior works, and is not the a priori obvious conclusion. In order to prove our results, we develop new ideas for proving computational indifferentiable separations.
An Analysis of the Algebraic Group Model 📺
The algebraic group model (AGM), formalized by Fuchsbauer, Kiltz, and Loss, has recently received significant attention. One of the appealing properties of the AGM is that it is viewed as being (strictly) weaker than the generic group model (GGM), in the sense that hardness results for algebraic algorithms imply hardness results for generic algorithms, and generic reductions in the AGM (namely, between the algebraic formulations of two problems) imply generic reductions in the~GGM. We highlight that as the GGM and AGM are currently formalized, this is not true: hardness in the AGM may not imply hardness in the GGM, and a generic reduction in the AGM may not imply a similar reduction in the~GGM.
Indifferentiability for Public Key Cryptosystems 📺
We initiate the study of indifferentiability for public key encryption and other public key primitives. Our main results are definitions and constructions of public key cryptosystems that are indifferentiable from ideal cryptosystems, in the random oracle model. Cryptosystems include: 1) Public key encryption; 2) Digital signatures; 3) Non-interactive key agreement. Our schemes are based on relatively standard public key assumptions. By being indifferentiable from an ideal object, our schemes automatically satisfy a wide range of security properties, including any property representable as a single-stage game, and can be composed to operate in higher-level protocols.
Impossibility of Order-Revealing Encryption in Idealized Models
An Order-Revealing Encryption (ORE) scheme gives a public procedure by which two ciphertexts can be compared to reveal the order of their underlying plaintexts. The ideal security notion for ORE is that only the order is revealed—anything else, such as the distance between plaintexts, is hidden. The only known constructions of ORE achieving such ideal security are based on cryptographic multilinear maps and are currently too impractical for real-world applications.In this work, we give evidence that building ORE from weaker tools may be hard. Indeed, we show black-box separations between ORE and most symmetric-key primitives, as well as public key encryption and anything else implied by generic groups in a black-box way. Thus, any construction of ORE must either (1) achieve weaker notions of security, (2) be based on more complicated cryptographic tools, or (3) require non-black-box techniques. This suggests that any ORE achieving ideal security will likely be somewhat inefficient.Central to our proof is a proof of impossibility for something we call information theoretic ORE, which has connections to tournament graphs and a theorem by Erdös. This impossibility proof will be useful for proving other black box separations for ORE.
A Ciphertext-Size Lower Bound for Order-Preserving Encryption with Limited Leakage
We consider a security definition of Chenette, Lewi, Weis, and Wu for order-revealing encryption (ORE) and order-preserving encryption (OPE) (FSE 2016). Their definition says that the comparison of two ciphertexts should only leak the index of the most significant bit on which they differ. While their work could achieve order-revealing encryption with short ciphertexts that expand the plaintext by a factor $$\approx 1.58$$, it could only find order-preserving encryption with longer ciphertexts that expanded the plaintext by a security-parameter factor. We give evidence that this gap between ORE and OPE is inherent, by proving that any OPE meeting the information-theoretic version of their security definition (for instance, in the random oracle model) must have ciphertext length close to that of their constructions. We extend our result to identify an abstract security property of any OPE that will result in the same lower bound.
Parameter-Hiding Order Revealing Encryption
Order-revealing encryption (ORE) is a primitive for outsourcing encrypted databases which allows for efficiently performing range queries over encrypted data. Unfortunately, a series of works, starting with Naveed et al. (CCS 2015), have shown that when the adversary has a good estimate of the distribution of the data, ORE provides little protection. In this work, we consider the case that the database entries are drawn identically and independently from a distribution of known shape, but for which the mean and variance are not (and thus the attacks of Naveed et al. do not apply). We define a new notion of security for ORE, called parameter-hiding ORE, which maintains the secrecy of these parameters. We give a construction of ORE satisfying our new definition from bilinear maps.