Abstract
Consider a server with a large set S of strings \(\{x_1,x_2\ldots ,x_N\}\) that would like to publish a small hash h of its set S such that any client with a string y can send the server a short message allowing it to learn y if \(y \in S\) and nothing otherwise. In this work, we study this problem of two-round private set intersection (PSI) with low (asymptotically optimal) communication cost, or what we call laconic private set intersection (\(\ell \)PSI) and its extensions. This problem is inspired by the recent general frameworks for laconic cryptography [Cho et al. CRYPTO 2017, Quach et al. FOCS’18].
We start by showing the first feasibility result for realizing \(\ell \)PSI based on the CDH assumption, or LWE with polynomial noise-to-modulus ratio. However, these feasibility results use expensive non-black-box cryptographic techniques leading to significant inefficiency. Next, with the goal of avoiding these inefficient techniques, we give a construction of \(\ell \)PSI schemes making only black-box use of cryptographic functions. Our construction is secure against semi-honest receivers, malicious senders and reusable in the sense that the receiver’s message can be reused across any number of executions of the protocol. The scheme is secure under the \(\phi \)-hiding, decisional composite residuosity and subgroup decision assumptions.
Finally, we show natural applications of \(\ell \)PSI to realizing a semantically-secure encryption scheme that supports detection of encrypted messages belonging to a set of “illegal” messages (e.g., an illegal video) circulating online. Over the past few years, significant effort has gone into realizing laconic cryptographic protocols. Nonetheless, our work provides the first black-box constructions of such protocols for a natural application setting.
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Notes
- 1.
Note that in the laconic setting we cannot prove malicious security against a receiver since it is information-theoretically impossible to extract its input. Thus, since the NIZK will only be computed by the sender, the protocol will remain laconic.
- 2.
We use the word reusability only in conjunction with malicious security, since in the semi-honest setting, reusability is satisfied by default.
- 3.
DV-NIZK only allows the designated prover to prove that it holds a witness for a certain \(\mathsf {NP}\) statement to a verifier in just one message.
- 4.
- 5.
\(\mathsf {Enc}\) also takes as input a public parameter \(\mathsf {pp}\), which we ignore here.
- 6.
\(\widetilde{\mathsf {C}}_0\) stands for the garbled circuit and \(\{\mathsf {lb}_{i,b}\}_i\) are the corresponding labels of inputs.
- 7.
We will not further discuss the small correctness-error of this protocol as our final protocol will not suffer from this defect.
- 8.
We remark that we use a PRF, not because we want uniform outputs, but to implicitly define the set of primes. A similar trick was used in [6].
- 9.
Note that \(\mathbb {NR}_N\) is not a cyclic group and we only assume this here for simplicity. Actually, if we choose N as a product of two safe primes, then we could find a cyclic subgroup \(\mathbb {J}_N\) which is the group of elements with Jacobi symbol 1, and its subgroup \(\mathbb {T}_N\) composing of \(N^\xi \)-th powers of \(\mathbb {J}_N\) has order \(\phi (N)/2\). Namely, just replace the group pair \((\mathbb {Z}^*_{N^{\xi +1}},\mathbb {NR}_N)\) with \((\mathbb {J}_N, \mathbb {T}_N)\) to fix this issue. Please refer to Sect. 3.1 and Sect. 6 for details.
- 10.
Via a standard rerandomization argument we can show that reusing the same random coins across different keys does not harm CPA security.
- 11.
Same as above.
- 12.
We assume that the verifier rejects if it fails to compute the discrete logarithm of \(k^{-1}\prod d_i^{\sigma _i}\).
- 13.
We note that the CDH construction of [8] satisfies a weaker notion of anonymity, in which only some part of the ciphertext is pseudorandom. But for ease of presentation we keep the notion as is, and remark that our \(\ell \)PSI construction works also with respect to that weaker notion.
- 14.
Here, we present a slightly different variant of the scheme in [15].
- 15.
We remark that we use a PPRF, not because we want uniform outputs, but to implicitly define the set of primes. A similar trick was used in [6].
- 16.
We proceed with an independent PKE scheme for the sake of simplicity.
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Acknowledgment
Pedro Branco thanks the support from DP-PMI and FCT (Portugal) through the grant PD/BD/135181/2017. This work is supported by Security and Quantum Information Group of Instituto de Telecomunicações, by the Fundação para a Ciência e a Tecnologia (FCT) through national funds, by FEDER, COMPETE 2020, and by Regional Operational Program of Lisbon, under UIDB/50008/2020.
Nico Döttling: This work is partially funded by the Helmholtz Association within the project “Trustworthy Federated Data Analytics” (TFDA) (funding number ZT-I-OO1 4).
Sanjam Garg is supported in part by DARPA under Agreement No. HR00112020026, AFOSR Award FA9550-19-1-0200, NSF CNS Award 1936826, and research grants by the Sloan Foundation, Visa Inc., and Center for Long-Term Cybersecurity (CLTC, UC Berkeley). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government or DARPA.
Mohammad Hajiabadi is supported in part by NSF CNS Award 2055564.
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Alamati, N., Branco, P., Döttling, N., Garg, S., Hajiabadi, M., Pu, S. (2021). Laconic Private Set Intersection and Applications. In: Nissim, K., Waters, B. (eds) Theory of Cryptography. TCC 2021. Lecture Notes in Computer Science(), vol 13044. Springer, Cham. https://doi.org/10.1007/978-3-030-90456-2_4
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