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Multiparty Reusable Non-interactive Secure Computation from LWE

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Advances in Cryptology – EUROCRYPT 2021 (EUROCRYPT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12697))

Abstract

Motivated by the goal of designing versatile and flexible secure computation protocols that at the same time require as little interaction as possible, we present new multiparty reusable Non-Interactive Secure Computation (mrNISC) protocols. This notion, recently introduced by Benhamouda and Lin (TCC 2020), is essentially two-round Multi-Party Computation (MPC) protocols where the first round of messages serves as a reusable commitment to the private inputs of participating parties. Using these commitments, any subset of parties can later compute any function of their choice on their respective inputs by just sending a single message to a stateless evaluator, conveying the result of the computation but nothing else. Importantly, the input commitments can be computed without knowing anything about other participating parties (neither their identities nor their number) and they are reusable across any number of desired computations.

We give a construction of mrNISC that achieves standard simulation security, as classical multi-round MPC protocols achieve. Our construction relies on the Learning With Errors (LWE) assumption with polynomial modulus, and on the existence of a pseudorandom function (PRF) in \(\mathsf {NC}^1\). We achieve semi-malicious security in the plain model and malicious security by further relying on trusted setup (which is unavoidable for mrNISC). In comparison, the only previously known constructions of mrNISC were either using bilinear maps or using strong primitives such as program obfuscation.

We use our mrNISC to obtain new Multi-Key FHE (MKFHE) schemes with threshold decryption:

  • In the CRS model, we obtain threshold MKFHE for \(\mathrm {NC}^1 \) based on LWE with only polynomial modulus and PRFs in \(\mathsf {NC}^1\), whereas all previous constructions rely on LWE with super-polynomial modulus-to-noise ratio.

  • In the plain model, we obtain threshold levelled MKFHE for \(\mathrm {P} \) based on LsWE with polynomial modulus, PRF in \(\mathrm {NC}^1 \), and NTRU, and another scheme for constant number of parties from LWE with sub-exponential modulus-to-noise ratio. The only known prior construction of threshold MKFHE (Ananth et al., TCC 2020) in the plain model restricts the set of parties who can compute together at the onset.

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Notes

  1. 1.

    The reconstruction of the output is “public” in the sense that it does not require any secrets. It is w.l.o.g. to consider public output reconstruction, as one can always consider the evaluator as a participant of MPC with a dummy input and uses the all zero string as its random tape.

  2. 2.

    Semi-malicious security is a strengthening of the semi-honest security wherein the adversary is allowed to choose its random tape arbitrarily. [10] showed that any protocol satisfying semi-malicious security can be made maliciously secure by additionally using Non-Interactive Zero-Knowledge proofs (NIZKs).

  3. 3.

    The CRS is needed for NIZK which exists from LWE with polynomial modulus [59].

  4. 4.

    DSPR stands for the decision small polynomial ratio assumption [52] which is used to prove the security of the NTRU encryption scheme.

  5. 5.

    It suffices to simulate only these computations that involve at least one honest party. Computations involving only corrupted parties can be viewed as part of the internal computation of the adversary.

  6. 6.

    The magnitude scales exponentially with the depth of \(g_2\), which is relatively small if we set the modulus to be sufficiently large.

  7. 7.

    It can be verified efficiently by checking whether it has small magnitude and \({\mathbf {D}^{(b)}}\boldsymbol{v} = \boldsymbol{u}\).

  8. 8.

    The common definition of a PRF in \(\mathsf {NC}^1\) is a PRF whose circuit representation is in \(\mathsf {NC}^1\) when viewed as a function of both the input and the seed. We actually need a slightly weaker condition, namely, that the circuit computing \(F_{x}(\cdot )=\mathsf {PRF}.\mathsf {Eval}(\cdot ,x)\) with the hardwired input x, as a function of the \(\mathsf {PRF}\) key is in \(\mathsf {NC}^1\).

  9. 9.

    For simplicity, we suppose that the set of parties participating in the computation is \(I = [n]\).

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Acknowledgments

Aayush Jain was supported by a Google PhD fellowship in the area of security and privacy (2018) and in part from DARPA SAFEWARE and SIEVE awards, NTT Research, NSF Frontier Award 1413955, and NSF grant 1619348, BSF grant 2012378, a Xerox Faculty Research Award, a Google Faculty Research Award, an equipment grant from Intel, and an Okawa Foundation Research Grant. This material is based upon work supported by the Defense Advanced Research Projects Agency through Award HR00112020024 and the ARL under Contract W911NF-15-C- 0205.

Ilan Komargodski is supported in part by an Alon Young Faculty Fellowship and by an ISF grant (No. 1774/20).

Huijia Lin was supported by NSF grants CNS-1528178, CNS-1929901, CNS-1936825 (CAREER), CNS-2026774, a Hellman Fellowship, a JP Morgan AI Research Award, a Simons Collaboration grant on the Theory of Algorithmic Fairness, the Defense Advanced Research Projects Agency (DARPA) and Army Research Office (ARO) under Contract No. W911NF-15-C-0236, and a subcontract No. 2017-002 through Galois.

The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense, DARPA, the National Science Foundation, or the U.S. Government.

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Authors and Affiliations

  1. Algorand Foundation, New York, USA

    Fabrice Benhamouda

  2. UCLA, Los Angeles, CA, 90095, USA

    Aayush Jain

  3. NTT Research, Sunnyvale, CA, 94085, USA

    Aayush Jain & Ilan Komargodski

  4. Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Ilan Komargodski

  5. University of Washington, Seattle, WA, 98195, USA

    Huijia Lin

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  1. Inria, Paris, France

    Anne Canteaut

  2. UCLouvain, Louvain-la-Neuve, Belgium

    François-Xavier Standaert

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Benhamouda, F., Jain, A., Komargodski, I., Lin, H. (2021). Multiparty Reusable Non-interactive Secure Computation from LWE. In: Canteaut, A., Standaert, FX. (eds) Advances in Cryptology – EUROCRYPT 2021. EUROCRYPT 2021. Lecture Notes in Computer Science(), vol 12697. Springer, Cham. https://doi.org/10.1007/978-3-030-77886-6_25

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  • DOI: https://doi.org/10.1007/978-3-030-77886-6_25

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-77885-9

  • Online ISBN: 978-3-030-77886-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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