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Round-Optimal Oblivious Transfer and MPC from Computational CSIDH

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Public-Key Cryptography – PKC 2023 (PKC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13940))

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Abstract

We present the first round-optimal and plausibly quantum-safe oblivious transfer (OT) and multi-party computation (MPC) protocols from the computational CSIDH assumption – the weakest and most widely studied assumption in the CSIDH family of isogeny-based assumptions. We obtain the following results:

  • The first round-optimal maliciously secure OT and MPC protocols in the plain model that achieve (black-box) simulation-based security while relying on the computational CSIDH assumption.

  • The first round-optimal maliciously secure OT and MPC protocols that achieves Universal Composability (UC) security in the presence of a trusted setup (common reference string plus random oracle) while relying on the computational CSIDH assumption.

Prior plausibly quantum-safe isogeny-based OT protocols (with/without setup assumptions) are either not round-optimal, or rely on potentially stronger assumptions.

We also build a 3-round maliciously-secure OT extension protocol where each base OT protocol requires only 4 isogeny computations. In comparison, the most efficient isogeny-based OT extension protocol till date due to Lai et al. [Eurocrypt 2021] requires 12 isogeny computations and 4 rounds of communication, while relying on the same assumption as our construction, namely the reciprocal CSIDH assumption.

S. Patranabis—Part of the work was done while the author was at VISA Research USA.

P. Sarkar—Supported by NSF Awards 1931714, 1414119, and the DARPA SIEVE program.

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Notes

  1. 1.

    The setup string is structured and it is sampled from a given distribution.

  2. 2.

    The random oracles in our protocol are local to each session.

  3. 3.

    We note that while prior works on OT from isogenies do not explicitly construct OT extension protocols, they do yield base OT protocols that can be converted in a generic manner into full-fledged OT extension protocols.

  4. 4.

    For standard two-round OT protocols, the setup algorithm need not output a trapdoor \(\textsf {td}\), but we include it for certain security properties described subsequently.

  5. 5.

    The recent work of [BDK+22] constructs a similar NIZK. But it is based on the decisional CSIDH assumption, and is hence insufficient for our purpose.

  6. 6.

    The verifier sends \((x_0, x_1)\) as the first round message by sampling \(g_0, g_1 \leftarrow _R\textit{G}\) and computing \(x_0 = g_0 \star x, x_1 = g_1 \star x\). The committer commits to bit b by sampling g and computing the commitment as \(z=g \star x_b\). The decommitment is (g, b). Bit b remains perfectly hidden. Binding follows from wU-EGA assumption since openings \((s_0, 0)\) and \((s_1, 1)\) for bits 0 and 1 help to find \(r=s_0\cdot s_1^{-1}\) such that \(x_1=r\star x_0\).

  7. 7.

    This was pointed out by the authors of [LGdSG21] in their Eurocrypt 2021 presentation.

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Acknowledgments

We thank the anonymous reviewers of IACR PKC 2023 for their helpful comments and suggestions. Pratik Sarkar is supported by NSF Awards 1931714, 1414119, and the DARPA SIEVE program.

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Authors and Affiliations

  1. LinkedIn, Seattle, USA

    Saikrishna Badrinarayanan

  2. Meta, Menlo Park, USA

    Daniel Masny

  3. Supra, Kolkata, India

    Pratyay Mukherjee

  4. IBM Research India, Bengaluru, India

    Sikhar Patranabis

  5. VISA Research, Palo Alto, USA

    Srinivasan Raghuraman

  6. Boston University, Boston, USA

    Pratik Sarkar

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  1. Saikrishna Badrinarayanan
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    Alexandra Boldyreva

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Vladimir Kolesnikov

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Badrinarayanan, S., Masny, D., Mukherjee, P., Patranabis, S., Raghuraman, S., Sarkar, P. (2023). Round-Optimal Oblivious Transfer and MPC from Computational CSIDH. In: Boldyreva, A., Kolesnikov, V. (eds) Public-Key Cryptography – PKC 2023. PKC 2023. Lecture Notes in Computer Science, vol 13940. Springer, Cham. https://doi.org/10.1007/978-3-031-31368-4_14

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