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Oblivious Transfer with Constant Computational Overhead

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Advances in Cryptology – EUROCRYPT 2023 (EUROCRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14004))

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Abstract

The computational overhead of a cryptographic task is the asymptotic ratio between the computational cost of securely realizing the task and that of realizing the task with no security at all.

Ishai, Kushilevitz, Ostrovsky, and Sahai (STOC 2008) showed that secure two-party computation of Boolean circuits can be realized with constant computational overhead, independent of the desired level of security, assuming the existence of an oblivious transfer (OT) protocol and a local pseudorandom generator (PRG). However, this only applies to the case of semi-honest parties. A central open question in the area is the possibility of a similar result for malicious parties. This question is open even for the simpler task of securely realizing many instances of a constant-size function, such as OT of bits.

We settle the question in the affirmative for the case of OT, assuming: (1) a standard OT protocol, (2) a slightly stronger “correlation-robust" variant of a local PRG, and (3) a standard sparse variant of the Learning Parity with Noise (LPN) assumption. An optimized version of our construction requires fewer than 100 bit operations per party per bit-OT. For 128-bit security, this improves over the best previous protocols by 1–2 orders of magnitude.

We achieve this by constructing a constant-overhead pseudorandom correlation generator (PCG) for the bit-OT correlation. Such a PCG generates N pseudorandom instances of bit-OT by locally expanding short, correlated seeds. As a result, we get an end-to-end protocol for generating N pseudorandom instances of bit-OT with o(N) communication, O(N) computation, and security that scales sub-exponentially with N.

Finally, we present applications of our main result to realizing other secure computation tasks with constant computational overhead. These include protocols for general circuits with a relaxed notion of security against malicious parties, protocols for realizing N instances of natural constant-size functions, and reducing the main open question to a potentially simpler question about fault-tolerant computation.

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Notes

  1. 1.

    Here the default security requirement is that any \(\textsf{poly} (N)\)-time adversary can only obtain a \({{\,\mathrm{\textsf{negl}}\,}}(N)\) advantage. Alternatively, using a separate security parameter \(\lambda \), the cN bound holds when N is sufficiently (but polynomially) larger than \(\lambda \).

  2. 2.

    See Sect. 2.1 for more details on our specific cost model. Briefly, functions and protocols are implemented as bounded fan-in Boolean circuits, and the computational cost is the number of gates. For concrete computational costs, we allow any bit operation over two-bit inputs.

  3. 3.

    In this work, OT refers by default to bit-OT, namely oblivious transfer of pairs of bits. However, as discussed below (cf. Sect. 5), our results apply to most other natural flavors of OT.

  4. 4.

    See the full version for an explicit attack.

  5. 5.

    Namely, we require \(\{P_j(\varDelta _j\oplus \pi _j(x))\}_{j\in N}\) is indistinguishable from random, where \(\varDelta _1,\dots ,\varDelta _N\) are pseudorandom with seed known to the adversary.

  6. 6.

    This should be contrasted with a more fine-grained measure of overhead considered in [17, 22, 36], which requires exponential security in \(\lambda \) (rather than super-polynomial), measures the overhead with respect to \(N+\lambda \), and requires the overhead to apply to all choices of N and \(\lambda \) (e.g., even when \(N=\lambda \)).

  7. 7.

    We refer the interested reader to this work for more details.

  8. 8.

    Instead of RA codes we could have used a code of Tillich and Zémor [68]; however the effect on the computational complexity is essentially nil.

  9. 9.

    In fact, the question is open even in the simpler special case of zero-knowledge functionalities. A solution for this special case would imply a solution for the general case by applying the GMW compiler [48] to a constant-overhead protocol with semi-honest security.

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Acknowledgements

E. Boyle supported by AFOSR Award FA9550-21-1-0046, ERC Project HSS (852952), and a Google Research Award. G. Couteau supported by the ANR SCENE. N. Gilboa supported by ISF grant 2951/20, ERC grant 876110, and a grant by the BGU Cyber Center. Y. Ishai supported by ERC Project NTSC (742754), BSF grant 2018393, and ISF grant 2774/20. L. Kohl is funded by NWO Gravitation project QSC. N. Resch supported in part by ERC H2020 grant No.74079 (ALGSTRONGCRYPTO). P. Scholl is supported by the Danish Independent Research Council under project number 0165-00107B (C3PO) and an Aarhus University Research Foundation starting grant.

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Authors and Affiliations

  1. IDC Herzliya and NTT Research, Sunnyvale, USA

    Elette Boyle

  2. IRIF, Paris, France

    Geoffroy Couteau

  3. Ben-Gurion University, Beersheba, Israel

    Niv Gilboa

  4. Technion, Haifa, Israel

    Yuval Ishai

  5. Cryptology Group, CWI Amsterdam, Amsterdam, The Netherlands

    Lisa Kohl

  6. University of Amsterdam, Amsterdam, The Netherlands

    Nicolas Resch

  7. Aarhus University, Aarhus, Denmark

    Peter Scholl

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  1. Elette Boyle
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Correspondence to Nicolas Resch .

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  1. Bar-Ilan University, Ramat Gan, Israel

    Carmit Hazay

  2. Simula UiB, Bergen, Norway

    Martijn Stam

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Boyle, E. et al. (2023). Oblivious Transfer with Constant Computational Overhead. In: Hazay, C., Stam, M. (eds) Advances in Cryptology – EUROCRYPT 2023. EUROCRYPT 2023. Lecture Notes in Computer Science, vol 14004. Springer, Cham. https://doi.org/10.1007/978-3-031-30545-0_10

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