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Annual International Conference on the Theory and Applications of Cryptographic Techniques

EUROCRYPT 2022: Advances in Cryptology – EUROCRYPT 2022 pp 303–334Cite as

Asymptotically Quasi-Optimal Cryptography

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13275))

Abstract

The question of minimizing the computational overhead of cryptography was put forward by the work of Ishai, Kushilevitz, Ostrovsky and Sahai (STOC 2008). The main conclusion was that, under plausible assumptions, most cryptographic primitives can be realized with constant computational overhead. However, this ignores an additive term that may depend polynomially on the (concrete) computational security parameter \(\lambda \). In this work, we study the question of obtaining optimal efficiency, up to polylogarithmic factors, for all choices of n and \(\lambda \), where n is the size of the given task. In particular, when \(n=\lambda \), we would like the computational cost to be only \(\tilde{O}(\lambda )\). We refer to this goal as asymptotically quasi-optimal (AQO) cryptography.

We start by realizing the first AQO semi-honest batch oblivious linear evaluation (BOLE) protocol. Our protocol applies to OLE over small fields and relies on the near-exponential security of the ring learning with errors (RLWE) assumption. Building on the above and on known constructions of AQO PCPs, we design the first AQO zero-knowledge (ZK) argument system for Boolean circuit satisfiability. Our construction combines a new AQO ZK-PCP construction that respects the AQO property of the underlying PCP along with a technique for converting statistical secrecy into soundness via OLE reversal. Finally, combining the above results, we get AQO secure computation protocols for Boolean circuits with security against malicious parties under RLWE.

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Notes

  1. 1.

    Throughout this paper, the security parameter \(\lambda \) refers to bits of concrete security, requiring that no adversary of circuit size \(2^{\lambda }\) can gain better than \(2^{-\lambda }\) advantage. This is a natural and robust notion of concrete security. An alternative notion that settles for negligible advantage is not as robust, analogously to relaxing standard security definitions by requiring that every polynomial-time adversary has o(1) advantage (rather than negligible in the sense of sub-polynomial).

  2. 2.

    We remark that the statements we want are proofs (of knowledge) of a short secret s such that \(As=t\) over a ring. On the other hand, the second type of protocols prove that there is a short secret s such that As equals a short multiple of t.

  3. 3.

    Recall that Batch-OT/OLE refers to multiple OT/OLE instances carried out in parallel.

  4. 4.

    We denote the Ring-LWE dimension by k, and the OLE batch-size by n.

  5. 5.

    We typically pick q to be a multiple of p so the rounding is not necessary.

  6. 6.

    The product of the first n primes is \(e^{O(n{\mathsf {log}}n)}\).

  7. 7.

    For example, in the classic MPC-in-the-head based ZKPCP [IKOS07], the verifier queries a random t subset out of n. Here Q contains all t subsets of [n].

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Acknowledgements

We thank Henry Corrigan-Gibbs for helpful comments and Hemanta Maji for answering our questions on [BGMN18]. C. Hazay was supported by the BIU Center for Research in Applied Cryptography and Cyber Security in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office, and by ISF grant No. 1316/18. Y. Ishai was supported in part by ERC Project NTSC (742754), BSF grant 2018393, and ISF grant 2774/20. L. de Castro and V. Vaikuntanathan were supported by grants from MIT-IBM Watson AI Labs and Analog Devices, by a Microsoft Trustworthy AI grant, and by DARPA under Agreement No. HR00112020023.

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Authors and Affiliations

  1. MIT, Cambridge, UK

    Leo de Castro & Vinod Vaikuntanathan

  2. Bar-Ilan University, Ramat Gan, Israel

    Carmit Hazay

  3. Technion, Haifa, Israel

    Yuval Ishai

  4. Georgetown University, Washington, USA

    Muthu Venkitasubramaniam

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  1. Leo de Castro
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  2. Carmit Hazay
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  4. Vinod Vaikuntanathan
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  5. Muthu Venkitasubramaniam
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Correspondence to Leo de Castro .

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  1. University of Haifa, Haifa, Haifa, Israel

    Orr Dunkelman

  2. University of Warsaw, Warsaw, Poland

    Stefan Dziembowski

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de Castro, L., Hazay, C., Ishai, Y., Vaikuntanathan, V., Venkitasubramaniam, M. (2022). Asymptotically Quasi-Optimal Cryptography. In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13275. Springer, Cham. https://doi.org/10.1007/978-3-031-06944-4_11

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