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Efficient Perfectly Secure Computation with Optimal Resilience

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Theory of Cryptography (TCC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13043))

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Abstract

Secure computation enables n mutually distrustful parties to compute a function over their private inputs jointly. In 1988 Ben-Or, Goldwasser, and Wigderson (BGW) demonstrated that any function can be computed with perfect security in the presence of a malicious adversary corrupting at most \(t< n/3\) parties. After more than 30 years, protocols with perfect malicious security, with round complexity proportional to the circuit’s depth, still require sharing a total of \(O(n^2)\) values per multiplication. In contrast, only O(n) values need to be shared per multiplication to achieve semi-honest security. Indeed sharing \(\varOmega (n)\) values for a single multiplication seems to be the natural barrier for polynomial secret sharing-based multiplication.

In this paper, we close this gap by constructing a new secure computation protocol with perfect, optimal resilience and malicious security that incurs sharing of only O(n) values per multiplication, thus, matching the semi-honest setting for protocols with round complexity that is proportional to the circuit depth. Our protocol requires a constant number of rounds per multiplication. Like BGW, it has an overall round complexity that is proportional only to the multiplicative depth of the circuit. Our improvement is obtained by a novel construction for weak VSS for polynomials of degree-2t, which incurs the same communication and round complexities as the state-of-the-art constructions for VSS for polynomials of degree-t.

Our second contribution is a method for reducing the communication complexity for any depth-1 sub-circuit to be proportional only to the size of the input and output (rather than the size of the circuit). This implies protocols with sublinear communication complexity (in the size of the circuit) for perfectly secure computation for important functions like matrix multiplication.

Gilad Asharov is sponsored by the Israel Science Foundation (grant No. 2439/20), 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 the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 891234.

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Notes

  1. 1.

    In the optimistic case the adversary does not deviate from the prescribed protocol. Thus, in the pessimistic case (when it does deviate from the protocol) the adversary might only make the execution more expensive.

  2. 2.

    We use two rounds of silence as an optimistic early stopping agreement on no complaints. We then combine this with a standard termination protocol that uses either the fast decision or the broadcast decision. It is easy to see that there will be no conflict between the two.

  3. 3.

    In that case, we simply give the adversary all inputs of all honest parties which makes any protocol vacuously secure as anything can be easily simulated, see Remark 3.2.

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Acknowledgments

Gilad Asharov would like to thank Ilan Komargodski and Ariel Nof for helpful discussions.

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Authors and Affiliations

  1. VMWare Research, Herzliya, Israel

    Ittai Abraham & Avishay Yanai

  2. Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel

    Gilad Asharov

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  1. Ittai Abraham
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Correspondence to Gilad Asharov .

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Editors and Affiliations

  1. Georgetown University, Washington, WA, USA

    Kobbi Nissim

  2. The University of Texas at Austin, Austin, TX, USA

    Brent Waters

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Abraham, I., Asharov, G., Yanai, A. (2021). Efficient Perfectly Secure Computation with Optimal Resilience. In: Nissim, K., Waters, B. (eds) Theory of Cryptography. TCC 2021. Lecture Notes in Computer Science(), vol 13043. Springer, Cham. https://doi.org/10.1007/978-3-030-90453-1_3

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