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On Statistical Security in Two-Party Computation

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Part of the Lecture Notes in Computer Science book series (LNSC,volume 12551)

Abstract

There has been a large body of work characterizing the round complexity of general-purpose maliciously secure two-party computation (\(\mathsf {2PC}\)) against probabilistic polynomial time adversaries. This is particularly true for zero-knowledge, which is a special case of \(\mathsf {2PC}\). In fact, in the special case of zero knowledge, optimal protocols with unconditional security against one of the two players have also been meticulously studied and constructed.

On the other hand, general-purpose maliciously secure \(\mathsf {2PC}\) with statistical or unconditional security against one of the two participants has remained largely unexplored so far. In this work, we initiate the study of such protocols, which we refer to as \(\mathsf {2PC}\) with one-sided statistical security. We settle the round complexity of \(\mathsf {2PC}\) with one-sided statistical security with respect to black-box simulation by obtaining the following tight results:

  • In a setting where only one party obtains an output, we design \(\mathsf {2PC}\) in 4 rounds with statistical security against receivers and computational security against senders.

  • In a setting where both parties obtain outputs, we design \(\mathsf {2PC}\) in 5 rounds with computational security against the party that obtains output first and statistical security against the party that obtains output last.

Katz and Ostrovsky (CRYPTO 2004) showed that \(\mathsf {2PC}\) with black-box simulation requires at least 4 rounds when one party obtains an output and 5 rounds when both parties obtain outputs, even when only computational security is desired against both parties. Thus in these settings, not only are our results tight, but they also show that statistical security is achievable at no extra cost to round complexity. This still leaves open the question of whether \(\mathsf {2PC}\) can be achieved with black-box simulation in 4 rounds with statistical security against senders and computational security against receivers. Based on a lower bound on computational zero-knowledge proofs due to Katz (TCC 2008), we observe that the answer is negative unless the polynomial hierarchy collapses.

This material is based on work supported in part by DARPA under Contract No. HR001120C0024. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government or DARPA.

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Notes

  1. 1.

    We note that this is without loss of generality, since any asymmetric functionality can be unconditionally computed from a symmetric one by having each party input a random value, and using it to mask the output.

  2. 2.

    Importantly, this is different from semi-malicious security [38, 39] where the adversary in addition to generating messages in the support of the distribution of all honestly generated messages, outputs the input and randomness that it used, on a special tape. On the other hand, simulating an explainable adversary is much more challenging: since in this case the adversary does not output any such special tape, and therefore the input and randomness must still be extracted from an explainable adversary by the simulator.

  3. 3.

    Alternatively, \(\mathsf {R} \) could withhold the garbled circuit decoding information, i.e. the correspondence between the output wire labels and the output of the circuit, from \(\mathsf {S} \) until the \(5^{th}\) round. This would achieve the same effect, but leads to a more complex analysis. For simplicity of analysis, we choose to garble an encrypted circuit in our formal presentation.

  4. 4.

    Such protocols have been used previously in the literature, most recently in [8].

  5. 5.

    We point out that Informal Theorem 2 follows from this theorem by exchanging the roles of \(\mathsf {S} \) and \(\mathsf {R} \).

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Acknowledgement

We thank Giulio Malavolta, Akshayaram Srinivasan and the anonymous TCC reviewers for useful suggestions.

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Correspondence to Dakshita Khurana .

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Khurana, D., Mughees, M.H. (2020). On Statistical Security in Two-Party Computation. In: Pass, R., Pietrzak, K. (eds) Theory of Cryptography. TCC 2020. Lecture Notes in Computer Science(), vol 12551. Springer, Cham. https://doi.org/10.1007/978-3-030-64378-2_19

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