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Just How Fair is an Unreactive World?

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Advances in Cryptology – ASIACRYPT 2023 (ASIACRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14443))

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Abstract

Fitzi, Garay, Maurer, and Ostrovsky (J. Cryptology 2005) showed that in the presence of a dishonest majority, no primitive of cardinality \(n - 1\) is complete for realizing an arbitrary n-party functionality with guaranteed output delivery. In this work, we show that in the presence of \(n - 1\) corrupt parties, no unreactive primitive of cardinality \(n - 1\) is complete for realizing an arbitrary n-party functionality with fairness. We show more generally that for \(t > \frac{n}{2}\), in the presence of t malicious parties, no unreactive primitive of cardinality t is complete for realizing an arbitrary n-party functionality with fairness. We complement this result by noting that \((t+1)\)-wise fair exchange is complete for realizing an arbitrary n-party functionality with fairness. In order to prove our results, we utilize the primitive of fair coin tossing and the notion of predictability. While this notion has been considered in some form in past works, we come up with a novel and non-trivial framework to employ it, one that readily generalizes from the setting of two parties to multiple parties, and also to the setting of unreactive functionalities.

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Notes

  1. 1.

    These channels may be implemented via a trusted third party, or hardware or cryptographic assumptions.

  2. 2.

    Cardinality refers to the number of parties interacting with a single instance of the ideal primitive.

  3. 3.

    In fact, some of their primitives are also complete for MPC with guaranteed output delivery. The upside of these primitives is that unlike [13], their primitive complexity is independent of the function being computed.

  4. 4.

    Note that for \(t < \frac{n}{2}\), no functionality is needed for MPC with fairness.

  5. 5.

    [10] specified [5]’s protocol for the 2-party case and analyzed the bias.

  6. 6.

    Our method of employing predictors and predictabilities to attack coin tossing protocols is distinct from other those considered in prior works (e.g. [2, 10, 20, 21]).

  7. 7.

    Let us assume that the local state contains all the randomness that the party will ever use through the course of the protocol.

  8. 8.

    Note that when \(t = n\), there is nothing to prove.

  9. 9.

    This can also be viewed as working in the \(\mathcal {F}_{\textsf{bc}}\)-hybrid model. See Sect. 2.4.

  10. 10.

    One way to model this is to consider circuits besides regular computational gates, additionally have “random” gates that simply produce random bits as output.

  11. 11.

    No internal state is retained between invocations of the functionality.

  12. 12.

    More precisely, as long as the channels are one-directional, such as OT channels, Cleve’s lower bound holds.

  13. 13.

    Note \(|\textsf{Pred}_{A, R} - \textsf{Pred}_{B,R}| = 0\), so the gap will not be in this term.

  14. 14.

    \(B_k/A_{k+1}\) is either not in the next unreactive functionality or it is the next broadcaster.

  15. 15.

    Note that after the first unreactive functionality is enabled, the predictor of the party being “kicked-out” is still an initial predictor.

  16. 16.

    This is a conjecture since the 1970s.

  17. 17.

    Note this is not a problem if \(t=n-1\).

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Authors and Affiliations

  1. Visa Research and MIT, Cambridge, USA

    Srinivasan Raghuraman

  2. Georgia Institute of Technology, Atlanta, USA

    Yibin Yang

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  1. Srinivasan Raghuraman
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  1. Nanyang Technological University, Singapore, Singapore

    Jian Guo

  2. Monash University, Melbourne, VIC, Australia

    Ron Steinfeld

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Raghuraman, S., Yang, Y. (2023). Just How Fair is an Unreactive World?. In: Guo, J., Steinfeld, R. (eds) Advances in Cryptology – ASIACRYPT 2023. ASIACRYPT 2023. Lecture Notes in Computer Science, vol 14443. Springer, Singapore. https://doi.org/10.1007/978-981-99-8736-8_14

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  • DOI: https://doi.org/10.1007/978-981-99-8736-8_14

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