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Completeness Theorems for Adaptively Secure Broadcast

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Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14081))

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Abstract

The advent of blockchain protocols has reignited the interest in adaptively secure broadcast; it is by now well understood that broadcasting over a diffusion network allows an adaptive adversary to corrupt the sender depending on the message it attempts to send and change it. Hirt and Zikas [Eurocrypt ’10] proved that this is an inherent limitation of broadcast in the simulation-based setting—i.e., this task is impossible against an adaptive adversary corrupting a majority of the parties (a task that is achievable against a static adversary).

The contributions of this paper are two-fold. First, we show that, contrary to previous perception, the above limitation of adaptively secure broadcast is not an artifact of simulation-based security, but rather an inherent issue of adaptive security. In particular, we show that: (1) it also applies to the property-based broadcast definition adapted for adaptive adversaries, and (2) unlike other impossibilities in adaptive security, this impossibility cannot be circumvented by adding a programmable random oracle, in neither setting, property-based or simulation-based.

Second, we turn to the resource-restricted cryptography (RRC) paradigm [Garay et al., Eurocrypt ’20], which has proven useful in circumventing impossibility results, and ask whether it also affects the above negative result. We answer this question in the affirmative, by showing that time-lock puzzles (TLPs)—which can be viewed as an instance of RRC—indeed allow for achieving the property-based definition and circumvent the impossibility of adaptively secure broadcast. The natural question is then, do TLPs also allow for simulation-based adaptively secure broadcast against corrupted majorities? We answer this question in the negative. However, we show that a positive result can be achieved via a non-committing analogue of TLPs in the programmable random-oracle model.

Importantly, and as a contribution of independent interest, we also present the first (limited) composition theorem in the resource-restricted setting, which is needed for the complexity-based, non-idealized treatment of TLPs in the context of other protocols.

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Notes

  1. 1.

    For the related consensus problem (aka Byzantine agreement), where all parties have an input, the best achievable bound is \(t < n/2\) [41].

  2. 2.

    In the context of collective coin tossing, the capability of the adversary to first learn the sender’s message and later to corrupt the sender and change its input has been referred to as strongly adaptive [55, 56, 60, 64].

  3. 3.

    Although in this work we focus on broadcast, corruption fairness can easily be defined—and is a natural requirement—for any adaptively secure MPC task.

  4. 4.

    In this work we refrain from using the term “strongly rushing,” because we believe it creates the misconception of an assumption on the adversary. We view the non-atomic multisend model as the “plain” model for a rushing adversary, a view which is consistent with the literature [20, 21], and atomic multisend as an assumption on the network which limits the adversary’s adaptivity.

  5. 5.

    We stress that the above is orthogonal to the synchrony assumption: Consider for example a synchronous setting, where a round takes 60 s (i.e., any message sent by an honest party is delivered within 60 s) and corrupting a party takes 30 s. Then delaying messages at the router gives the adversary time to corrupt the sender and crash it based on messages it sends, dropping all pending messages.

  6. 6.

    We view non-atomic multisend as the “plain” model for a rushing adversary, a view which is consistent with the literature on security models for MPC [20, 21].

  7. 7.

    Terminology taken from [43].

  8. 8.

    In fact, we conjecture that it might be impossible to capture all natural properties of adaptive security in one property-based definition, i.e., without effectively resorting to the simulation-based paradigm.

  9. 9.

    It might be useful to make a distinction here between corruption fairness and input independence: The latter requires that the adversary cannot bias corrupted parties’ input based on the honest parties’ input, and, unlike corruption fairness, applies both to static and adaptive adversaries.

  10. 10.

    Note that in our model the adversary can corrupt a party after sending a message and drop the message from the network, but this is done independently of the content of the message; therefore, we require all other parties to broadcast dummy messages.

  11. 11.

    In most broadcast protocols from the literature (e.g., [28, 35, 46, 83, 84]), the sender starts by sending its input to all parties, meaning that \(R=1\).

  12. 12.

    The UC composition theorem in [22] applies to balanced environments, i.e., environments that do not give honest parties much more resources than to the adversary. In [22] the focus is on running time, whereas in this work it is on parallel running time; hence, by abusing the terminology from [22], one can say that the environment in our protocol is not balanced with respect to parallel running time.

  13. 13.

    The environment can take its time running \(\hat{B}_\pi \) after the protocol terminates.

  14. 14.

    Classical correlated randomness setup assumes efficient sampling and distribution mechanisms. By removing such restrictions here we can even capture non-programmable random oracle, as an exponential-space correlated randomness functionality that samples the entire random table of the RO.

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Acknowledgments

Ran Cohen’s research is supported in part by NSF grant no. 2055568. Juan Garay’s research is supported in part by NSF grants no. 2001082 and 2055694. Vassilis Zikas’s research is supported in part by NSF grant no. 2055599 and by Sunday Group. The authors are also supported by the Algorand Centres of Excellence programme managed by Algorand Foundation. 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 Algorand Foundation.

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  1. Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel

    Ran Cohen

  2. Texas A&M University, College Station, USA

    Juan Garay

  3. Purdue University, West Lafayette, USA

    Vassilis Zikas

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Cohen, R., Garay, J., Zikas, V. (2023). Completeness Theorems for Adaptively Secure Broadcast. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14081. Springer, Cham. https://doi.org/10.1007/978-3-031-38557-5_1

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