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Theory of Cryptography Conference

TCC 2020: Theory of Cryptography pp 473–501Cite as

Topology-Hiding Communication from Minimal Assumptions

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

Abstract

Topology-hiding broadcast (THB) enables parties communicating over an incomplete network to broadcast messages while hiding the topology from within a given class of graphs. THB is a central tool underlying general topology-hiding secure computation (THC) (Moran et al. TCC’15). Although broadcast is a privacy-free task, it was recently shown that THB for certain graph classes necessitates computational assumptions, even in the semi-honest setting, and even given a single corrupted party.

In this work we investigate the minimal assumptions required for topology–hiding communication—both Broadcast or Anonymous Broadcast (where the broadcaster’s identity is hidden). We develop new techniques that yield a variety of necessary and sufficient conditions for the feasibility of THB/THAB in different cryptographic settings: information theoretic, given existence of key agreement, and given existence of oblivious transfer. Our results show that feasibility can depend on various properties of the graph class, such as connectivity, and highlight the role of different properties of topology when kept hidden, including direction, distance, and/or distance-of-neighbors to the broadcaster.

An interesting corollary of our results is a dichotomy for THC with a public number of at least three parties, secure against one corruption: information-theoretic feasibility if all graphs are 2-connected; necessity and sufficiency of key agreement otherwise.

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Notes

  1. 1.

    Such protocols exist in the honest-majority setting assuming key agreement, and thus under this assumption, THB implies THC. In the information-theoretic setting THC can be strictly stronger, as we will see.

  2. 2.

    That is, the Quadratic Residuosity assumption, the Decisional Diffie-Hellman assumption, and the Learning With Errors assumption, respectively.

  3. 3.

    The lower bound of [3] holds for 4-party 2-secure THB with respect to a small class of 4-node graphs, namely, a square, and a square with any of its edges removed.

  4. 4.

    LaVigne et al. [17] recently studied THC in a non-synchronous setting, demonstrating many barriers.

  5. 5.

    A graph is k-connected if and only if every pair of nodes is connected by k vertex-disjoint paths.

  6. 6.

    If the class of graphs contains a 2-path, then oblivious transfer is necessary for secure computation [14].

  7. 7.

    Note that OT is strictly stronger than KA in terms of black-box reductions, since OT implies KA in a black-box way, but the converse does not hold [10].

  8. 8.

    If the neighbor sends the message in the first round that the party learns it, then its distance is one less of the party’s distance. If the neighbor sends after the party learned it, then its distance equals the party’s distance. If the neighbor does not send, then its distance is one more than the party’s distance.

  9. 9.

    An infinitely often key agreement guarantees correctness and security for infinitely many \(\lambda \in {\mathbb {N}}\) (where \(\lambda \) stands for the security parameter).

  10. 10.

    In particular, the “left/right” orientation can be deduced locally from each node’s neighbor set.

  11. 11.

    An infinitely often OT protocol guarantees correctness and security for infinitely many \(\lambda \in {\mathbb {N}}\) (where \(\lambda \) stands for the security parameter).

  12. 12.

    The result of [18] was limited to graphs of small diameter to allow an arbitrary number of corruptions. With a single corruption the same construction can support all graphs.

  13. 13.

    THB exists trivially for any graph class in which each party’s neighborhood uniquely identifies the graph topology.

  14. 14.

    In fact, for this step we will only need for the smaller graph class \(\{\mathsf {B}-\mathsf {C}, \mathsf {A}-\mathsf {B}-\mathsf {C}, \mathsf {B}-\mathsf {C}-\mathsf {A}\}\subset {\mathcal {G}_{\mathsf {2-vs-3}}}\).

  15. 15.

    The standard notation in the literature is st-orientation; to avoid confusion with the notation t that stands for the corruption threshold, we use \(\sigma \tau \)-orientation instead.

  16. 16.

    In fact, the upper bound holds for a large body of graph classes, where only distance need be hidden.

  17. 17.

    In fact, we critically exploit the fact that if a party has a single neighbor (and thus no guaranteed local honest majority), she knows that neighbor’s neighborhood is majority honest.

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Acknowledgments

We thank the anonymous reviewers of TCC 2020 for pointing to the connection between anonymous communication and key agreement in [13]. M. Ball’s research is supported in part by an IBM Research PhD Fellowship.

M. Ball and T. Malkin’s work is supported in part by JPMorgan Chase & Co. as well as the U.S. Department of Energy (DOE), Office of Science, Office of Advanced Scientific Computing Research under award number DE-SC-0001234. E. Boyle’s research is supported in part by ISF grant 1861/16, AFOSR Award FA9550-17-1-0069, and ERC Starting Grant 852952 (HSS). R. Cohen’s research is supported by NSF grant 1646671. L. Kohl’s research is supported by ERC Project NTSC (742754). P. Meyer’s research is supported in part by ISF grant 1861/16, AFOSR Award FA9550-17-1-0069, and ERC Starting Grant 852952 (HSS).

Any views or opinions expressed herein are solely those of the authors listed, and may differ from the views and opinions expressed by JPMorgan Chase & Co. or its affiliates. This material is not a product of the Research Department of J.P. Morgan Securities LLC. This material should not be construed as an individual recommendation for any particular client and is not intended as a recommendation of particular securities, financial instruments or strategies for a particular client. This material does not constitute a solicitation or offer in any jurisdiction.

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Authors and Affiliations

  1. Columbia University, New York, USA

    Marshall Ball & Tal Malkin

  2. IDC Herzliya, Herzliya, Israel

    Elette Boyle, Pierre Meyer & Tal Moran

  3. Northeastern University, Boston, USA

    Ran Cohen & Tal Moran

  4. Technion, Haifa, Israel

    Lisa Kohl

  5. École Normale Supérieure de Lyon, Lyon, France

    Pierre Meyer

  6. Spacemesh, Tel Aviv, Israel

    Tal Moran

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  1. Marshall Ball
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Correspondence to Pierre Meyer .

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Editors and Affiliations

  1. Cornell Tech, New York, NY, USA

    Prof. Rafael Pass

  2. Institute of Science and Technology Austria, Klosterneuburg, Austria

    Krzysztof Pietrzak

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Ball, M. et al. (2020). Topology-Hiding Communication from Minimal Assumptions. 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_17

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