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Locally Verifiable Distributed SNARGs

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Theory of Cryptography (TCC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14369))

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Abstract

The field of distributed certification is concerned with certifying properties of distributed networks, where the communication topology of the network is represented as an arbitrary graph; each node of the graph is a separate processor, with its own internal state. To certify that the network satisfies a given property, a prover assigns each node of the network a certificate, and the nodes then communicate with one another and decide whether to accept or reject. We require soundness and completeness: the property holds if and only if there exists an assignment of certificates to the nodes that causes all nodes to accept. Our goal is to minimize the length of the certificates, as well as the communication between the nodes of the network. Distributed certification has been extensively studied in the distributed computing community, but it has so far only been studied in the information-theoretic setting, where the prover and the network nodes are computationally unbounded.

In this work we introduce and study computationally bounded distributed certification: we define locally verifiable distributed \(\textsf{SNARG}\)s (s), which are an analog of \(\textsf{SNARG}\)s for distributed networks, and are able to circumvent known hardness results for information-theoretic distributed certification by requiring both the prover and the verifier to be computationally efficient (namely, PPT algorithms).

We give two constructions: the first allows us to succinctly certify any network property in \({\textsf{P}}\), using a global prover that can see the entire network; the second construction gives an efficient distributed prover, which succinctly certifies the execution of any efficient distributed algorithm. Our constructions rely on non-interactive batch arguments for \({\textsf{NP}}\) (\(\textsf{BARG}\)s) and on \(\textsf{RAM}~\textsf{SNARG}\)s, which have recently been shown to be constructible from standard cryptographic assumptions.

R. Oshman’s research is supported by ISF grant no. 2801/20. E. Boyle’s research is supported in part by AFOSR Award FA9550-21-1-0046 and ERC Project HSS (852952). R. Cohen’s research is supported in part by NSF grant no. 2055568 and 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 authors and do not necessarily reflect the views of Algorand Foundation. T. Moran’s research is supported by ISF grant no. 2337/22.

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Notes

  1. 1.

    As just one example of many, in [KP98] it is shown that one can construct a k-dominating set of the network graph in \(\tilde{O}(k)\) communication per edge, and this is used to construct a minimum-weight spanning tree in \(\tilde{O}(\sqrt{n})\) communication per edge.

  2. 2.

    In general, the nodes of the network may have inputs, on which the property may depend, but for simplicity we ignore inputs for the time being and discuss only properties of the graph topology itself.

  3. 3.

    Assuming the underlying network is connected, which is a standard assumption in the area; otherwise additional information, such as the size of the network, is required.

  4. 4.

    In fact, as we mentioned in Sect. 1, a centralized prover can also be implemented by a distributed algorithm where one node learns the entire network graph and then generates the certificates. This is easy to do in polynomial rounds and message length.

  5. 5.

    The schemes we construct actually satisfy adaptive soundness: there is no PPT algorithm that can, with non-negligible probability, output a network graph and certificates for all the nodes, such that the property does not hold for the network graph but all of the nodes accept.

  6. 6.

    We consider only connected networks, since in disconnected networks one can never hope to carry out any computation involving more than one connected component. Also, it is fairly standard to assume an undirected graph topology, i.e., bidirectional communication links, although directed networks are also considered sometimes (for instance, in [BFO22]).

  7. 7.

    For the centralized case, we denote \(\mathcal {P}(\textsf{crs},G,x)\) instead of \((\textsf{crs};G;x)\) as we have one entity that receives the entire input.

  8. 8.

    A \(\textsf{RAM}\) machine M is given query access to an input x and an unbounded random-access memory array, and returns some output y. Each query to the input x or the memory is considered a unit-cost operation.

  9. 9.

    This requires that \(G'\) not be connected, but that is not necessarily a problem for the prover, depending on the property \(\mathcal {L}\).

  10. 10.

    This step is a little delicate, and relies on the fact that in recent \(\textsf{RAM}~\textsf{SNARG}\) constructions (e.g., [CJJ21b, KLVW23]), completeness holds for any digest d that opens to the input instance at every location the RAM machine reads from.

  11. 11.

    For simplicity we assume that nodes can query the communication infrastructure for a consistent order of their neighbors (e.g., by “port number”); thus the relative ordering \(I_{v\rightarrow u}\) does not count against v’s storage. This is a standard assumption in the area. In the general case, the port numbers themselves, which may stand for MAC addresses or similar, do not necessarily need to be consecutive numbers from \(1,\ldots ,\deg (v)\), but we can order v’s neighbors in order of increasing port number.

  12. 12.

    We believe that this additional temporary storage requirement can be avoided using incrementally verifiable computation, but we have not gone through the details.

  13. 13.

    As explained above, we actually require that this opening show that \(\textsf{h}\textsf{Read}_{r,i}\) and \(\textsf{h}\textsf{Read}_{r,i+1}\) only differ in the location d and \(\textsf{h}\textsf{Read}_{r,i+1}\) opens to m in that location.

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Acknowledgments

We would like to thank Omer Paneth for fruitful and illuminating iscussions.

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Authors and Affiliations

  1. Tel-Aviv University, Tel Aviv, Israel

    Eden Aldema Tshuva & Rotem Oshman

  2. Reichman University, Herzliya, Israel

    Elette Boyle, Ran Cohen & Tal Moran

  3. NTT Research, Sunnyvale, USA

    Elette Boyle

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  1. Eden Aldema Tshuva
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Correspondence to Eden Aldema Tshuva .

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  1. Apple, Cupertino, CA, USA

    Guy Rothblum

  2. NTT Research, Sunnyvale, CA, USA

    Hoeteck Wee

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Aldema Tshuva, E., Boyle, E., Cohen, R., Moran, T., Oshman, R. (2023). Locally Verifiable Distributed SNARGs. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14369. Springer, Cham. https://doi.org/10.1007/978-3-031-48615-9_3

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