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Cryptography from Planted Graphs: Security with Logarithmic-Size Messages

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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

We study the following broad question about cryptographic primitives: is it possible to achieve security against arbitrary \(\textsf{poly}(n)\)-time adversary with \(O(\log n)\)-size messages? It is common knowledge that the answer is “no” unless information-theoretic security is possible. In this work, we revisit this question by considering the setting of cryptography with public information and computational security.

We obtain the following main results, assuming variants of well-studied intractability assumptions:

  • A private simultaneous messages (PSM) protocol for every \(f:[n]\times [n]\rightarrow \{0,1\}\) with \((1+\epsilon )\log n\)-bit messages, beating the known lower bound on information-theoretic PSM protocols. We apply this towards non-interactive secure 3-party computation with similar message size in the preprocessing model, improving over previous 2-round protocols.

  • A secret-sharing scheme for any “forbidden-graph” access structure on n nodes with \(O(\log n)\) share size.

  • On the negative side, we show that computational threshold secret-sharing schemes with public information require share size \(\varOmega (\log \log n)\). For arbitrary access structures, we show that computational security does not help with 1-bit shares.

The above positive results guarantee that any adversary of size \(n^{o(\log n)}\) achieves an \(n^{-\varOmega (1)}\) distinguishing advantage. We show how to make the advantage negligible by slightly increasing the asymptotic message size, still improving over all known constructions.

The security of our constructions is based on the conjectured hardness of variants of the planted clique problem, which was extensively studied in the algorithms, statistical inference, and complexity theory communities. Our work provides the first applications of such assumptions to improving the efficiency of mainstream cryptographic primitives, gives evidence for the necessity of such assumptions, and suggests new questions in this domain that may be of independent interest.

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Notes

  1. 1.

    Here and elsewhere, \(\log n\) stands for \(\log _2 n\).

  2. 2.

    We use \(\mathcal {D}\equiv H\) to denote the distribution that always outputs the subgraph H.

  3. 3.

    If the parties use independent randomness, an adversary can run a residual function attack. Check the full version of the paper [ABI+23a, Section 5.1] for more details.

  4. 4.

    We “XOR” two graphs by XORing their adjacency matrices.

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Acknowledgements

We thank Uriel Feige, Prasad Raghavendra, and Daniel Reichman for helpful discussions and literature pointers. Damiano Abram was supported by a GSNS travel grant from Aarhus University and by the Aarhus University Research Foundation (AUFF). Amos Beimel was supported by ERC Project NTSC (742754) and ISF grant 391/21. Yuval Ishai and Varun Narayanan were supported by ERC Project NTSC (742754), BSF grant 2018393, and ISF grant 2774/20. Work of Varun Narayanan was done while working at Technion, Israel Institute of Technology. Eyal Kushilevitz was supported by BSF grant 2018393 and ISF grant 2774/20.

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Authors and Affiliations

  1. Aarhus University, Aarhus, Denmark

    Damiano Abram

  2. Ben-Gurion University, Beersheba, Israel

    Amos Beimel

  3. Technion, Haifa, Israel

    Yuval Ishai & Eyal Kushilevitz

  4. University of California, Los Angeles, USA

    Varun Narayanan

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  1. Damiano Abram
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Correspondence to Yuval Ishai .

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  1. Apple, Cupertino, CA, USA

    Guy Rothblum

  2. NTT Research, Sunnyvale, CA, USA

    Hoeteck Wee

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Abram, D., Beimel, A., Ishai, Y., Kushilevitz, E., Narayanan, V. (2023). Cryptography from Planted Graphs: Security with Logarithmic-Size Messages. 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_11

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