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A Note on the Complexity of Private Simultaneous Messages with Many Parties

Authors Marshall Ball, Tim Randolph

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Author Details

Marshall Ball
  • Courant Institute of Mathematical Sciences, New York University, NY, USA
Tim Randolph
  • Columbia University, New York, NY, USA

Funding

  • Ball, Marshall: Part of this work was performed while the author was a student at Columbia University and a postdoc at University of Washington. This material is based upon work supported by the National Science Foundation under Grant #2030859 to the Computing Research Association for the CIFellows Project. 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 the National Science Foundation nor the Computing Research Association.
  • Randolph, Tim: This material is based upon work supported by the following grants: NSF CCF-1563155, NSF IIS-1838154, NSF CCF-1814873, NSF CCF-1703925, NSF CCF-2106429, and NSF CCF-2107187.

Acknowledgements

The authors thank Tal Malkin for helpful discussion, and several anonymous reviewers for helpful comments on an earlier draft.

Cite As Get BibTex

Marshall Ball and Tim Randolph. A Note on the Complexity of Private Simultaneous Messages with Many Parties. In 3rd Conference on Information-Theoretic Cryptography (ITC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 230, pp. 7:1-7:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITC.2022.7

Abstract

For k = ω(log n), we prove a Ω(k²n / log(kn)) lower bound on private simultaneous messages (PSM) with k parties who receive n-bit inputs. This extends the Ω(n) lower bound due to Appelbaum, Holenstein, Mishra and Shayevitz [Journal of Cryptology, 2019] to the many-party (k = ω(log n)) setting. It is the first PSM lower bound that increases quadratically with the number of parties, and moreover the first unconditional, explicit bound that grows with both k and n. This note extends the work of Ball, Holmgren, Ishai, Liu, and Malkin [ITCS 2020], who prove communication complexity lower bounds on decomposable randomized encodings (DREs), which correspond to the special case of k-party PSMs with n = 1. To give a concise and readable introduction to the method, we focus our presentation on perfect PSM schemes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic protocols
  • Theory of computation → Communication complexity
  • Security and privacy → Information-theoretic techniques
Keywords
  • Secure computation
  • Private Simultaneous Messages

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References

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