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


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

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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)


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
  • Secure computation
  • Private Simultaneous Messages


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