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

For k = ω (log n ), we prove a Ω( k 2 n/ 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. Theory of computation Communication complexity;