Close the communication-complexity gap for worst-case Boolean functions

Determine the exact private simultaneous-messages communication complexity of the Boolean functions with the highest complexity, thereby closing the gap between the known upper and lower bounds for worst-case functions.

Background

The paper places its results in the broader study of communication complexity for multiparty Private Simultaneous Messages (PSM) protocols. General upper bounds are known for arbitrary Boolean functions, while lower bounds of order Omega(n2/log n) are known for almost all functions and for some explicit functions. The authors identify closing the remaining gap between the upper and lower bounds for the most difficult functions as a major unresolved problem.

References

Closing the gap between upper and lower bounds for the ``worst'' functions remains a major open question.

— On the Complexity of Symmetric Boolean Functions in Multiparty Private Simultaneous Messages  (2610.21216 - Ball et al., 15 Oct 2026) in Section 1, Introduction, paragraph discussing general PSM communication complexity