Determine the exact communication complexity of securely computing OR

Determine the exact private simultaneous-messages communication complexity of the Boolean function \(OR_n\), improving the currently known bounds between \(\omega(n)\) and \(O(n\log n)\).

Background

The paper proves a superlinear lower bound C(ORn)=ω(n)C(OR_n)=\omega(n) for perfectly correct, support-private PSM protocols computing ORnOR_n. It also recalls a known perfectly private protocol with communication complexity O(nlog⁡n)O(n\log n). Because these bounds do not match, the exact communication cost of securely computing ORnOR_n remains unresolved.

References

Thus, settling the exact cost of securely computing $OR$ remains open.

— On the Complexity of Symmetric Boolean Functions in Multiparty Private Simultaneous Messages  (2610.21216 - Ball et al., 15 Oct 2026) in Section 1, subsection “Our Results,” paragraph following the OR lower-bound theorem