Tightness of synchronous communication lower bounds

Determine whether the communication lower bounds established for error-free synchronous protocols against general $Q^d$ adversaries are tight, and, if they are, construct protocols matching those bounds.

Background

The paper proves lower bounds of order Ω(Loutn1+1/d)\Omega(L_{\mathsf{out}} n^{1+1/d}) for several agreement tasks against QdQ^d adversaries in synchronous networks, under error-free Byzantine security. In contrast, the authors show that analogous asynchronous lower bounds are tight for sufficiently large outputs by giving termination protocols with matching asymptotic communication complexity.

The unresolved issue is whether comparable upper bounds exist in the synchronous setting, either for the specific projective-geometry adversary structures used in the paper or for arbitrary QdQ^d-satisfying adversary structures. The authors conjecture that the synchronous lower bounds are tight, but do not provide matching protocols.

References

An open question we leave for future work is whether the lower bounds we have proven for error-free synchronous protocols are tight (not necessarily just for $Z_proj{n,d}$ but for $Qd$ adversaries in general), and if so, to come up with protocols that match them. While the combination of byzantine faults and error-free security might make it quite difficult to design such protocols, we conjecture that our lower bounds for synchronous protocols are tight like their asynchronous counterparts.

— Multivalued Consensus: General Adversaries Require More Communication  (2608.17998 - Mizrahi et al., 18 Aug 2026) in Discussion, subsection “Synchronous Tightness”