Determine the exact small-advantage communication complexity of VSP

Determine whether the classical two-way communication advantage for the Haar-distributed exact Vector-in-Subspace problem satisfies an upper bound linear in the transcript length at rate O((L+1)n^{-1/2}), and thereby establish the corresponding small-advantage threshold and the Ω(√n) constant-advantage complexity.

Background

The paper uses an Ω(n{1/3}) lower bound for two-way Vector-in-Subspace communication, while the known upper bound is O(√n). A one-way protocol is shown to have advantage O((L+1)/√n) at every message length, but no analogous result is known for interactive protocols. Such a result would sharpen the EFI security exponent and match the known constant-advantage upper bound.

References

A two-way analogue of \Cref{prop:oneway-rate}, linear in $L$ at rate $O((L+1)n{-1/2})$, would sharpen \Cref{thm:EFI-security} to its exact exponent and would in particular give $\Omega(\sqrt n)$ at constant advantage, matching Raz. Whether a spectral argument of that kind survives interaction is open.

— EFI Pairs Without One-Way Puzzles: Oracle Separations from Communication Complexity  (2609.11901 - Mantri, 10 Sep 2026) in Section 5.1, The one-way rate for VSP_n; Section 7, Discussion and open problems