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Super-constant lower bounds for randomized complexity of Integer Inner Product

Prove that for fixed t > 2, the public-coin randomized communication complexity R(IIP_{t,n}) grows super-constantly with n, i.e., R(IIP_{t,n}) = ω(1).

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Background

Despite strong structural obstacles for equality-oracle protocols and large γ₂-norm lower bounds for IIP_t (t > 2), no ω(1) lower bound is known for its randomized public-coin complexity. Establishing any super-constant lower bound would already separate IIP_t from BPP_0.

This problem is a weaker target toward the conjecture that IIP_t ∉ BPP_0.

References

We do not know how to prove any $\omega(1)$ lower bound for $R(\IIP_{t,n})$.

— Structure in Communication Complexity and Constant-Cost Complexity Classes (2401.14623 - Hatami et al., 26 Jan 2024) in Following Conjecture on IIP_t, Section 5.1 (The power of randomness: BPP)