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Integer Inner Product not in constant-cost BPP

Prove that for every fixed integer t > 2, the integer inner product communication problem family IIP_t is not in BPP_0; equivalently, no public-coin randomized protocol of constant cost solves IIP_{t,n} for all n.

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Background

The integer inner product problem IIP_t has a polylogarithmic-cost randomized protocol (via modular hashing) but is known to be outside PEQ and to have large γ₂-norm for t > 2. Whether IIP_t admits constant-cost randomized protocols remains open.

A resolution would yield a concrete separation within the constant-cost landscape and clarify the power of randomization beyond equality-oracle protocols.

References

\begin{conjecture}[{See Conjecture 6.4] For $t>2$, $\IIP_t \not\in BPP_0$. \end{conjecture}

— Structure in Communication Complexity and Constant-Cost Complexity Classes (2401.14623 - Hatami et al., 26 Jan 2024) in Conjecture [See CHHS23], Section 5.1 (The power of randomness: BPP)