Average-case search-to-decision reduction for independent spaces

Construct an average-case search-to-decision reduction for the independent space problem for cubic polynomials, analogous to the known reduction for planted independent sets.

Background

The paper defines decision and search versions of both the independent space problem and the planted independent space problem. While a worst-case search-to-decision reduction is available, the authors state that the corresponding average-case reduction is nontrivial and unresolved. Such a reduction would clarify the relationship between distinguishing planted instances and recovering the hidden independent space.

References

The average-case search-to-decision reduction for the independent space problem looks rather non-trivial, and we leave this to future work.

— Verifiable quantum advantage based on polynomials with planted structures  (2609.39918 - Bläser et al., 30 Sep 2026) in Section 2, subsection “Worst-case hardness and search-to-decision reductions”