Efficient extraction from identical-challenge prediction measurements

Develop a uniform polynomial-time procedure for constructing or implementing local full-string extraction POVMs from efficient local prediction measurements for the identical-challenge non-local search-to-decision problem over \(\mathbb F_2\), thereby enabling a polynomial-time reduction from computational search security.

Background

The main theorem proves only an information-theoretic implication: if Bob and Charlie can jointly predict a uniformly random parity with noticeable advantage when given the same challenge, then local POVMs exist with which they jointly recover the entire hidden string with related probability. However, the proof does not provide an efficient, uniform black-box method for constructing or implementing those extraction POVMs from efficient prediction measurements.

Such an efficient extraction procedure is needed to convert the information-theoretic result into a polynomial-time reduction from computational search security. The paper mentions a candidate reduction for the weaker target that negligible search probability implies prediction probability at most 0.6, but explicitly states that the candidate has not been fully verified and is not claimed as a result.

References

The present proof establishes the information-theoretic existence of local extraction POVMs, but it does not give a uniform polynomial-time procedure for constructing or implementing them from the prediction measurements. Such an efficient extraction procedure is necessary for a polynomial-time reduction from computational search security. During the development of this work, ChatGPT using GPT-5.6 Sol Pro suggested a candidate efficient reduction for the weaker target p_{}=\operatorname{negl}(n) \quad\Longrightarrow\quad p_{}\le 0.6 for all sufficiently large n. This candidate argument has not been fully verified and is not claimed as a result of this paper.

— Non-Local Search-to-Decision Reduction over F2  (2608.19091 - Ananth, 19 Aug 2026) in Paragraph “Open problems,” subsection “Efficient extraction” (Introduction)

It remains open to prove an analogue of the present theorem over finite fields larger than \mathbb F_2. ChatGPT using GPT-5.6 Sol Pro suggested candidate arguments in certain restricted regimes, including when the shared states have additional prescribed structure and when the prediction advantage is assumed to be non-negligible. These arguments have not been fully verified, and a theorem covering arbitrary state ensembles and the full negligible-advantage regime remains open.

— Non-Local Search-to-Decision Reduction over F2  (2608.19091 - Ananth, 19 Aug 2026) in Paragraph “Open problems,” subsection “Extension to larger fields” (Introduction)