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The Planted Orthogonal Vectors Problem

Authors David Kühnemann , Adam Polak , Alon Rosen

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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Average-case complexity
  • fine-grained complexity
  • orthogonal vectors

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Abstract

In the k-Orthogonal Vectors (k-OV) problem we are given k sets, each containing n binary vectors of dimension d = n^o(1), and our goal is to pick one vector from each set so that at each coordinate at least one vector has a zero. It is a central problem in fine-grained complexity, conjectured to require n^{k-o(1)} time in the worst case.
We propose a way to plant a solution among vectors with i.i.d. p-biased entries, for appropriately chosen p, so that the planted solution is the unique one. Our conjecture is that the resulting k-OV instances still require time n^{k-o(1)} to solve, on average.
Our planted distribution has the property that any subset of strictly less than k vectors has the same marginal distribution as in the model distribution, consisting of i.i.d. p-biased random vectors. We use this property to give average-case search-to-decision reductions for k-OV.

Cite As Get BibTex

David Kühnemann, Adam Polak, and Alon Rosen. The Planted Orthogonal Vectors Problem. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 95:1-95:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.95

Author Details

David Kühnemann
  • University of Amsterdam, The Netherlands
Adam Polak
  • Bocconi University, Milan, Italy
Alon Rosen
  • Bocconi University, Milan, Italy

Funding

Work supported by European Research Council (ERC) under the EU’s Horizon 2020 research and innovation programme (Grant agreement No. 101019547) and Cariplo CRYPTONOMEX grant. Part of this work was done when the first author was visiting Bocconi University.

Acknowledgements

We are grateful to Andrej Bogdanov, Antoine Joux, Moni Naor, Nicolas Resch, Nikolaj Schwartzbach, and Prashant Vasudevan for insightful discussions.

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