Nondeterministic Interactive Refutations for Nearest Boolean Vector

Authors Andrej Bogdanov , Alon Rosen

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Author Details

Andrej Bogdanov
  • University of Ottawa, Canada
Alon Rosen
  • Bocconi University, Milano, Italy
  • Reichman University, Herzliya, Israel


We are grateful to Chris Jones for useful advice and feedback. Part of this work was done while the first author was affiliated with and the second author visited the Chinese University of Hong Kong.

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Andrej Bogdanov and Alon Rosen. Nondeterministic Interactive Refutations for Nearest Boolean Vector. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Most n-dimensional subspaces 𝒜 of ℝ^m are Ω(√m)-far from the Boolean cube {-1, 1}^m when n < cm for some constant c > 0. How hard is it to certify that the Nearest Boolean Vector (NBV) is at least γ √m far from a given random 𝒜? Certifying NBV instances is relevant to the computational complexity of approximating the Sherrington-Kirkpatrick Hamiltonian, i.e. maximizing x^TAx over the Boolean cube for a matrix A sampled from the Gaussian Orthogonal Ensemble. The connection was discovered by Mohanty, Raghavendra, and Xu (STOC 2020). Improving on their work, Ghosh, Jeronimo, Jones, Potechin, and Rajendran (FOCS 2020) showed that certification is not possible in the sum-of-squares framework when m ≪ n^1.5, even with distance γ = 0. We present a non-deterministic interactive certification algorithm for NBV when m ≫ n log n and γ ≪ 1/mn^1.5. The algorithm is obtained by adapting a public-key encryption scheme of Ajtai and Dwork.

Subject Classification

ACM Subject Classification
  • Theory of computation → Interactive proof systems
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Cryptographic primitives
  • average-case complexity
  • statistical zero-knowledge
  • nondeterministic refutation
  • Sherrington-Kirkpatrick Hamiltonian
  • complexity of statistical inference
  • lattice smoothing


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