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Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One

Author Noah Stephens-Davidowitz



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Noah Stephens-Davidowitz

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Noah Stephens-Davidowitz. Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 19:1-19:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.19

Abstract

We show the first dimension-preserving search-to-decision reductions for approximate SVP and CVP. In particular, for any gamma <= 1 + O(log n/n), we obtain an efficient dimension-preserving reduction from gamma^{O(n/log n)}-SVP to gamma-GapSVP and an efficient dimension-preserving reduction from gamma^{O(n)}-CVP to gamma-GapCVP. These results generalize the known equivalences of the search and decision versions of these problems in the exact case when gamma = 1. For SVP, we actually obtain something slightly stronger than a search-to-decision reduction - we reduce gamma^{O(n/log n)}-SVP to gamma-unique SVP, a potentially easier problem than gamma-GapSVP.
Keywords
  • Lattices
  • SVP
  • CVP

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