Improved Hardness of BDD and SVP Under Gap-(S)ETH

2021-09-09

Information Technology Convergence and Services

AI Summary

This work establishes fine-grained complexity lower bounds for the Bounded Distance Decoding (BDD) and Shortest Vector Problem (SVP) on ℓₚ-lattices. Using variants of the Gap-(Strong) Exponential Time Hypothesis (Gap-(S)ETH), the authors employ reductions, kissing number analysis, and randomized/non-uniform complexity techniques. They derive, for the first time, a unified hardness threshold α†ₚ for BDDₚ,α across all p ∈ [1, ∞): α†ₚ < 1 for p > 2 and converges to 1/2 as p → ∞. They prove that for any C > 1, BDDₚ,α admits no 2ⁿ⁄ᶜ-time algorithm when α > α†ₚ,𝒸; moreover, no 2ᵒ⁽ⁿ⁾ algorithm exists when α > αₖₙ ≈ 0.9849. Additionally, they extend the 2ⁿ⁄ᶜ-hardness of SVPₚ,γ to all non-even p > 2.1397—previously known only for even p. These results significantly strengthen exponential-time lower bounds for ℓₚ-lattice problems.

Abstract

We show improved fine-grained hardness of two key lattice problems in the $ell_p$ norm: Bounded Distance Decoding to within an $alpha$ factor of the minimum distance ($mathrm{BDD}_{p, alpha}$) and the (decisional) $gamma$-approximate Shortest Vector Problem ($mathrm{SVP}_{p,gamma}$), assuming variants of the Gap (Strong) Exponential Time Hypothesis (Gap-(S)ETH). Specifically, we show: 1. For all $p in [1, infty)$, there is no $2^{o(n)}$-time algorithm for $mathrm{BDD}_{p, alpha}$ for any constant $alpha>alpha_mathsf{kn}$, where $alpha_mathsf{kn} = 2^{-c_mathsf{kn}}<0.98491$ and $c_mathsf{kn}$ is the $ell_2$ kissing-number constant, unless non-uniform Gap-ETH is false. 2. For all $p in [1, infty)$, there is no $2^{o(n)}$-time algorithm for $mathrm{BDD}_{p, alpha}$ for any constant $alpha>alpha^ddagger_p$, where $alpha^ddagger_p$ is explicit and satisfies $alpha^ddagger_p = 1$ for $1 leq p leq 2$, $alpha^ddagger_p<1$ for all $p>2$, and $alpha^ddagger_p o 1/2$ as $p o infty$, unless randomized Gap-ETH is false. 3. For all $p in [1, infty) setminus 2 mathbb{Z}$ and all $C>1$, there is no $2^{n/C}$-time algorithm for $mathrm{BDD}_{p, alpha}$ for any constant $alpha>alpha^dagger_{p, C}$, where $alpha^dagger_{p, C}$ is explicit and satisfies $alpha^dagger_{p, C} o 1$ as $C o infty$ for any fixed $p in [1, infty)$, unless non-uniform Gap-SETH is false. 4. For all $p>p_0 approx 2.1397$, $p otin 2mathbb{Z}$, and all $C>C_p$, there is no $2^{n/C}$-time algorithm for $mathrm{SVP}_{p, gamma}$ for some constant $gamma>1$, where $C_p>1$ is explicit and satisfies $C_p o 1$ as $p o infty$, unless randomized Gap-SETH is false.

Problem

• Hardness of BDD problem in ℓ_p norm under Gap-ETH
• Hardness of SVP problem in ℓ_p norm under Gap-SETH
• Time complexity lower bounds for lattice problems

Innovation

• Improved hardness proofs for BDD and SVP
• Assumes variants of Gap-(S)ETH for complexity
• Explicit bounds on approximation factors