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

09/09/2021
by   Huck Bennett, et al.
0

We show improved fine-grained hardness of two key lattice problems in the ℓ_p norm: Bounded Distance Decoding to within an α factor of the minimum distance (BDD_p, α) and the (decisional) γ-approximate Shortest Vector Problem (SVP_p,γ), assuming variants of the Gap (Strong) Exponential Time Hypothesis (Gap-(S)ETH). Specifically, we show: 1. For all p ∈ [1, ∞), there is no 2^o(n)-time algorithm for BDD_p, α for any constant α > α_𝗄𝗇, where α_𝗄𝗇 = 2^-c_𝗄𝗇 < 0.98491 and c_𝗄𝗇 is the ℓ_2 kissing-number constant, unless non-uniform Gap-ETH is false. 2. For all p ∈ [1, ∞), there is no 2^o(n)-time algorithm for BDD_p, α for any constant α > α^_p, where α^_p is explicit and satisfies α^_p = 1 for 1 ≤ p ≤ 2, α^_p < 1 for all p > 2, and α^_p → 1/2 as p →∞, unless randomized Gap-ETH is false. 3. For all p ∈ [1, ∞) ∖ 2, all C > 1, and all ε > 0, there is no 2^(1-ε)n/C-time algorithm for BDD_p, α for any constant α > α^†_p, C, where α^†_p, C is explicit and satisfies α^†_p, C→ 1 as C →∞ for any fixed p ∈ [1, ∞), unless non-uniform Gap-SETH is false. 4. For all p > p_0 ≈ 2.1397, p ∉ 2, and all ε > 0, there is no 2^(1-ε)n/C_p-time algorithm for SVP_p, γ for some constant γ = γ(p, ε) > 1 and explicit constant C_p > 0 where C_p → 1 as p →∞, unless randomized Gap-SETH is false.

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