Huck Bennett, Chris Peikert, Yi Tang
We show improved fine-grained hardness of two key lattice problems in the $\ell_p$ norm: Bounded Distance Decoding to within an $α$ factor of the minimum distance ($\mathrm{BDD}_{p, α}$) and the (decisional) $γ$-approximate Shortest Vector Problem ($\mathrm{SVP}_{p,γ}$), 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 $\mathrm{BDD}_{p, α}$ for any constant $α> α_\mathsf{kn}$, where $α_\mathsf{kn} = 2^{-c_\mathsf{kn}}$ and $c_\mathsf{kn}$ is the $\ell_2$ kissing-number constant, assuming $c_\mathsf{kn} > 0$ and that non-uniform Gap-ETH holds. 2. For all $p \in [1, \infty)$, there is no $2^{o(n)}$-time algorithm for $\mathrm{BDD}_{p, α}$ for any constant $α> α^\ddagger_p$, where $α^\ddagger_p$ is explicit and satisfies $α^\ddagger_p = 1$ for $1 \leq p \leq 2$, $α^\ddagger_p < 1$ for all $p > 2$, and $α^\ddagger_p \to 1/2$ as $p \to \infty$, 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 $\mathrm{BDD}_{p, α}$ for any constant $α> α^\dagger_{p, C}$, where $α^\dagger_{p, C}$ is explicit and satisfies $α^\dagger_{p, C} \to 1$ as $C \to \infty$ for any fixed $p \in [1, \infty)$, assuming $c_\mathsf{kn} > 0$ and that non-uniform Gap-SETH holds. 4. For all $p > p_0 \approx 2.1397$, $p \notin 2\mathbb{Z}$, and all $C > C_p$, there is no $2^{n/C}$-time algorithm for $\mathrm{SVP}_{p, γ}$ for some constant $γ> 1$, where $C_p > 1$ is explicit and satisfies $C_p \to 1$ as $p \to \infty$, unless randomized Gap-SETH is false.