We prove new hardness results for fundamental lattice problems under the Exponential Time Hypothesis (ETH). Building on a recent breakthrough by Bitansky et al.\ \cite{BHIRW24}, who gave a polynomial-time reduction from $\mathsf{3SAT}$ to the (gap) $\mathsf{MAXLIN}$ problem-a class of CSPs with linear equations over finite fields-we derive ETH hardness for several lattice problems. First, we show that for any $p \in [1, \infty)$, there exists an explicit constant $γ> 1$ such that $\mathsf{CVP}_{p,γ}$ (the $\ell_p$-norm approximate Closest Vector Problem) does not admit a $2^{o(n)}$-time algorithm unless ETH is false. Our reduction is deterministic and proceeds via a direct reduction from (gap) $\mathsf{MAXLIN}$ to $\mathsf{CVP}_{p,γ}$. Our main contribution is a randomized ETH hardness result for $\mathsf{SVP}_{p,γ}$ (the $\ell_p$-norm approximate Shortest Vector Problem) for all $p \in (2, \infty)$. This result relies on a novel geometric property of the integer lattice $\mathbb{Z}^n$ in the $\ell_p$ norm, which says that for any $p \in (2, \infty)$, the number of lattice vectors close to $\frac{1}{2}\vec{1}_n$ (in the $\ell_p$ norm) is exponentially larger than the number of short vectors (namely those close to the origin). We establish this property via a new inequality for the Theta function, which we use to get a randomized reduction from $\mathsf{CVP}_{p,γ}$ to $\mathsf{SVP}_{p,γ'}$. Finally, we also use our ideas to give some minor improvements over prior reductions from $\mathsf{3SAT}$ to $\mathsf{BDD}_{p,α}$ (the Bounded Distance Decoding Problem), yielding better ETH hardness results for $\mathsf{BDD}_{p,α}$ for any $p \in [1, \infty)$ and $α> α_p^{\ddagger}$, where $α_p^{\ddagger}$ is an explicit threshold depending on $p$.

翻译：我们在指数时间假设（ETH）下证明了基本格问题的新硬度结果。基于Bitansky等人近期突破性工作（\cite{BHIRW24}），该工作给出了从$\mathsf{3SAT}$到（间隙）$\mathsf{MAXLIN}$问题（一类有限域上线性方程组的约束满足问题(CSPs)）的多项式时间归约，我们推导了若干格问题的ETH硬度。首先，我们证明对于任意$p \in [1, \infty)$，存在显式常数$\gamma > 1$，使得$\mathsf{CVP}_{p,\gamma}$（$\ell_p$范数近似最近向量问题）不存在$2^{o(n)}$时间算法，除非ETH为假。我们的归约是确定性的，通过从（间隙）$\mathsf{MAXLIN}$到$\mathsf{CVP}_{p,\gamma}$的直接归约实现。主要贡献是：对于所有$p \in (2, \infty)$，我们得到了$\mathsf{SVP}_{p,\gamma}$（$\ell_p$范数近似最短向量问题）的随机化ETH硬度结果。该结果依赖于$\ell_p$范数下整数格$\mathbb{Z}^n$的一个新颖几何性质：对于任意$p \in (2, \infty)$，接近$\frac{1}{2}\vec{1}_n$（在$\ell_p$范数下）的格向量数量指数级大于短向量（即接近原点的向量）数量。我们通过Theta函数的新不等式建立了该性质，并利用其得到从$\mathsf{CVP}_{p,\gamma}$到$\mathsf{SVP}_{p,\gamma'}$的随机化归约。最后，我们还利用这些思想改进了从$\mathsf{3SAT}$到$\mathsf{BDD}_{p,\alpha}$（有界距离解码问题）的先前归约，从而对任意$p \in [1, \infty)$和$\alpha > \alpha_p^{\ddagger}$（其中$\alpha_p^{\ddagger}$是依赖于$p$的显式阈值）得到了更好的$\mathsf{BDD}_{p,\alpha}$ETH硬度结果。