AI 中文总结
该研究针对最短向量问题提出随机算法，利用周期高斯函数的中点海森矩阵性质，结合随机子陪集等优化技术，将经典时间复杂度降至 $2^{0.6039n+o(n)}$，优于此前最优算法。
AI 中文摘要
我们针对最短向量问题（SVP）提出了随机算法。对于n维格 $\mathcal L$，我们的算法在经典计算机上求解SVP的时间为 $2^{0.6039n+o(n)}$，在量子计算机上的时间为 $2^{0.5411n+o(n)}$，空间复杂度为 $2^{0.5n+o(n)}$，优于Aggarwal、Dadush、Regev和Stephens-Davidowitz[STOC'15]提出的此前最优算法，该算法的时间和空间复杂度均为 $2^{n+o(n)}$。我们的算法大量利用了周期高斯函数在半最短向量处的海森矩阵的性质：对于格 $\mathcal L$ 中的最短向量 $v$，其在 $v/2$ 处的海森矩阵具有接近 $v$ 的特征向量，可用于通过（预处理）有界距离解码算法恢复 $v$。在模 $\mathcal L$ 的周期性下，候选中点由 $\mathcal L/2\mathcal L$ 中的奇偶类索引。我们的算法通过使用离散高斯样本估计对应的海森矩阵来搜索最短向量的类。我们利用随机子陪集和各种采样技术优化算法，最终达到该复杂度。这些优化技术可能具有独立的研究价值。
英文摘要
We present randomized algorithms for the shortest vector problem (SVP). For the $n$-dimensional lattice $\mathcal L$, our algorithms solve SVP in time $2^{0.6039n+o(n)}$ classically and $2^{0.5411n+o(n)}$ quantumly and space $2^{0.5n+o(n)}$, improving the previous best algorithm running in $2^{n+o(n)}$ time and space of Aggarwal, Dadush, Regev, and Stephens-Davidowitz [STOC'15]. Our algorithms heavily use the property of the Hessian of the periodic Gaussian function at the half shortest vector: For a shortest vector $v \in \mathcal L$, the Hessian at $v/2$ has the eigenvector close to $v$, which can be used to recover $v$ using the (preprocessing) bounded distance decoding algorithm. Given the periodicity modulo $\mathcal L$, the candidate midpoints are indexed by the parity classes in $\mathcal L/2\mathcal L$. Our algorithm searches for the class of a shortest vector by estimating the corresponding Hessians using discrete Gaussian samples. We optimize the algorithm using random sublattice cosets and various sampling technique, achieving the final complexity. The optimization techniques may be of independent interest.
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