发表机构
Tohoku Gakuin University（东北学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了Haar随机格上直接离散高斯SVP搜索的最优样本指数，给出精确公式及逆命题，并证明本原约化保持该速率，为随机格搜索提供查询保证。
AI 中文摘要
我们确定了在Haar随机幺模格上直接离散高斯SVP搜索的最优样本指数，并计数零输出。输出是一个采样向量，可选地除以其格坐标的最大公约数。对于每个固定的近似因子$1\leq\gamma<\sqrt e$，指数为$\gamma^2/(2e)-\log\gamma$（以自然对数单位计）；当$\gamma\geq\sqrt e$时指数为零。对于精确SVP，这给出以2为底的$0.2653689\ldots$。逆命题允许从格和所有先前输出中选择任意正宽度，且固定宽度达到阈值以上的每个指数。Aggarwal、Dadush、Regev和Stephens-Davidowitz已给出每个格上精确SVP的宽度在最优因子的两倍以内。我们确定了显式的Haar典型速率，并证明本原约化保持该速率。一个显式的有限维逆命题同时控制所有宽度，包括从长倍数恢复；结合有限达到界，它在两个方向上都产生查询保证。证明使用随机格矩和关于宽度优化的逐点高斯界。我们考虑采样误差，并在与Pouly和Shen的随机格搜索及后续算法的比较中，将样本需求与生成成本分开。
英文摘要
We determine the optimal sample exponent for direct discrete Gaussian SVP search on Haar random unimodular lattices, counting zero outputs. The output is one sampled vector, optionally divided by the greatest common divisor of its lattice coordinates. For every fixed approximation factor $1\leqγ<\sqrt e$, the exponent is $γ^2/(2e)-\logγ$ in natural logarithmic units; it is zero for $γ\geq\sqrt e$. For exact SVP this gives $0.2653689\ldots$ in base two. The converse permits arbitrary positive widths chosen from the lattice and all previous outputs, and a fixed width attains every exponent above the threshold. Aggarwal, Dadush, Regev, and Stephens-Davidowitz already give a width within a factor two of optimal for exact SVP on each lattice. We determine the explicit Haar typical rate and show that primitive reduction preserves it. An explicit finite dimensional converse controls all widths simultaneously, including recovery from long multiples; together with finite attainment bounds, it yields query guarantees in both directions. The proof uses random lattice moments and a pointwise Gaussian bound optimized over the width. We account for sampling error and separate sample requirements from generation costs in comparisons with the random lattice search of Pouly and Shen and later algorithms.
Comments30 pages, 1 fugure
