Faster SVP in Polynomial Space

2026-09-18

AI Summary

本文通过一种随机算法改进了在多项式空间内解决精确欧几里得最短向量问题的时间复杂度，主要方法是利用差分样本表示和低空间碰撞搜索。

Abstract

Kannan's algorithm, as analyzed by Hanrot and Stehlé in 2007, solves the exact Euclidean shortest vector problem in polynomial space and $n^{\frac{n}{2e}+o(n)}$ time. In the classical setting with polynomial space, we obtain the first improvement on this bound via a randomized algorithm that runs in $n^{\frac{n}{4e}+o(n)}$ time. The main idea is to represent a fixed shortest vector in many ways as a difference of samples, thereby enabling the low-space collision search of Lyu and Zhu (SODA 2023) to replace exhaustive enumeration in the original analysis.

Problem

• shortest vector problem
• polynomial space
• time complexity

Innovation

• randomized algorithm
• polynomial space
• shortest vector problem
• collision search