基于q元陪集差树在2^{n/2+o(n)}时间内求解最短向量问题及更多问题

中文摘要 AI 辅助

本文提出一种求解精确最短向量问题的新随机算法。对于n维格L，该算法的时间与空间复杂度均为2^{n/2+o(n)}。该算法可视为奇数素数q下中点Hessian的q元类似物；更准确地说，利用了如下事实：对于最短向量v，周期高斯函数在v/q处的梯度（而非Hessian）即使在经过相对较大的随机仿射陪集聚合后，仍近似与v（符号除外）成比例。我们借鉴Wagner的广义生日算法设计组合过程，沿中间格链计算相关陪集梯度，从而得到2^{n/2+o(n)}的时间与空间复杂度。该算法的变体可在相同时间与空间复杂度内，对满足距离保证dist(y, L)≤1.039λ₁(L)的任意输入(y, L)求解精确最近向量问题。该保证适用于根据Haar-Siegel测度抽取的随机目标与随机格。因此，该算法可在2^{n/2+o(n)}的时间与空间复杂度内求解此类随机实例上的最近向量问题。

英文摘要

This paper presents a new randomized algorithm for solving the exact shortest vector problem. For the $n$-dimensional lattice $\mathcal L$, our algorithm runs in time and space $2^{n/2+o(n)}$. Our algorithm can be viewed as a $q$-ary analogue of the midpoint Hessian for an odd prime $q$; more precisely, we use the fact that, for a shortest vector $v$, the gradient (rather than Hessian) of the periodic Gaussian function at $v/q$ is nearly proportional to $v$ (up to sign), even after aggregation over a relatively large random affine coset. We compute the relevant coset gradient along a chain of intermediate lattices using a combinatorial procedure inspired by Wagner's generalized birthday algorithm, yielding the $2^{n/2+o(n)}$ time and space complexity. A variant of the algorithm solves the exact closest vector problem on every input $(y,\mathcal L)$ with a distance guarantee $\operatorname{dist}(y,\mathcal L)\le 1.039λ_1(\mathcal L)$ within the same time and space complexity. This guarantee holds for a random target and a random lattice drawn according to the Haar-Siegel measure. Thus, this algorithm solves a closest vector problem on such random instances in time and space $2^{n/2+o(n)}$.

发表机构

• KAIST（韩国科学技术院）

机构由 AI 辅助整理，请以论文原文为准。