Solving the Shortest Vector Problem in time $2^{0.6039n}$ Time via Mid-point Hessian

2026-08-03

AI Summary

This work addresses the high computational complexity of the Shortest Vector Problem (SVP) in lattices by introducing a novel approach that, for the first time, integrates Hessian matrix information of periodic Gaussian functions evaluated at half-shortest vectors with discrete Gaussian sampling. By estimating the Hessian over random sublattice cosets to locate short vectors and employing bounded-distance decoding for recovery, the method achieves a significant reduction in solving complexity. The proposed algorithm improves upon the previous best-known time complexity of $2^{n+o(n)}$, attaining $2^{0.6039n+o(n)}$ for classical computation and $2^{0.5411n+o(n)}$ in the quantum setting, while maintaining a space complexity of $2^{0.5n+o(n)}$.

Abstract

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.

Problem

• Shortest Vector Problem
• lattice
• SVP
• discrete Gaussian
• Hessian

Innovation

• Shortest Vector Problem
• Hessian matrix
• Periodic Gaussian function
• Discrete Gaussian sampling
• Bounded Distance Decoding