Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search

TL;DR

This paper applies Grover's quantum search algorithm to lattice algorithms to significantly improve the asymptotic quantum complexity of solving the shortest vector problem, impacting post-quantum cryptography.

Contribution

It introduces a method to enhance lattice algorithms with quantum search, achieving faster asymptotic solutions for the shortest vector problem.

Findings

Quantum algorithms solve SVP in time $2^{1.799n + o(n)}$

Classical algorithms take time $2^{2.465n + o(n)}$

Quantum approach outperforms classical in both provable and heuristic scenarios

Abstract

By applying Grover's quantum search algorithm to the lattice algorithms of Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and Stehl\'{e}, we obtain improved asymptotic quantum results for solving the shortest vector problem. With quantum computers we can provably find a shortest vector in time $2^{1.799n + o(n)}$, improving upon the classical time complexity of $2^{2.465n + o(n)}$ of Pujol and Stehl\'{e} and the $2^{2n + o(n)}$ of Micciancio and Voulgaris, while heuristically we expect to find a shortest vector in time $2^{0.312n + o(n)}$, improving upon the classical time complexity of $2^{0.384n + o(n)}$ of Wang et al. These quantum complexities will be an important guide for the selection of parameters for post-quantum cryptosystems based on the hardness of the shortest vector problem.