Lattice sieving via quantum random walks

TL;DR

This paper improves the quantum algorithm for solving the Shortest Vector Problem in lattices, reducing its heuristic running time and exploring quantum memory trade-offs using quantum random walks with local sensitive filtering.

Contribution

It introduces a novel quantum random walk approach to lattice sieving, surpassing previous quantum algorithms for SVP with improved heuristic complexity.

Findings

Heuristic quantum running time improved to 2^{0.2570 d + o(d)}

Quantum memory and RAM trade-offs quantified

Quantum random walks with local sensitive filtering enhance sieving algorithms

Abstract

Lattice-based cryptography is one of the leading proposals for post-quantum cryptography. The Shortest Vector Problem (SVP) is arguably the most important problem for the cryptanalysis of lattice-based cryptography, and many lattice-based schemes have security claims based on its hardness. The best quantum algorithm for the SVP is due to Laarhoven [Laa16 PhD] and runs in (heuristic) time $2^{0.2653d + o(d)}$. In this article, we present an improvement over Laarhoven's result and present an algorithm that has a (heuristic) running time of $2^{0.2570 d + o(d)}$ where $d$ is the lattice dimension. We also present time-memory trade-offs where we quantify the amount of quantum memory and quantum random access memory of our algorithm. The core idea is to replace Grover's algorithm used in [Laa16 PhD] in a key part of the sieving algorithm by a quantum random walk in which we add a layer of local sensitive filtering.