Quantum Algorithms for Variants of Average-Case Lattice Problems via Filtering

TL;DR

This paper presents polynomial-time quantum algorithms for certain variants of lattice problems, including SIS, LWE, and EDCP, using a novel filtering technique to solve LWE given LWE-like quantum states.

Contribution

The paper introduces a new filtering method to solve LWE with LWE-like quantum states, enabling quantum algorithms for variants of SIS, LWE, and EDCP not previously solvable in polynomial time.

Findings

Quantum algorithms for SIS, LWE, and EDCP variants.

Filtering technique for solving LWE with LWE-like states.

Extension of previous EDCP results with quantum algorithms.

Abstract

We show polynomial-time quantum algorithms for the following problems: (*) Short integer solution (SIS) problem under the infinity norm, where the public matrix is very wide, the modulus is a polynomially large prime, and the bound of infinity norm is set to be half of the modulus minus a constant. (*) Learning with errors (LWE) problem given LWE-like quantum states with polynomially large moduli and certain error distributions, including bounded uniform distributions and Laplace distributions. (*) Extrapolated dihedral coset problem (EDCP) with certain parameters. The SIS, LWE, and EDCP problems in their standard forms are as hard as solving lattice problems in the worst case. However, the variants that we can solve are not in the parameter regimes known to be as hard as solving worst-case lattice problems. Still, no classical or quantum polynomial-time algorithms were known for the variants of SIS and LWE we consider. For EDCP, our quantum algorithm slightly extends the result of Ivanyos et al. (2018). Our algorithms for variants of SIS and EDCP use the existing quantum reductions from those problems to LWE, or more precisely, to the problem of solving LWE given LWE-like quantum states. Our main contribution is solving LWE given LWE-like quantum states with interesting parameters using a filtering technique.