Classical Hardness of Learning with Errors

TL;DR

This paper proves that the Learning with Errors (LWE) problem remains classically hard, matching worst-case lattice problems, and introduces techniques that clarify the relationship between problem parameters.

Contribution

It establishes classical hardness of LWE with polynomial modulus, extending prior quantum-based results, and provides new insights into the parameter tradeoffs.

Findings

LWE is classically as hard as worst-case lattice problems.

New techniques elucidate the dimension-modulus tradeoff.

Results improve understanding of LWE's complexity landscape.

Abstract

We show that the Learning with Errors (LWE) problem is classically at least as hard as standard worst-case lattice problems, even with polynomial modulus. Previously this was only known under quantum reductions. Our techniques capture the tradeoff between the dimension and the modulus of LWE instances, leading to a much better understanding of the landscape of the problem. The proof is inspired by techniques from several recent cryptographic constructions, most notably fully homomorphic encryption schemes.