On Ideal Lattices and Learning with Errors over Rings

TLDR

The learning‑with‑errors problem distinguishes noisy linear equations from uniform ones, is provably hard via worst‑case lattice problems, yet suffers from quadratic overhead that limits practical cryptographic use. The authors aim to determine whether exploiting algebraic structure can make LWE and its applications truly efficient. They introduce ring‑LWE, an algebraic variant, and prove it is pseudorandom and as hard as worst‑case ideal‑lattice problems for polynomial‑time quantum algorithms. Ring‑LWE enables the first practical lattice‑based public‑key cryptosystem with efficient security reduction, and improves efficiency of many other LWE‑based applications.

Abstract

The “learning with errors” (LWE) problem is to distinguish random linear equations, which have been perturbed by a small amount of noise, from truly uniform ones. The problem has been shown to be as hard as worst-case lattice problems, and in recent years it has served as the foundation for a plethora of cryptographic applications. Unfortunately, these applications are rather inefficient due to an inherent quadratic overhead in the use of LWE. A main open question was whether LWE and its applications could be made truly efficient by exploiting extra algebraic structure, as was done for lattice-based hash functions (and related primitives). We resolve this question in the affirmative by introducing an algebraic variant of LWE called ring-LWE , and proving that it too enjoys very strong hardness guarantees. Specifically, we show that the ring-LWE distribution is pseudorandom, assuming that worst-case problems on ideal lattices are hard for polynomial-time quantum algorithms. Applications include the first truly practical lattice-based public-key cryptosystem with an efficient security reduction; moreover, many of the other applications of LWE can be made much more efficient through the use of ring-LWE.