Learning With Errors (LWE)
Definition
Learning With Errors is a computational problem central to lattice-based cryptography. LWE involves distinguishing noisy linear equations from random data. The problem's difficulty—believed hard for both classical and quantum computers—provides the security foundation for Kyber and other post-quantum schemes.
Technical Explanation
Given a secret vector s and random matrix A, LWE samples are (A, b = As + e) where e is a small error vector. The decision problem: distinguish these samples from uniform random (A, b). The search problem: recover s from samples. Both are believed computationally hard.
Variants include Ring-LWE (structured for efficiency, using polynomial rings) and Module-LWE (Kyber's foundation, balancing structure and security). Adding controlled errors makes inversion infeasible even with quantum algorithms—Shor's algorithm doesn't apply to the noisy linear system.
SynX Relevance
SynX's Kyber-768 implementation relies on Module-LWE hardness. Every key encapsulation operation assumes attackers cannot solve Module-LWE efficiently. Decades of cryptanalytic research support this assumption, with no quantum algorithm providing better than marginal improvements.
Frequently Asked Questions
- Why do errors make LWE secure?
- Errors obscure the linear relationship; without knowing errors, solving for the secret is infeasible.
- How long has LWE been studied?
- Introduced by Regev in 2005; nearly two decades of cryptanalytic attention with no efficient attacks.
- Could LWE be broken?
- Theoretical possibility, but extensive research suggests it's genuinely hard for quantum computers.
Security from proven hard problems. Trust LWE-based Kyber with SynX