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Also known as: LWE
A lattice-based hard problem where learning a secret from noisy linear equations is computationally infeasible, underpinning many post-quantum schemes.
DESCRIPTION
Learning with Errors (LWE) formalizes the task of recovering a secret vector from linear equations perturbed by small random noise. Its hardness is related to worst-case lattice problems, providing strong security foundations. Variants such as Ring-LWE and Module-LWE improve efficiency for practical cryptosystems. LWE supports key encapsulation, signatures, and even fully homomorphic encryption constructions. It is central to NIST-selected algorithms like Kyber and Dilithium through related module-lattice assumptions.
ALTERNATIVE DEFINITIONS
Wikipedia
In computational learning theory, the learning with errors (LWE) problem is the problem of distinguishing noisy linear equations from random data.
Emphasizes the decision version and its framing in learning theory.
Oded Regev
We are interested in the Learning with Errors (LWE) problem: given access to random linear equations perturbed by small errors, the goal is to recover the secret or distinguish the samples from random.
Authoritative survey defining both search and decision formulations.
NIST
Several third-round candidates are based on structured lattice problems such as Module-LWE and Module-SIS, which offer efficiency advantages over plain LWE and SIS.
Connects LWE variants to practical standardization candidates and efficiency considerations.
RELATED TERMS
Lattice-Based Cryptography Ring-LWE Module-LWE Post-Quantum Cryptography Regev Cryptosystem Kyber Dilithium Homomorphic Encryption
COMPETING TERMS
Short Integer Solution Discrete Logarithm Problem
KNOWLEDGE AREAS
TARGET AUDIENCE
Security Architect Cybersecurity Engineer Software Developer Chief Information Security Officer
EXTERNAL SOURCES
- Wikipedia - Learning with errors
- CRYSTALS-Kyber Team - Kyber KEM based on Module-LWE
- CRYSTALS-Dilithium Team - Dilithium signatures based on Module-LWR/SIS
- IACR ePrint - Regev’s LWE survey preprint
TOP QUESTIONS
What is Learning with Errors in simple terms?
It’s the challenge of solving noisy linear equations. The added noise makes recovering the secret computationally hard.
Category: Basics
Why is LWE important to PQC?
LWE provides strong worst-case to average-case hardness links and underpins efficient KEMs and signatures selected by standard bodies.
Category: Relevancy
How do Ring-LWE and Module-LWE relate to LWE?
They structure LWE over rings or modules to gain performance while retaining believed hardness properties.
Category: Comparison
What schemes use LWE or its variants?
Kyber (KEM) and Dilithium (signatures) rely on module-lattice assumptions related to LWE and SIS.
Category: Examples
What is a common misunderstanding about LWE?
That it’s purely theoretical; in practice, efficient parameterizations are implemented in real libraries and protocols.
Category: Misunderstanding
What are the risks of relying on LWE?
Improper parameters or weak noise distributions can reduce security. Ongoing cryptanalysis must inform safe choices.
Category: Risks
How can developers get started with LWE-based crypto?
Use reference implementations from standardization teams, follow parameter recommendations, and integrate via vetted libraries.
Category: HowTo
When was LWE introduced?
LWE was introduced by Oded Regev in the mid-2000s and has since become foundational in lattice cryptography.
Category: History
What’s the biggest debate around LWE?
Security margin versus efficiency trade-offs and the exact hardness of structured variants continue to be studied.
Category: Opinion