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Related papers: The Polynomial Learning With Errors Problem and th…

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Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with…

Quantum Physics · Physics 2019-03-27 Alex B. Grilo , Iordanis Kerenidis , Timo Zijlstra

The Polynomial Learning With Errors problem (PLWE) serves as the background of two of the three cryptosystems standardized in August 2024 by the National Institute of Standards and Technology to replace non-quantum resistant current…

Cryptography and Security · Computer Science 2025-07-01 Iván Blanco Chacón , Raúl Durán Díaz , Rodrigo Martín Sánchez-Ledesma

The Learning with Errors (\LWE) problem has been widely utilized as a foundation for numerous cryptographic tools over the years. In this study, we focus on an algebraic variant of the \LWE problem called \emph{Group ring} \LWE ($\GRLWE$).…

Cryptography and Security · Computer Science 2026-03-11 Jiaqi Liu , Fang-Wei Fu

The "Ring Learning with Errors" (RLWE) problem was formulated as a variant of the "Learning with Errors" (LWE) problem, with the purpose of taking advantage of an additional algebraic structure in the underlying considered lattices; this…

Cryptography and Security · Computer Science 2018-02-05 Alberto Pedrouzo-Ulloa , Juan Ramón Troncoso-Pastoriza , Fernando Pérez-González

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…

Computational Complexity · Computer Science 2013-06-04 Zvika Brakerski , Adeline Langlois , Chris Peikert , Oded Regev , Damien Stehlé

The Learning with Errors (LWE) problem is the fundamental backbone of modern lattice based cryptography, allowing one to establish cryptography on the hardness of well-studied computational problems. However, schemes based on LWE are often…

Information Theory · Computer Science 2020-08-06 Charles Grover , Cong Ling , Roope Vehkalahti

Recent work showed that ML-based attacks on Learning with Errors (LWE), a hard problem used in post-quantum cryptography, outperform classical algebraic attacks in certain settings. Although promising, ML attacks struggle to scale to more…

Machine Learning · Computer Science 2025-08-26 Eshika Saxena , Alberto Alfarano , François Charton , Zeyuan Allen-Zhu , Emily Wenger , Kristin Lauter

We introduce a continuous analogue of the Learning with Errors (LWE) problem, which we name CLWE. We give a polynomial-time quantum reduction from worst-case lattice problems to CLWE, showing that CLWE enjoys similar hardness guarantees to…

Computational Complexity · Computer Science 2020-10-27 Joan Bruna , Oded Regev , Min Jae Song , Yi Tang

Lattice-based cryptography is a foundation for post-quantum security, with the Learning with Errors (LWE) problem as a core component in key exchange, encryption, and homomorphic computation. Structured variants like Ring-LWE (RLWE) and…

Cryptography and Security · Computer Science 2025-02-12 Dongfang Zhao

At ASIACRYPT 2018, a digital attack based on linear least squares was introduced for a variant of the learning with errors (LWE) problem which omits modular reduction known as the integer learning with errors problem (ILWE). In this paper,…

Cryptography and Security · Computer Science 2025-12-10 Kyle Yates , Antsa Pierrottet , Abdullah Al Mamun , Ryann Cartor , Mashrur Chowdhury , Shuhong Gao

Learning with Errors (LWE) is a hard math problem underlying recently standardized post-quantum cryptography (PQC) systems for key exchange and digital signatures. Prior work proposed new machine learning (ML)-based attacks on LWE problems…

Cryptography and Security · Computer Science 2024-02-05 Samuel Stevens , Emily Wenger , Cathy Li , Niklas Nolte , Eshika Saxena , François Charton , Kristin Lauter

The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far…

Cryptography and Security · Computer Science 2015-08-11 Yara Elias , Kristin E. Lauter , Ekin Ozman , Katherine E. Stange

The Learning with Errors (LWE) problem is a hard math problem in lattice-based cryptography. In the simplest case of binary secrets, it is the subset sum problem, with error. Effective ML attacks on LWE were demonstrated in the case of…

Cryptography and Security · Computer Science 2026-04-07 Alberto Alfarano , Eshika Saxena , Emily Wenger , François Charton , Kristin Lauter

Learning with Errors (LWE) is a hard math problem used in post-quantum cryptography. Homomorphic Encryption (HE) schemes rely on the hardness of the LWE problem for their security, and two LWE-based cryptosystems were recently standardized…

Cryptography and Security · Computer Science 2023-10-30 Cathy Yuanchen Li , Emily Wenger , Zeyuan Allen-Zhu , Francois Charton , Kristin Lauter

The Ring Learning-With-Errors (LWE) problem, whose security is based on hard ideal lattice problems, has proven to be a promising primitive with diverse applications in cryptography. There are however recent discoveries of faster algorithms…

Cryptography and Security · Computer Science 2017-06-22 Qi Cheng , Jun Zhang , Jincheng Zhuang

Learning with Errors (LWE) problems are the foundations for numerous applications in lattice-based cryptography and are provably as hard as approximate lattice problems in the worst case. Here we present a reduction from LWE problem to…

Quantum Physics · Physics 2013-06-05 Fada Li , Wansu Bao , Xiangqun Fu , Yuchao Zhang , Tan Li

Currently deployed public-key cryptosystems will be vulnerable to attacks by full-scale quantum computers. Consequently, "quantum resistant" cryptosystems are in high demand, and lattice-based cryptosystems, based on a hard problem known as…

Cryptography and Security · Computer Science 2023-04-25 Emily Wenger , Mingjie Chen , François Charton , Kristin Lauter

Our main result is a quantum public-key encryption scheme based on the Extrapolated Dihedral Coset problem (EDCP) which is equivalent, under quantum polynomial-time reductions, to the Learning With Errors (LWE) problem. For limited number…

Quantum Physics · Physics 2021-05-28 Javad Doliskani

We show direct and conceptually simple reductions between the classical learning with errors (LWE) problem and its continuous analog, CLWE (Bruna, Regev, Song and Tang, STOC 2021). This allows us to bring to bear the powerful machinery of…

Cryptography and Security · Computer Science 2022-11-03 Aparna Gupte , Neekon Vafa , Vinod Vaikuntanathan

Learning with Errors (LWE) is a hard math problem underpinning many proposed post-quantum cryptographic (PQC) systems. The only PQC Key Exchange Mechanism (KEM) standardized by NIST is based on module~LWE, and current publicly available PQ…

Cryptography and Security · Computer Science 2023-11-01 Cathy Li , Jana Sotáková , Emily Wenger , Mohamed Malhou , Evrard Garcelon , Francois Charton , Kristin Lauter
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