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The cryptosystem based on the Learning-with-Errors (LWE) problem is considered as a post-quantum cryptosystem, because it is not based on the factoring problem with large primes which is easily solved by a quantum computer. Moreover, the…

Systems and Control · Computer Science 2021-01-11 Junsoo Kim , Hyungbo Shim , Kyoohyung Han

The Learning with Errors (LWE) problem underlies modern lattice-based cryptography and is assumed to be quantum hard. Recent results show that estimating entanglement entropy is as hard as LWE, creating tension with quantum gravity and…

Quantum Physics · Physics 2025-10-20 Yunfei Wang , Xin Jin , Junyu Liu

We study three problems that involve identifying homogeneous halfspaces under Gaussian distributions: agnostic learning, one-sided reliable learning, and fairness auditing. In each of these problems, we are given labeled examples…

Machine Learning · Computer Science 2026-04-30 Jizhou Huang , Brendan Juba

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

We initiate the study of multi-party computation for classical functionalities (in the plain model) with security against malicious polynomial-time quantum adversaries. We observe that existing techniques readily give a polynomial-round…

Quantum Physics · Physics 2020-11-23 Amit Agarwal , James Bartusek , Vipul Goyal , Dakshita Khurana , Giulio Malavolta

The hardness of the learning with errors (LWE) problem is one of the most fruitful resources of modern cryptography. In particular, it is one of the most prominent candidates for secure post-quantum cryptography. Understanding its quantum…

Cryptography and Security · Computer Science 2019-05-24 Zvika Brakerski , Elena Kirshanova , Damien Stehlé , Weiqiang Wen

This paper aims to provide an efficient implementation of encrypted linear dynamic controllers that perform recursive multiplications on a Ring-Learning With Errors (Ring-LWE) based cryptosystem. By adopting a system-theoretical approach,…

Systems and Control · Electrical Eng. & Systems 2026-01-01 Donghyeon Song , Yeongjun Jang , Joowon Lee , Junsoo Kim

This paper introduces a privacy-preserving distributed learning framework via private-key homomorphic encryption. Thanks to the randomness of the quantization of gradients, our learning with error (LWE) based encryption can eliminate the…

Cryptography and Security · Computer Science 2024-02-05 Guangfeng Yan , Shanxiang Lyu , Hanxu Hou , Zhiyong Zheng , Linqi Song

We develop a generic reduction procedure for active learning problems. Our approach is inspired by a recent polynomial-time reduction of the exact learning problem for weighted automata over integers to that for weighted automata over…

Formal Languages and Automata Theory · Computer Science 2025-10-14 Quentin Aristote , Sam van Gool , Daniela Petrişan , Mahsa Shirmohammadi

In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [Eisentr\"ager-Hallgren-Lauter, Elias-Lauter-Ozman-Stange] gives rise to…

Number Theory · Mathematics 2015-09-24 Yara Elias , Kristin E. Lauter , Ekin Ozman , Katherine E. Stange

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

Chen, Liu, and Zhandry [CLZ22] introduced the problems $S|LWE\rangle$ and $C|LWE\rangle$ as quantum analogues of the Learning with Errors problem, designed to construct quantum algorithms for the Inhomogeneous Short Integer Solution…

Quantum Physics · Physics 2026-05-12 André Chailloux , Paul Hermouet

In this work, we propose an open-source, first-of-its-kind, arithmetic hardware library with a focus on accelerating the arithmetic operations involved in Ring Learning with Error (RLWE)-based somewhat homomorphic encryption (SHE). We…

Cryptography and Security · Computer Science 2020-07-06 Rashmi Agrawal , Lake Bu , Alan Ehret , Michel A. Kinsy

We study the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for cyclotomic number fields,namely: we prove that both problems are equivalent via a polynomial noise increase as long as the…

Number Theory · Mathematics 2020-04-16 Iván Blanco Chacón

Arora & Ge introduced a noise-free polynomial system to compute the secret of a Learning With Errors (LWE) instance via linearization. Albrecht et al. later utilized the Arora-Ge polynomial model to study the complexity of Gr\"obner basis…

Cryptography and Security · Computer Science 2025-04-01 Matthias Johann Steiner

The shortest vector problem (SVP) over ideal lattices is closely related to the Ring-LWE problem, which is widely used to build post-quantum cryptosystems. Power-of-two cyclotomic fields are frequently adopted to instantiate Ring-LWE. Pan…

Cryptography and Security · Computer Science 2026-01-16 Gaohao Cui , Jianing Li , Jincheng Zhuang

In this paper we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of PAC learning. In order to attain our results, we establish connections…

Quantum Physics · Physics 2019-09-20 Srinivasan Arunachalam , Alex B. Grilo , Aarthi Sundaram

The Ring-Learning With Errors (RLWE) problem forms the backbone of highly efficient Fully Homomorphic Encryption (FHE) schemes. A significant component of the RLWE public key and ciphertext of the form $(b,a)$ is the uniformly random…

Cryptography and Security · Computer Science 2026-02-24 Ilan Rosenfeld , Noam Kleinburd , Hillel Chapman , Dror Reuven

Recently Brakerski, Christiano, Mahadev, Vazirani and Vidick (FOCS 2018) have shown how to construct a test of quantumness based on the learning with errors (LWE) assumption: a test that can be solved efficiently by a quantum computer but…

Quantum Physics · Physics 2021-10-05 Shuichi Hirahara , François Le Gall

The Learning With Errors ($\mathsf{LWE}$) problem asks to find $\mathbf{s}$ from an input of the form $(\mathbf{A}, \mathbf{b} = \mathbf{A}\mathbf{s}+\mathbf{e}) \in (\mathbb{Z}/q\mathbb{Z})^{m \times n} \times…

Cryptography and Security · Computer Science 2024-05-15 Thomas Debris-Alazard , Pouria Fallahpour , Damien Stehlé