### On Ideal Lattices and Learning with Errors Over Rings

Vadim Lyubashevsky, Chris Peikert, and Oded Regev

##### 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 \emph{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.

Available format(s)
Category
Foundations
Publication info
Published elsewhere. Full version of paper appearing in Eurocrypt 2010
Keywords
latticesLWEringsalgebraic number theory
Contact author(s)
cpeikert @ cc gatech edu
History
2013-06-26: last of 2 revisions
See all versions
Short URL
https://ia.cr/2012/230

CC BY

BibTeX

@misc{cryptoeprint:2012/230,
author = {Vadim Lyubashevsky and Chris Peikert and Oded Regev},
title = {On Ideal Lattices and Learning with Errors Over Rings},
howpublished = {Cryptology ePrint Archive, Paper 2012/230},
year = {2012},
note = {\url{https://eprint.iacr.org/2012/230}},
url = {https://eprint.iacr.org/2012/230}
}

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