### How to Find Ternary LWE Keys Using Locality Sensitive Hashing

Elena Kirshanova and Alexander May

##### Abstract

Let $As = b + e \bmod q$ be an LWE-instance with ternary keys $s,e \in \{0, \pm 1\}^n$. Let $s$ be taken from a search space of size $\mathcal{S}$. A standard Meet-in-the-Middle attack recovers $s$ in time $\mathcal{S}^{0.5}$. Using the representation technique, a recent improvement of May shows that this can be lowered to approximately $\mathcal{S}^{0.25}$ by guessing a sub-linear number of $\Theta(\frac{n}{\log n})$ coordinates from $e$. While guessing such an amount of $e$ can asymptotically be neglected, for concrete instantiations of e.g. NTRU, BLISS or GLP the additional cost of guessing leads to complexities around $\mathcal{S}^{0.3}$. We introduce a locality sensitive hashing (LSH) technique based on Odlyzko's work that avoids any guessing of $e$'s coordinates. This LSH technique involves a comparably small cost such that we can significantly improve on previous results, pushing complexities towards the asymptotic bound $\mathcal{S}^{0.25}$. Concretely, using LSH we lower the MitM complexity estimates for the currently suggested NTRU and NTRU Prime instantiations by a factor in the range $2^{20}-2^{49}$, and for BLISS and GLP parameters by a factor in the range $2^{18}-2^{41}$.

Available format(s)
Category
Public-key cryptography
Publication info
Published elsewhere. IMACC
Keywords
Ternary LWECombinatorial attackRepresentationsLSH
Contact author(s)
elena kirshanova @ rub de
alex may @ rub de
History
2021-09-21: revised
See all versions
Short URL
https://ia.cr/2021/1255

CC BY

BibTeX

@misc{cryptoeprint:2021/1255,
author = {Elena Kirshanova and Alexander May},
title = {How to Find Ternary LWE Keys Using Locality Sensitive Hashing},
howpublished = {Cryptology ePrint Archive, Paper 2021/1255},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/1255}},
url = {https://eprint.iacr.org/2021/1255}
}

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